Approximation by light maps and parametric Lelek maps

APPRO XIMA TION BY LIGHT MAPS AN D P ARAMETRIC LELEK MAPS T ARAS BANAKH AN D VESKO V ALO V Abstract. The class o f metrizable space s M with the follow- ing approximation prop er ty is introduced and in vestigated: M ∈ AP ( n, 0) if for every ε > 0 and a map g : I n → M there exists a 0-dimensiona l map g ′ : I n → M which is ε -ho motopic to g . It is shown that this clas s has v er y nice prop erties. F or ex ample, if M i ∈ AP ( n i , 0), i = 1 , 2, then M 1 × M 2 ∈ AP ( n 1 + n 2 , 0). More- ov er, M ∈ AP ( n, 0) if and only if ea ch p o int of M has a lo cal base of neighbo rho o ds U with U ∈ AP ( n, 0). Using the pr o p er- ties of AP ( n, 0 )-spaces, we ge neralize so me results of Levin a nd Kato-Mats uha shi concerning the existence of residual sets of n - dimensional Lelek maps. 1. Intr oduction All spaces in the pap er are assumed to b e metrizable and all maps con tinuous . By C ( X , M ) w e denote all maps from X into M . Un- less stated o therwise, all function spaces are endo w ed with the source limitation top olog y . One of the imp ortant prop erties of the n -dimensional cub e I n or the Euclidean space R n , widely exploited in dimension theory , is that an y map from an n -dimensional c o mpactum X in to I n (resp., R n ) can b e ap- pro ximated b y maps with 0-dimensional ﬁb ers. The a im of this article is to in tro duce and inv estigate the class of spaces having that prop ert y . More precisely , w e say that a space M has the ( n, 0) -appr oximation pr op erty ( br., M ∈ AP ( n, 0) ) if for ev ery ε > 0 and a map g : I n → M there exists a 0-dimensional map g ′ : I n → M whic h is ε -homotopic to g . Here, g ′ is ε -homotopic to g means that there is an ε - small ho - motop y h : I n × I → M connecting g and g ′ . It is easily seen that this deﬁnition doesn’t dep end on the metric generating the to p ology of M . If M is LC n , then M ∈ AP ( n, 0)) if and o nly if f or ev ery ε > 0 and g ∈ C ( I n , M ) there exists a 0-dimensional map g ′ whic h is ε - close 1991 Mathematics Subje ct Classiﬁc ation. Primary 54 F45; Secondar y 55M10 . Key wor ds and phr a ses. dimensio n, n -dimensional ma ps , n -dimensional Lelek maps, dendrites, Can tor n - ma nifolds, gener al p osition pr op erties. The s e c ond author w as partially suppor ted by NSERC Grant 261914- 03. 1 2 to g . W e show that the class of AP ( n, 0)- spaces is quite la rge and it has v ery nice prop erties. F or example, if M i ∈ AP ( n i , 0), i = 1 , 2, then M 1 × M 2 ∈ AP ( n 1 + n 2 , 0). W e also pro v e that M ∈ AP ( n, 0) if and only if eac h point of M has a lo cal base of ne ig h b o rho o ds U with U ∈ AP ( n, 0). M oreo ver, ev ery path- connected compactum M ∈ AP ( n, 0) is a ( V n )-con tinuum in the sense of P . Alexandroﬀ [1]. In particular, according to [1 0], an y suc h M is a Can tor n -manifo ld, as w ell as a strong Can t or n -manifo ld in the since o f Had ˇ ziiv ano v [9]. All complete LC 0 -spaces without isolated p oin ts are AP (1 , 0), as w ell as, ev ery manifold mo deled on the n -dimensional Menger cub e or the n -dimensional N¨ ob eling space has the AP ( n, 0)-prop erty . The class of AP ( n, 0)-spaces is v ery natural for obtaining results ab out approximation b y dimensionally restricted maps. W e presen t suc h a result ab out n -dimensional Lelek maps. Recall that a map g : X → M b et wee n compact spaces is said to b e an n -dimensional L elek map [17 ] if the union of all non-trivial con t in ua contained in the ﬁb ers of g is of dimension ≤ n . F or con v enience, if n ≤ 0, b y an n - dimensional Lelek map w e simply mean a 0- dimensional map. Ev ery n -dimensional Lelek map g b et w een compacta is a t most n -dimensional b ecause the comp onen t s of eac h ﬁb er of g is at most n -dimensional (w e sa y that g is n -dimensional if all ﬁb ers of g are a t most n -dimensional). Lelek [15] constructed suc h a map from I n +1 on to a dendrite. Levin [17] impro v ed this result by sho wing that the space C ( X, I n ) of a ll maps from X into I n con tains a dense G δ -subset consisting of ( m − n )- dimensional Lelek maps for an y compactum X with dim X = m ≤ n . Recen tly Kato and Matsuhashi [12 ] established a v ersion of the Levin result with I n replaced by more general class of spaces. This class consists of complete metric AN R - spaces having a piecewise em b edding dimension ≥ n . In the presen t pap er w e g eneralize the result o f Kato- Matsuhashi in t wo directions. First, w e establish a parametric version of their theorem and second, we sho w that this parametric vers io n ho lds for AP ( n, 0) - spaces (see Section 6 where it is show n tha t the class of AP ( n, 0 )- spaces con tains prop erly the class of AN R ’s with piecew ise embedding dimensions ≥ n ). Here is our result ab out parametric Lelek maps (a more general v ersion is establishe d in Section 3; moreov er w e a lso presen t a v ersion of Theorem 1.1 when M has a prop erty w eaker than AP ( n, 0), see Theorem 4.5). Theorem 1.1. L et f : X → Y b e a p erfe c t map with dim △ ( f ) ≤ m , wher e X a nd Y ar e metric sp ac es. If M ∈ AP ( n, 0) is c omp letely Approximation by ligh t maps 3 metrizable, then ther e exists a G δ -set H ⊂ C ( X, M ) such that ev- ery simp l i c ial ly fa c torizable map in C ( X, M ) is h o m otopic al ly appr ox- imate d by m a ps fr om H and for e v e ry g ∈ H and y ∈ Y the r estriction g | f − 1 ( y ) is an ( m − n ) -dime n sional L elek map. Let us explain the notions in Theorem 1.1 . A ma p g ∈ C ( X, M ) is homotopically appro ximated by maps from H means that for ev ery function ε ∈ C ( X, ( 0 , 1]) there exists g ′ ∈ H whic h is ε -homotopic to g . Here, the maps g and g ′ are said to b e ε -homoto pic, if there is a homoto py h : X × I → M connecting g and g ′ suc h that eac h set h ( { x } × I ) has a diameter < ε ( x ), x ∈ X . The function space C ( X , M ) app earing in this theorem is endo we d with the source limitation topolo gy whose neighbor ho o d base at a giv en function f ∈ C ( X , M ) consists o f the sets B  ( f , ε ) = { g ∈ C ( X, M ) :  ( g , f ) < ε } , where  is a ﬁxed compatible metric on M and ε : X → ( 0 , 1] runs o v er con tinuous p ositive functions o n X . The sym b ol  ( f , g ) < ε means that  ( f ( x ) , g ( x )) < ε ( x ) for all x ∈ X . Since X is metrizable, the source limitation top ology do esn’t dep end on the metric ρ [14] and it has the Baire prop erty pro vided M is completely metrizable [20]. W e sa y that a map g : X → M is simplicially factorizable [2] if there exists a simplicial complex L and t w o maps g 1 : X → L and g 2 : L → M suc h that g = g 2 ◦ g 1 . In eac h of the followin g cases t he set o f simplicially factorizable maps is dense in C ( X, M ) (see [2, Prop osition 4]): (i) M is an AN R ; (ii) dim X ≤ k a nd M is LC k − 1 ; (iii) X is a C -space and M is lo cally con tractible. The dimension dim △ ( f ) w as deﬁne d in [2]: dim △ ( f ) of a map f : X → Y is equal to the smallest cardinal n umber τ f o r whic h there is a map g : X → I τ suc h that the diag onal pro duct f ∆ g : X → Y × I τ is a 0 - dimensional map. F o r any p erfect map f : X → Y betw een metric spaces w e hav e: (i) dim( f ) ≤ dim △ ( f ); (ii) dim △ ( f ) = dim( f ) if Y is a C -space, see [21] a nd [27]; (iii) dim △ ( f ) ≤ dim( f ) + 1 if the spaces X , Y are compact, see [16]. Since eve ry metric space admitting a p erfect ﬁnite-dimensional map on to a C -space is a lso a C -space [11], Theorem 1.1 implies the fo llo wing (here, H is homot o pically dense in C ( X, M ) if eve ry g ∈ C ( X , M ) is homotopically approx imated by maps from H ): 4 Corollary 1.2. L et f : X → Y b e a p erfe ct m -di m ensional map b e- twe en metric sp ac es with Y b eing a C -sp ac e. If M ∈ A ( n, 0) is c o m - pletely metrizable, then in e ach of the fo l lowing c ases ther e exists a ho- motopic al ly d e nse G δ -subset H ⊂ C ( X, M ) c onsisting of maps g such that g | f − 1 ( y ) is an ( m − n ) -dime n sional L elek map for every y ∈ Y : • M is l o c al ly c ontr actible; • X is ﬁni te dim ensional a nd M ∈ LC k − 1 for k = dim X . The pap er is orga nized as follows. In Section 2 some preliminary results ab out AE ( n, 0)-spaces are presen ted. In Section 3 w e establish a generalized v ersion of Theorem 1.1. Section 4 is devoted to almost AP ( n, 0)-spaces. The ﬁnal t wo sections con tain some in teresting prop- erties of AP ( n, 0)- spaces. In Section 5 w e pro ve t ha t AP ( n, 0)-pro p ert y has a lo cal nature, as w ell as, M 1 × M 2 ∈ AP ( n 1 + n 2 , 0) provided each M i ∈ AP ( n i , 0) is completely metrizable. It is a lso established in this section that ev ery path connected compactum is a ( V n )-con tinuum. Section 6 is dev oted to the in terpla y o f AP ( n, 0)-pro p ert y and the gen- eral p osition prop erties m − D D { n,k } -prop erties. In particular, it is sho wn that ev ery p oint o f a lo cally pat h- connected AP ( n, 0)-space X is a homolog ical Z n − 1 -p oint in X . Another result fro m this section states that ev ery completely metrizable space p ossessing the disjoint ( n − 1)-disks prop erty is a n AP ( n, 0)-space. 2. Pre liminar y res ul ts about AP ( n, 0) -s p aces In this section w e established some preliminary results on AP ( n, 0)- spaces whic h are g oing to b e used in next sections. Supp ose M is a metric space a nd ε > 0. W e write d n ( M ) < ε if M can b e co ve red b y an op en family γ suc h that diam U < ε for all U ∈ γ and or d ( γ ) ≤ n + 1 (the last inequalit y means that at most n + 1 elemen ts of γ can hav e a common p oin t). It is easily seen that if F ⊂ M is closed and d n ( F ) < ε , then F is co v ered b y an op en in M fa mily γ of mesh < ε and or d ≤ n + 1. W e also agree to denote b y cov ( M ) the family of all op en co v ers o f M . Let us b egin with the follo wing tec hnical lemma. Lemma 2.1. L et Z = A ∪ B b e a c omp actum, wher e A and B ar e close d subsets of Z . Supp ose C ⊂ Z is close d such that dim C ∩ A ≤ 0 and d 0 ( C ∩ B ) < ε . Then d 0 ( C ) < ε . Pr o of. Since d 0 ( C ∩ B ) < ε , t here exists a disjoin t op en f a mily γ = { W 1 , .., W k } in B of mesh < ε . W e extend ev ery W i to an op en set ˜ W i in Z suc h that ˜ W i ∩ B = W i and ˜ γ = { ˜ W i : i = 1 , .., k } is a disjoin t family of mesh < ε (this can b e done b ecause B is closed Approximation by light maps 5 in Z ). Observ e that C 1 = C \ S i = k i =1 ˜ W i is a closed subset of C ∩ A disjoin t from B . Sinc e dim C ∩ A ≤ 0, there exists a clop en set C 2 in C ∩ A disjoint fr o m B and con ta ining C 1 . Obviously , C 2 is clop en in C . Moreo ve r, dim C 2 ≤ 0. Hence, there exists a cov er ω = { V j : j = 1 , .., m } ∈ cov ( C 2 ) consisting of clop en subsets of C 2 with diam V j < ε for ev ery j . Then ω 1 = ω ∪ { ˜ W i ∩ C \ C 2 } is a disjoint op en co ver of C and mesh ( ω 1 ) < ε . So , d 0 ( C ) < ε .  The pro o f of next lemma is extracted from [2, Theorem 4]. Lemma 2.2. L et G ⊂ C ( X, M ) , wher e ( M ,  ) is a c omplete metric sp ac e. Supp ose ( U ( i ) i ≥ 1 is a se quenc e of op en subsets of C ( X , M ) such that • for any g ∈ G , i ≥ 1 and any function η ∈ C ( X, (0 , 1]) ther e exists g i ∈ B  ( g , η ) ∩ U ( i ) ∩ G which is η -hom o topi c to g . Then, for any g ∈ G and ε : X → (0 , 1] ther e exists g ′ ∈ T ∞ i =1 U ( i ) and an ε -homotopy c onne cting g and g ′ . Mor e over, g ′ | A = g 0 | A f o r som e g 0 ∈ C ( X , M ) and A ⊂ X pr ovide d g i | A = g 0 | A for a l l i . Pr o of. F or ﬁxed g ∈ G and ε ∈ C ( X, (0 , 1 ]) let g 0 = g and ε 0 = ε/ 3. W e shall construct by induction a sequence ( g i : X → M ) i ≥ 1 ⊂ G , a sequence ( ε i ) i ≥ 1 of p ositiv e functions, and a sequence ( H i : X × [0 , 1] → M ) i ≥ 1 of ε i − 1 -homotopies satisfying t he conditions: • H i +1 ( x, 0) = g i ( x ) and H i +1 ( x, 1) = g i +1 ( x ) for ev ery x ∈ X ; • g i +1 ∈ B  ( g i , ε i ) ∩ U ( i + 1) ∩ G ; • ε i +1 ≤ ε i / 2; • B  ( g i +1 , 3 ε i +1 ) ⊂ U ( i + 1 ) . Assume that, fo r some i , w e ha ve already constructed maps g 1 , . . . , g i , p ositiv e n um b ers ε 1 , . . . , ε i , and homotopies H 1 , . . . , H i satisfying the ab ov e conditions. According to the hypotheses, there exists a map g i +1 ∈ B  ( g i , ε i ) ∩ U ( i + 1) ∩ G suc h that g i +1 is ε i -homotopic to g i . Let H i +1 : X × [0 , 1] → M be an ε i -homotopy connecting t he maps g i and g i +1 . Since the set U ( i + 1) is op en in C ( X, M ), there is a p ositiv e function ε i +1 ≤ ε i / 2 suc h t hat B  ( g i +1 , 3 ε i +1 ) ⊂ U ( i + 1 ). This completes the inductiv e step. It follo ws from the construction that the function sequ ence ( g i ) i ≥ 1 con v erg es uniformly to some con tin uous function g ′ : X → M . Ob- viously ,  ( g ′ , g i ) ≤ P ∞ j = i ε j ≤ 2 ε i for ev ery i . Hence, according to the c hoice of the seque nces ( ε i ) and ( g i ), g ′ ∈ U ( i ) for ev ery i ≥ 1. So, g ′ ∈ T ∞ i =1 U ( i ). Moreov er, the ε i − 1 -homotopies H i comp ose an 6 ε -homotopy H : X × [0 , 1] → M H ( x, t ) =    H i  x, 2 i ( t − 1 + 1 2 i − 1 )  if t ∈ [1 − 1 2 i − 1 , 1 − 1 2 i ], i ≥ 1; g ′ ( x ) if t = 1 . connecting g 0 = g and g ′ . If follo ws f rom our construction that g ′ | A = g 0 | A if g i | A = g 0 | A for all i ≥ 1.  Prop osition 2.3. L et M b e a c ompletely metrizable sp ac e with the AP ( n, 0) -pr op erty and X b e an n -dimensio nal c omp actum. Then E = { g ∈ C ( X , M ) : dim g ≤ 0 } is a G δ -subset of C ( X , M ) such that every simplicial ly factorizable map is ho motopic al ly ap p r oximate d by maps fr om E . Pr o of. W e ﬁx a metric on X and a complete metric  o n M . F o r ev ery ε > 0 let C ( X, M ; ε ) b e the set all ma ps g ∈ C ( X, M ) with d 0 ( g − 1 ( a )) < ε for ev ery a ∈ M . It is easily seen that C ( X, M ; ε ) is op en in C ( X , M ) and E = T ∞ i =1 C ( X, M ; 1 /i ). So, E is a G δ -subset of C ( X, M ). Claim. F or every simplicial ly factorizable map h ∈ C ( X , M ) and p ositive numb ers η and δ ther e exists a simplicial ly factorizable map h ′ ∈ C ( X, M ; η ) wh i c h is δ - h omotopic to h . T o prov e the claim, ﬁx a simplicially f actorizable map h ∈ C ( X , M ) and p ositive n umbers δ , η . Then there exists a simplicial complex L and maps q 1 : X → L , q 2 : L → M with h = q 2 ◦ q 1 . W e can assume that L is ﬁnite and q 2 ( L ) ⊂ ∪ γ , where γ = { B  ( y , δ / 4) : y ∈ g ( X ) } . Because L is an AN R and dim X ≤ n , there is a p olyhedron K with dim K ≤ n a nd t wo maps f : X → K , α : K → L suc h tha t f is an η - map and q 1 and α ◦ f are q − 1 2 ( γ )-homotopic. So, h and q 2 ◦ α ◦ f are δ / 2-ho mo t o pic. Moreov er, there is a co v er ω ∈ cov ( K ) suc h that diam f − 1 ( W ) < η for ev ery W ∈ ω . Let θ b e a Lebesgue n umber of ω . It remains to ﬁnd a map h ∗ : K → M whic h is δ / 2-homoto pic to q 2 ◦ α and h ∗ ∈ C ( K , M ; θ ) (then h ′ = h ∗ ◦ f w ould b e a simplicially factorizable map δ -homotopic to h and h ′ ∈ C ( X , M ; η )). T o ﬁnd suc h a map h ∗ : K → M , let { σ 1 , .., σ m } b e an en umeration of the simplexes of K and K i = S j = i j =1 σ j . W e are going to construct b y induction maps h i : K → M , i = 0 , .., m , satisfying the follo wing conditions: • h 0 = q 2 ◦ α ; • h i | K i b elongs to C ( K i , M ; θ ), 1 ≤ i ≤ m ; • h i and h i +1 are ( δ / 2 m )- homotopic, i = 0 , .., m − 1. Approximation by light maps 7 Let V = { B  ( y , δ / 4 m ) : y ∈ M } ∈ co v ( M ) and assume that h i has already b een constructed. Since h i | K i ∈ C ( K i , M ; θ ), ev ery ﬁb er h − 1 i ( y ) ∩ K i of h i | K i , y ∈ h i ( K 1 ), is co v ered b y a ﬁnite o p en and disjoin t family Ω( y ) in K i with mesh(Ω( y )) < θ . Using that h i | K i is a p erf ect map, we ﬁnd a cov er V i ∈ cov ( M ) suc h that V i is a star-reﬁnemen t of V and if S t ( y , V i ) ∩ h i ( K i ) 6 = ∅ for some y ∈ M , then there is z ∈ h i ( K i ) with h − 1 i ( S t ( y , V i )) ∩ K i ⊂ Ω( z ). Sinc e M has the AP ( n, 0)-prop ert y , there exists a 0-dimensional map p i : σ i +1 → M whic h is V i -homotopic to h i | σ i +1 . By the Homotop y Extension Theorem, p i can b e extended to a map h i +1 : K → M b eing V i -homotopic to h i . Then h i +1 is ( δ / 2 m )- homotopic to h i . T o sho w that h i +1 | K i +1 b elongs to C ( K i +1 , M ; θ ), we observ e that h − 1 i +1 ( y ) ∩ K i +1 =  h − 1 i +1 ( y ) ∩ K i  ∪  h − 1 i +1 ( y ) ∩ σ i +1  , y ∈ M . According to our construction, we hav e h i  h − 1 i +1 ( y ) ∩ K i  ⊂ S t ( y , V i ). Hence, h − 1 i +1 ( y ) ∩ K i is contained in Ω( z ) fo r some z ∈ h i ( K i ). Therefore, d 0  h − 1 i +1 ( y ) ∩ K i  < θ . Since h − 1 i +1 ( y ) ∩ σ i +1 is 0-dimensional, Lemma 2.1 implies that d 0 ( h − 1 i +1 ( y ) ∩ K i +1 ) < θ . Ob viously the map h ∗ = h m is δ / 2-homotopic to q 2 ◦ α and h ∗ ∈ C ( K, M ; θ ). This completes the pro of of the claim. T o ﬁnish the pro of of the prop osition, w e apply Lemma 2.2 with G b eing the set of a ll simplicially factorizable maps fro m C ( X, M ) and U ( i ) = C ( X , M ; 1 /i ), i ≥ 1 (we can apply Lemma 2.2 b ecause of the claim). Hence, for ev ery simplicially fa ctorizable map g ∈ C ( X , M ) and a p ositiv e n um b er ε there exists a map g ′ ∈ T ∞ i =1 U ( i ) whic h is ε -homotopic to g . Finally , w e observ e that T ∞ i =1 U ( i ) consists of 0- dimensional maps.  Prop osition 2.4. L et M b e a c ompletely metrizable sp ac e with the AP ( n, 0) -pr op erty, X a c omp actum and Z a n F σ -subset of X with dim Z ≤ n − 1 . Then ther e exists a G δ -subset H ⊂ C ( X , M ) with the fol lo wing pr op erties: • Z is c ontaine d in the union of trivial c omp onents of the ﬁb ers of g for al l g ∈ H ; • for every simplicial ly factorizable map g ∈ C ( X, M ) and every ε > 0 ther e e x i s ts g ′ ∈ H which is ε -homotopic to g . Pr o of. W e represen t Z a s the union of a n increasing sequence ( Z i ) i ≥ 1 with all Z i b eing closed in Z . F or ev ery ε > 0 let H ( Z i , ε ) = { g ∈ C ( X, M ) : F ( g , ε ) ∩ Z i = ∅} , where F ( g , ε ) = ∪{ C : C is a comp onen t of a ﬁb er of g with diam C ≥ ε } . 8 It is easily seen that eac h H ( Z i , ε ) is op en in C ( X , M ). So, the set H = T ∞ i =1 H ( Z i , 1 /i ) is G δ and Z is con tained in the union of trivial comp onen ts of the ﬁb ers of g for ev ery g ∈ H . T o pro v e the second item of our prop osition, w e ﬁrst consider the particular case when X is a p olyhedron a nd Z is a (compact) subpoly- hedron of X . Claim 1 . Supp ose that, in addition to hyp otheses of Pr op osition 2 . 4 , X i s a p olyhe dr on and Z i s a subp olyhe dr on of X . Then, for every g 0 ∈ C ( X, M ) and δ > 0 ther e e x ists g ∈ H = T ∞ i =1 H ( Z , 1 /i ) which is δ -ho motopic to g 0 . The pro of of Claim 1 is a slight mo diﬁcation of the pro of of [12, Theorem 2.2]. Let g 0 ∈ C ( X, M ) and δ > 0. W e tak e an op en neigh b or ho o d W of Z in X , a retraction r : W → Z and a function α : X → I such that α − 1 (0) = Z , α − 1 (1) = X \ W . Since X is a com- pact AN R , w e can c ho ose W so small that g 0 | W is δ / 2-homotopic to ( g 0 ◦ r ) | W . Nex t, denote by π : Z × I → Z the pro jection a nd consider the map ϕ : W → Z × I , ϕ ( x ) = ( r ( x ) , α ( x )). Obvious ly , ( g 0 ◦ r ) | W = ( g 0 ◦ π ◦ ϕ ) | W . So, ( g 0 ◦ π ◦ ϕ ) | W is δ / 2-homotopic t o g 0 | W . Because Z is a p o lyhedron, so is Z × I . Hence, eve r y map from C ( Z × I , M ) is simplicially factorizable. Moreo v er, dim Z × I ≤ n and M ∈ AP ( n, 0). Therefore, b y Prop osition 2.3, there exists a 0 - dimensional ma p h : Z × I → M whic h is δ / 2-homotopic to the map g 0 ◦ π . Then ( h ◦ ϕ ) | W is δ / 2-ho mo t o pic to ( g 0 ◦ π ◦ ϕ ) | W . Conseq uently , ( h ◦ ϕ ) | W is δ -homotopic to g 0 | W . By the Homotop y Extension Theo- rem, there exits a map g ∈ C ( X , M ) suc h that g is δ -ho mo t opic to g 0 and g | U = ( h ◦ ϕ ) | U , where U is an o p en neigh b orho o d of Z in X with U ⊂ W . T o ﬁnish the pro of o f Claim 1, it remains to show that g ∈ H . Let C b e a sub contin uum of g − 1 ( y ) fo r some y ∈ M and let Z ∩ C 6 = ∅ . W e are going to pro ve that C ⊂ Z . Otherwise , there w ould b e a sub con tinuum C ′ ⊂ C ∩ U suc h that C ′ ∩ Z 6 = ∅ and C ′ \ Z 6 = ∅ . Then g ( C ′ ) = h ( ϕ ( C ′ )) = y and, according to the deﬁnition of α , ϕ ( C ′ ) is a non-degenerate con t inuum in h − 1 ( y ). Since h is 0-dimensional, this is a contradiction. Hence, C ⊂ Z . Using ag ain that dim( h ) ≤ 0, w e conclude that ϕ ( C ) is a p oin t. On the other hand ϕ ( C ) = C × { 0 } ⊂ Z × I . Therefore, C should b e a trivial contin uum. No w, consider the general case of Prop o sition 2.4. Claim 2 . L et g 0 ∈ C ( X, M ) b e a sim p licial ly factorizabl e map and δ , η p ositive numb ers. Then for any i ther e ex ists a sim plicial ly factoriz- able map g ∈ H ( Z i , η ) which is δ -homotopic to g 0 . Approximation by light maps 9 Since g 0 is simplicially facto r izable, there exist a ﬁnite simplicial complex L and maps q 1 : X → L , q 2 : L → M with g 0 = q 2 ◦ q 1 . L et V = { B  ( y , δ / 4) : y ∈ M } ∈ cov ( M ) and W = q − 1 2 ( V ) ∈ cov ( L ). Next, c ho o se a ﬁnite cov er U ∈ co v ( X ) of X with mesh( U ) < η suc h that: • at most n elemen t s of the family γ = { U ∈ U : U ∩ Z i 6 = ∅} can ha ve a common p oint; • there ex ists a map h : N ( U ) → L suc h that h ◦ f U is W - homotopic to q 1 , where N ( U ) is the nerve o f U and f U : X → N ( U ) is the natural map. Then q 2 ◦ h ◦ f U is δ / 2-homotopic to g 0 and the subp olyhedron K of N ( U ) generated b y the fa mily γ is of dimension ≤ n − 1 . So, according to Claim 1, there exists a map g 1 : N ( U ) → M suc h tha t g 1 is δ / 2-homotopic to q 2 ◦ h and K is con tained in the union of trivial comp onen ts o f the ﬁb ers of g 1 . Since all ﬁb ers of f U are o f diameter < η and f U ( Z i ) ⊂ K , the map g = g 1 ◦ f U b elongs to H ( Z i , η ). Moreo v er, g is ob viously simplicially f actorizable and δ -homotopic to g 0 . This completes the pro of o f Claim 2 . Finally , the pro of o f Prop o sition 2 .4 follows from Lemma 2.2 (with G b eing the set of all simplicially factorizable maps from C ( X , M ) and U ( i ) = H ( Z i , 1 /i ) for ev ery i ≥ 1) and Claim 2.  3. P arametric Le le k maps In this section w e are going to prov e Theorem 1.1. Every where in this section w e supp ose that ( M ,  ) is a giv en complete metric space and µ = { W ν : ν ∈ Λ } , µ 1 = { G ν : ν ∈ Λ } are lo cally ﬁnite op en co vers of M such that (*) G ν ⊂ W ν and W ν ∈ AP ( n ν , 0) with 0 ≤ n ν for ev ery ν ∈ Λ. Ob viously , Theorem 1.1 follows directly from next theorem. Theorem 3.1. L et f : X → Y b e a p erfe ct map b etwe en metrizable sp ac es with dim △ ( f ) ≤ m . Supp ose ( M ,  ) is a c omp l e te metric sp ac e and µ , µ 1 two lo c al ly ﬁnite op en c overs of M satisfying c ondition ( ∗ ) . Then ther e i s a G δ -set H ⊂ C ( X, M ) s uch that any simplicial ly fac- torizable map in C ( X, M ) c an b e hom otopic al ly appr oximate d by maps fr om H an d every g ∈ H has the fol lowin g pr op erty: for any y ∈ Y and ν ∈ Λ the r estriction g |  f − 1 ( y ) ∩ g − 1 ( G ν )  is an ( m − n ν ) -dimensiona l L elek map fr om f − 1 ( y ) ∩ g − 1 ( G ν ) into G ν . The pro of of Theorem 3 .1 consists of few pro p ositions. W e can as- sume that n ν ≤ m for all ν ∈ Λ. F or ev ery y ∈ Y , ε > 0, ν ∈ Λ and g ∈ C ( X, M ) w e denote b y F ν ( g , ε, y ) the union of all con tinua C of diameter ≥ ε suc h that C ⊂ f − 1 ( y ) ∩ g − 1 ( z C ) for some z C ∈ G ν . Note 10 that eac h F ν ( g , ε, y ) is compact as a closed subset of f − 1 ( y ). Then, for a ﬁxed η > 0 , let H ( y , ε, η ) b e the set of all g ∈ C ( X, M ) suc h that d m − n ν  F ν ( g , ε, y )  < η for ev ery ν ∈ Λ. The la st inequalit y means that F ν ( g , ε, y ) can b e co v ered b y an op en family γ in X suc h that mesh( γ ) < η and no more m − n ν + 1 elemen ts o f γ ha v e a com- mon po in t. If V ⊂ Y , then H ( V , ε, η ) denotes the in tersection of all H ( y , ε, η ), y ∈ V . Lemma 3.2. F or every y ∈ Y and every g ∈ H ( y , ε, η ) ther e exists a neighb orho o d V y of y in Y and δ y > 0 such that if y ′ ∈ V y and  ( h ( x ) , g ( x )) < δ y for al l x ∈ f − 1 ( y ′ ) , then h ∈ H ( y ′ , ε, η ) . Pr o of. Since µ 1 is lo cally ﬁnite and f − 1 ( y ) is compact, fo r eve r y y ∈ Y and g ∈ C ( X , M ) there exists a neigh b orho o d O g ( y ) of g ( f − 1 ( y )) suc h that the family Λ g ( y ) = { ν ∈ Λ : O g ( y ) ∩ G ν 6 = ∅} is ﬁnite. Assume t he lemma is not true for some y ∈ Y and g ∈ H ( y , ε, η ). Then there exists a lo cal base of neighborho o ds V i of y , p oin ts y i ∈ V i and functions g i ∈ C ( X , M ) suc h that g i | f − 1 ( y i ) is 1 /i -close t o g | f − 1 ( y i ) but g i 6∈ H ( y i , ε, η ). It is easily seen that for some k a nd all i ≥ k we hav e g i ( f − 1 ( y i )) ⊂ O g ( y ) . Consequen tly , for ev ery i ≥ k there exists ν ( i ) ∈ Λ g ( y ) suc h that d m − n ν ( i )  F ν ( i ) ( g i , ε, y i )  ≥ η (b ecause g i 6∈ H ( y i , ε, η )) . So, each F ν ( i ) ( g i , ε, y i ) do esn’t ha v e any op en cov er of mesh < η and order ≤ m − n ν ( i ) + 1. Since the family Λ g ( y ) is ﬁnite, there exists ν (0) ∈ Λ g ( y ) with ν ( i ) = ν (0) for inﬁnitely man y i . With out loss of generality , we ma y suppose that ν ( i ) = ν (0) for all i ≥ k . This implies that g i ( f − 1 ( y i )) ∩ G 6 = ∅ , i ≥ k , where G = G ν (0) . Consequen tly , g ( f − 1 ( y )) ∩ G 6 = ∅ . Since g ∈ H ( y , ε, η ), F ν (0) ( g , ε, y ) can b e co vered by an o p en fa mily γ in X of order ≤ m − n ν (0) + 1 and mesh( γ ) < η . Let U = ∪ γ . T o obtain a contradiction, it suﬃce to sho w tha t F ν ( i ) ( g i , ε, y i ) ⊂ U for some i ≥ k . Indeed, otherwise fo r ev ery i ≥ k there w ould exist p oin ts x i ∈ F ν ( i ) ( g i , ε, y i ) \ U , z i ∈ G , and a compo nen t C i of f − 1 ( y i ) ∩ g − 1 i ( z i ) con taining x i with diam C i ≥ ε . Using that P = f − 1 ( { y i } ∞ i = k ∪ { y } ) is a compactum, w e can supp o se that { x i } ∞ i = k con v erg es to a p oint x 0 ∈ f − 1 ( y ), { z i } ∞ i = k con v erg es to a p oint z 0 ∈ G and { C i } ∞ i = k (considered as a sequence in the space of all closed subsets of P equipp ed with the Vietoris top olo gy) con verges to a closed set C ⊂ f − 1 ( y ) ∩ g − 1 ( z 0 )). It is easily seen that C is connected and dia m C ≥ ε . Hence, C ⊂ F ν (0) ( g , ε, y ) ⊂ U . So, x i ∈ C i ⊂ U for some i , a con tradiction.  No w, we ar e in a p osition to show that the sets H ( Y , ε, η ) are op en in C ( X , M ). Approximation by light maps 11 Prop osition 3.3. F or any close d set F ⊂ Y and any ε, η > 0 , the set H ( F , ε , η ) is op en in C ( X , M ) . Pr o of. Let g 0 ∈ H ( F , ε, η ). Then, b y Lemma 3.2, fo r ev ery y ∈ F there exist a neigh b orho o d V y and a p ositiv e δ y ≤ 1 suc h tha t h ∈ H ( y ′ , ε, η ) pro vided y ′ ∈ V y and h | f − 1 ( y ′ ) is δ y -close to g 0 | f − 1 ( y ′ ). The family { V y ∩ Y : y ∈ F } can b e supp osed to b e lo cally ﬁnite in F . Consider the set-v alued low er semi-con tin uous map ϕ : F → (0 , 1 ], ϕ ( y ) = ∪{ (0 , δ z ] : y ∈ V z } . By [22, Theorem 6.2, p.116], ϕ admits a con tin uous selection β : F → (0 , 1]. Let β : Y → (0 , 1 ] b e a contin uous extension of β and α = β ◦ f . It remains only to sho w t hat if g ∈ C ( X, M ) with   g 0 ( x ) , g ( x )  < α ( x ) for a ll x ∈ X , then g ∈ H ( F , ε, η ). So, w e tak e suc h a g and ﬁx y ∈ F . Then there exists z ∈ F with y ∈ V z and α ( x ) ≤ δ z for all x ∈ f − 1 ( y ). Hence,   g ( x ) , g 0 ( x )  < δ z for eac h x ∈ f − 1 ( y ). According to the c hoice of V z and δ z , g ∈ H ( y , ε, η ). Therefore, H ( F , ε, η ) is op en in C ( X, M ).  T o prov e Theorem 3.1 it suﬃces to sho w tha t if g ∈ C ( X, M ) is a simplicially factorizable map and δ ∈ C ( X, (0 , 1]), then for any ε, η > 0 there exists a simplicially factorizable map g εη ∈ H ( Y , ε, η ) whic h is δ -homotopic to g . Indeed, since a n y set H ( Y , ε, η ) is op en, Lemma 2 .2 w ould imply that ev ery simplicially factorizable map is homotopically a ppro ximated b y simplicially f actorizable ma ps from H = T ∞ i,j =1 H ( Y , 1 /i, 1 / j ). But for ev ery g ∈ H , y ∈ Y , ν ∈ Λ and i ≥ 1 w e ha ve d m − n ν  F ν ( g , 1 /i, y )  = 0 . So, dim F ν ( g , 1 /i, y ) ≤ m − n ν . Then dim F ν ( g , y ) ≤ m − n ν , where F ν ( g , y ) = S ∞ i =1 F ν ( g , 1 /i, y ) . On the other hand, F ν ( g , y ) is the union of all no n- trivial contin ua contained in the ﬁb ers of g | g − 1 ( G ν ) ∩ f − 1 ( y ). Therefore, H consists of maps g suc h that g | g − 1 ( G ν ) ∩ f − 1 ( y ) : g − 1 ( G ν ) ∩ f − 1 ( y ) → G ν is ( m − n ν )- dimensional Lelek map for ev ery ν ∈ Λ and y ∈ Y . Next prop osition show s that all simplicially fa cto r izable maps in C ( X, M ) can b e homotopically a ppro ximated b y simplicially factor- izable maps from H ( Y , ε, η ) provide d the set H p ( L ) is homotopically dense in C ( N , M ) , where p : N → L is any p erfect m -dimensional P L -map b et w een t w o simplicial complexes N , L equipp ed with the C W -top ology and H p ( L ) is the set of all maps q : N → M suc h that q | q − 1 ( G ν ) ∩ p − 1 ( z ) : q − 1 ( G ν ) ∩ p − 1 ( z ) → G ν is an ( m − n ν )-dimensional Lelek map for ev ery ν ∈ Λ and z ∈ L . Recall that p : N → L is a P L -map (resp., a simplicial map) if p maps ev ery simplex σ of N in to (resp., on to) some simplex of L and p is linear on σ . Prop osition 3.4. L et X , Y , f and M satisfy the hyp otheses of Th e o- r em 3 . 1 . Supp ose the set H p ( L ) is homotopic al ly dense in C ( N , M ) for 12 any p erfe ct m -dime nsional P L -map p : N → L . The n fo r any simpli- cial ly factorizable map g ∈ C ( X , M ) and any δ ∈ C ( X , (0 , 1]) , ε, η > 0 ther e exists a simp licial ly fa c torizable h ∈ H ( Y , ε, η ) such that h is δ -ho motopic to g . Pr o of. F or ﬁxed δ ∈ C ( X , (0 , 1]) a nd a simplicially fa ctorizable map g ∈ C ( X, M ) w e are going to ﬁnd a simplicially factor izable h ∈ H ( Y , ε, η ) suc h that  ( g ( x ) , h ( x )) < δ ( x ) for all x ∈ X , where ε and η are ar- bitrary p ositiv e reals. Since g is simplicially fa ctorizable, there exists a simplicial complex D and maps g D : X → D , g D : D → M with g = g D ◦ g D . The metric  induces a con tinuous pseudometric  D on D ,  D ( x, y ) =  ( g D ( x ) , g D ( y )). By [5] and [24], D b eing a stratiﬁable AN R is a neighborho o d retract of a lo cally conv ex space. Hence, w e can apply [2 , Lemma 8.1] to ﬁnd an op en cov er U of X satisfying the follo wing condition: if α : X → K is a U -map in to a paracompact space K (i.e., α − 1 ( ω ) reﬁnes U fo r some ω ∈ cov ( K )), then there exists a map q ′ : G → D , where G is a n op en neighborho o d of α ( X ) in K , suc h tha t g D and q ′ ◦ α are δ / 2-ho motopic with resp ect to the metric  D . Let U 1 b e an op en co v er of X r eﬁning U with mesh U 1 < min { ε, η } and inf δ ( U ) > 0 for all U ∈ U 1 . Next, according to [2, Theorem 6 ], there exists an op en co v er V of Y suc h that: for any V -map β : Y → L into a simplicial complex L we can ﬁnd an U 1 -map α : X → K in to a simplicial complex K and a p erfect m -dimensional P L -map p : K → L with β ◦ f = p ◦ α . W e can assume that V is lo cally ﬁnite. T ake L to b e the nerv e of the co v er V and β : Y → L the corresponding natura l map. Then there are a simplicial complex K and maps p and α satisfying the ab ov e conditions. Hence, the follow ing diag r am is comm utat ive: X α − − − → K f   y   y p Y β − − − → L Since K is para compact, the choice of the co ver U guaran tees the exis- tence of a map q D : G → D , where G ⊂ K is a n op en neigh b orho o d o f α ( X ), suc h that g D and h D = q D ◦ α a re δ / 2-homotopic with resp ect to  D . Then, a ccording to the deﬁnition o f  D , h ′ = g D ◦ q D ◦ α is δ / 2- homotopic to g with resp ect to  . Replacing the t riangulation of K by a suitable sub division, w e may additiona lly assume that no simplex of K meets b oth α ( X ) and K \ G . So, the union N of all simplexe s σ ∈ K with σ ∩ α ( X ) 6 = ∅ is a sub complex of K and N ⊂ G . Moreov er, Approximation by light maps 13 since N is closed in K , p : N → L is a p erfect m -dimensional P L - map. Therefore, w e ha ve the following commutativ e diagram, where q = g D ◦ q D : L Y ◗ ◗ s β X ❄ f ✲ h ′ ◗ ◗ s α N ❄ p ✚ ✚ ❃ q M No w, we shall construct a con tinuous function δ 1 : N → (0 , 1] with δ 1 ◦ α ≤ δ . Since α is a U 1 -map, there is a n op en co ve r V 1 of N suc h that the cov er α − 1 ( V 1 ) = { α − 1 ( V ) : V ∈ V 1 } reﬁnes U 1 . Because inf δ ( U ) > 0 for any U ∈ U 1 , inf δ ( α − 1 ( V )) > 0 for any V ∈ V 1 . W e can a ssume that V 1 is lo cally ﬁnite and consider the low er semi-con tin uo us set- v alued map ϕ : N → (0 , 1] deﬁned b y ϕ ( z ) = ∪{ (0 , inf δ ( α − 1 ( V )) ] : z ∈ V ∈ V 1 } . T hen, by [2 2, Theorem 6.2, p. 116], ϕ a dmits a contin uous selection δ 1 : N → (0 , 1]. Ob viously , δ 1 ( z ) ≤ inf δ ( α − 1 ( z )) f o r all z ∈ N . Hence, δ 1 ◦ α ≤ δ . Since, according to our assumption, H p ( L ) is homotopically dense in C ( N , M ), there exists a map q 1 ∈ H p ( L ) suc h that q 1 is δ 1 / 2-homotopic to q . Let h = q 1 ◦ α . Then h and q ◦ α a r e δ / 2-homo t o pic because δ 1 ◦ α ≤ δ . On the other ha nd, q ◦ α = h ′ is δ / 2-homotopic to g . Hence, g and h ar e δ -homoto pic. Moreo v er, h is o b viously simplicially factorizable. It remains to sho w that h ∈ H ( Y , ε, η ). T o this end, w e ﬁx y ∈ Y and ν ∈ Λ, and consider the set F ν ( h, ε, y ). Recall that F ν ( h, ε, y ) is the union of a ll contin ua of diameter ≥ ε suc h that C ⊂ f − 1 ( y ) ∩ h − 1 ( a C ) f or some a C ∈ G ν . F or any suc h con tinuum C w e ha v e α ( C ) ⊂ p − 1 ( β ( y )) ∩ q − 1 1 ( a C ). Since t he diameters of all ﬁb ers of α are < ε (recall that α is an U 1 -map), α ( C ) is a non- trivial con tin- uum in p − 1 ( β ( y )) ∩ q − 1 1 ( a C ). Therefore, α ( C ) ⊂ F ν ( q 1 , β ( y )), where F ν ( q 1 , β ( y )) denotes the unio n of a ll non-trivial con tinua whic h are con- tained in the ﬁb ers of the restriction q y ν = q 1 |  p − 1 ( β ( y )) ∩ q − 1 1 ( G ν )  . Actually , w e pro ved that α  F ν ( h, ε, y )  ⊂ F ν ( q 1 , β ( y )). Since q 1 ∈ H p ( L ), q y ν : p − 1 ( β ( y )) ∩ q − 1 1 ( G ν ) → G ν is ( m − n ν )-dimensional Lelek map. Consequen tly , dim F ν ( q 1 , β ( y )) ≤ m − n ν . So, there exists an op en cov er γ of F ν ( q 1 , β ( y )) of order ≤ m − n ν + 1 (suc h a cov er γ exists b ecause F ν ( q 1 , β ( y )) is metrizable as a subset of the metrizable compactum p − 1 ( β ( y )). W e can supp o se that γ is so small t hat α − 1 ( γ ) reﬁnes U 1 . But mesh ( U 1 ) < η . Consequen tly , α − 1 ( γ ) ∩ F ν ( h, ε, y ) is an 14 op en cov er of F ν ( h, ε, y ) of o rder ≤ m − n ν + 1 and mesh < η . This means that d m − nν  F ν ( h, ε, y )  < η . Therefore, w e found a simplicially factorizable map h ∈ H ( Y , ε, η ) whic h is δ -ho motopic to g .  In next tw o lemmas w e supp ose that p : N → L is an m -dimensional P L -map b et we en ﬁnite simplicial complexes. As eve r ywhere in this section, ( M ,  ) is a complete metric space p ossess ing tw o lo cally ﬁnite op en co v ers µ = { W ν : ν ∈ Λ } and µ 1 = { G ν : ν ∈ Λ } suc h that G ν ⊂ W ν , W ν ∈ AP ( n ν , 0) a nd 0 ≤ n ν ≤ m for ev ery ν ∈ Λ. F or giv en ε, η > 0 and y ∈ L w e denote by H p ( y , ε, η ) the set of g ∈ C ( N , M ) suc h that d m − n ν  F ν ( g , ε, y )  < η for ev ery ν ∈ Λ. Here, F ν ( g , ε, y ) is the union of a ll con t in ua C ⊂ p − 1 ( y ) ∩ g − 1 ( z C ) with z C ∈ G ν and diam C ≥ ε . If B ⊂ L , then H p ( B , ε, η ) stands for the interse ction of all H p ( y , ε, η ), y ∈ B . Lemma 3.5. L et B ⊂ L b e a sub c omplex of L and g 0 ∈ C ( N , M ) b e a map such that g 0 ∈ H p ( B ) = T i ≥ 1 H p ( B , 1 /i, 1 /i ) . Then, for e v ery ε > 0 , q ≥ 1 and g ∈ C ( N , M ) with g | p − 1 ( B ) = g 0 | p − 1 ( B ) ther e exists a map g q ∈ C ( N , M ) extending g 0 | p − 1 ( B ) such that g q is ε -hom otopic to g and g q ∈ H p ( y , 1 /q , 1 / q ) for al l y ∈ L . Pr o of. W e ﬁx q a nd g ∈ C ( N , M ) with g | p − 1 ( B ) = g 0 | p − 1 ( B ). Then, b y Lemma 3.2, f or ev ery y ∈ B there exists a neigh b or ho o d V y in L and δ y > 0 such t ha t an y h ∈ C ( N , M ) b elongs to H p ( V y , 1 /q , 1 /q ) provid ed h | p − 1 ( V y ) is δ y -close to g | p − 1 ( V y ) (we can apply Lemma 3.2 b ecause g | p − 1 ( B ) = g 0 | p − 1 ( B ) yields g ∈ H p ( y , 1 /q , 1 / q ) for ev ery y ∈ B ) . Let { V y ( i ) } i ≤ s b e a ﬁnite subfamily of { V y : y ∈ B } cov ering B and V = S 1 ≤ i ≤ s V y ( i ) . Since µ is lo cally ﬁnite in M , g ( N ) meets only ﬁnitely man y W j = W ν ( j ) , j = 1 , .., k . F or an y j ≤ k let P j = g − 1 ( G ν ( j ) ) and U 1 j , U 2 j b e op en subs ets of N suc h that P j ⊂ U 1 j ⊂ U 1 j ⊂ U 2 j ⊂ U 2 j ⊂ g − 1 ( W j ). W e also c ho ose ε 0 > 0 satisfying the follow ing condition: • If h ∈ C ( N , M ) with   g ( x ) , h ( x )  < ε 0 for ev ery x ∈ N , then { ν ∈ Λ : h ( N ) ∩ W ν 6 = ∅} is contained in { ν ( j ) : j = 1 , .., k } , and for all j w e ha v e h − 1 ( G ν ( j ) ) ⊂ U 1 j and h ( U 2 j ) ⊂ W j . Let δ 0 = min { ε, ε 0 , δ y ( i ) : i ≤ s } . Considering suitable sub divisions of N and L , w e can supp ose that p is a simplicial map and the following conditions hold: • Ev ery simplex σ ∈ L in tersecting the set L \ V do es not meet B ; • Ev ery simplex τ ∈ N inte r secting the set U 1 j do es not meet N \ U 2 j , j = 1 , .., k . Approximation by light maps 15 Let L 0 = ∪{ σ ∈ L : σ ∩ L \ V 6 = ∅} and N j = ∪{ τ ∈ N : τ ∩ U 1 j 6 = ∅} , 1 ≤ j ≤ k . O bviously L 0 is a subp olyhedron o f L disjoin t fr o m B and con taining L \ V . Moreov er, eac h N j is a subp olyhedron of N suc h that U 1 j ⊂ N j ⊂ U 2 j . Now, for ev ery j ≤ k consider the map p j = p | N j : N j → p ( N j ). Since p is m -dimensional, b y [21] or [26], there exists an ( n ν ( j ) − 1)- dimensional sigma-compact set Z j = S i ≥ 1 Z ij ⊂ N j suc h that eac h Z ij is compact and dim p − 1 j ( y ) \ Z j ≤ m − n ν ( j ) for all y ∈ p ( N j ). Denote T ij = { h ∈ C ( N , M ) : F j ( h, 1 /q ) ∩ Z ij = ∅} , where F j ( h, 1 /q ) is the union of all contin ua C ⊂ N j of diameter ≥ 1 /q whic h are con tained in ﬁb ers of the map h | N j . It is easily seen that T ij are op en in C ( N , M ). Claim. F o r any h ∈ B  ( g , δ 0 ) , η > 0 and i, j ≥ 1 ther e exists h ij ∈ T ij ∩ B  ( h, η ) ∩ B  ( g , δ 0 ) which is η -homotopic to h . Indeed, h ∈ B  ( g , δ 0 ) and δ 0 ≤ ε 0 imply that h ( N j ) ⊂ W j and h ( N ) ∩ W ν = ∅ if ν 6 = ν ( j ) for all j . Let η 1 = min { η , δ 0 −   g ( x ) , h ( x )  : x ∈ N } . Since h | N j is simplicially factorizable (as a map whose domain is a p olyhedron), W j ∈ AP ( n ν ( j ) , 0) and Z ij is a compact subset of N j with dim Z ij ≤ n ν ( j ) − 1, according to Proposition 2.4, there is a map h ′ : N j → W j whic h is η 1 -homotopic to h | N j and the union of all non-trivial comp onents of the ﬁb ers of h ′ is disjoin t from Z ij . By the Homotop y Extension Theorem, h ′ admits an extension h ij ∈ C ( N , M ) with h ij b eing η 1 -homotopic to h . Ob viously , h ij ∈ T ij and h ij ∈ B  ( g , δ 0 ). The ab ov e claim allo ws us to apply Lemma 2.2 fo r the set B  ( g , δ 0 ) and the sequence { T ij } i,j ≥ 1 to obtain a map h 1 ∈ C ( N , M ) such that h 1 ∈ T ∞ i,j =1 T ij and h 1 is δ 0 -homotopic to g . Let h 2 : p − 1 ( L 0 ∪ B ) → M b e deﬁned b y h 2 ( x ) = g 0 ( x ) if x ∈ p − 1 ( B ) and h 2 ( x ) = h 1 ( x ) if x ∈ p − 1 ( L 0 ). Obviously , h 2 is δ 0 -homotopic t o g . Since L 0 ∪ B is a subp o lyhedron of L and p is a simplicial map, p − 1 ( L 0 ∪ B ) is a sub- p olyhedron of N . Then, b y the Homotopy Extension Theorem, there exists a map g q ∈ C ( N , M ) extending h 2 with g q b eing δ 0 -homotopic to g . It remains only to sho w that g q ∈ H p ( y , 1 /q , 1 / q ) for all y ∈ L . This is true if y ∈ V . Indeed, then y b elongs to some V y ( i ) . Since g q | V y ( i ) is δ 0 -close to g | V y ( i ) and δ 0 ≤ δ y ( i ) , g q ∈ H p ( y , 1 /q , 1 / q ) according to the c hoice of V y ( i ) and δ y ( i ) . If y ∈ L 0 , then g q | p − 1 ( y ) = h 1 | p − 1 ( y ). Since h 1 is δ 0 -close to g and δ 0 ≤ ε 0 , h − 1 1 ( G ν ( j ) ) ⊂ U 1 j ⊂ N j for ev ery j ≤ k . So, p − 1 ( y ) ∩ g − 1 q ( G ν ( j ) ) = p − 1 ( y ) ∩ h − 1 1 ( G ν ( j ) ) = p − 1 j ( y ) ∩ g − 1 q ( G ν ( j ) ), 16 j ≤ k . On the other hand, h 1 ∈ T ∞ i,j =1 T ij implies that eve r y restric- tion h 1 | h − 1 1 ( G ν ( j ) ) : h − 1 1 ( G ν ( j ) ) → G ν ( j ) has the fo llowing prop erty: the union of all non-trivial comp onen ts of the ﬁb ers of h 1 | h − 1 1 ( G ν ( j ) ) is con tained in N j \ Z j . Hence, the union o f all non-trivial comp onen ts of the ﬁb ers of g q | p − 1 ( y ) ∩ g − 1 q ( G ν ( j ) ) is contained in p − 1 j ( y ) \ Z j . Since dim p − 1 j ( y ) \ Z j ≤ m − n ν ( j ) , ev ery g q | p − 1 ( y ) ∩ g − 1 q ( G ν ( j ) ) is an ( m − n ν ( j ) )- dimensional Lelek map. But g q ( N ) do esn’t meet an y G ν except for ν ∈ { ν ( j ) : j = 1 , .., k } . The refo r e, g q ∈ H p ( y , 1 /i, 1 /i ) for ev ery i ≥ 1.  Lemma 3.6. L et B ⊂ L b e a sub c omplex of L a nd g 0 ∈ C ( N , M ) b e a map such that g 0 ∈ H p ( B ) = T i ≥ 1 H p ( B , 1 /i, 1 /i ) . Then , for every δ > 0 ther e exists g 0 ∈ H p ( L ) which is δ -homotopic to g 0 and g 0 | p − 1 ( B ) = g 0 | p − 1 ( B ) Pr o of. Eac h set H p ( L, 1 /i, 1 /i ) is op en in C ( N , M ) according to Prop o- sition 3 .3. So, by Lemma 3.5 , w e can apply Lemma 2.2 (with U ( i ) b eing in our cas e H p ( L, 1 /i, 1 /i ) and A = p − 1 ( B )) to ﬁnd a map g 0 ∈ T i ≥ 1 H p ( L, 1 /i, 1 /i ) which is δ -homotopic to g 0 and g 0 | p − 1 ( B ) = g 0 | p − 1 ( B ). Finally , g 0 ∈ H p ( L ) b ecause H p ( L ) = T i ≥ 1 H p ( L, 1 /i, 1 /i ).  Next prop osition completes the pro of of Theorem 3.1. W e supp ose that p : N → L is a p erfect m -dimensional P L -map b etw een simplicial complexes, ( M ,  ) a complete metric space and µ = { W ν : ν ∈ Λ } , µ 1 = { G ν : ν ∈ Λ } are lo cally ﬁnite op en co vers of M with G ν ⊂ W ν and W ν ∈ AP ( n ν , 0) for ev ery ν ∈ Λ, where n ν ≤ m are in tegers. If B ⊂ A ⊂ L , denote b y H p ( A, B ) the set of the maps g ∈ C ( p − 1 ( A ) , M ) suc h that g | g − 1 ( G ν ) ∩ p − 1 ( y ) : g − 1 ( G ν ) ∩ p − 1 ( y ) → G ν is ( m − n ν )- dimensional Lelek map fo r eve r y ν ∈ Λ and y ∈ B . When A = B , w e write H p ( B ) instead of H p ( A, B ). Prop osition 3.7. L et p : N → L and ( M ,  ) b e as ab ove. Th e n the set H p ( L ) is homotopic al ly d ense in C ( N , M ) . Pr o of. As usual, the simplicial complexes N and L are equipp ed with the C W -to p ology . But when consider a diameter of a ny subset of N w e mean the diameter with resp ect to the standard metric g enerating the metric to p ology of N . According to the notatio ns in this section, for ev ery sets B ⊂ A ⊂ L and ε, η > 0, let H p ( A, B , ε, η ) b e the set of all g ∈ C ( p − 1 ( A ) , M ) suc h t hat any d m − n ν  F ν ( g , ε, y )  < η for all y ∈ B a nd ν ∈ Λ. Although the do main of g is the set A (not the whole space N ), w e use the same notat io n F ν ( g , ε, y ) to denote Approximation by light maps 17 the union of a ll contin ua C ⊂ p − 1 ( y ) ∩ g − 1 ( z C ) with z C ∈ G ν and diam C ≥ ε . Let us also denote b y H p ( A, B ) the set of the maps g ∈ C ( p − 1 ( A ) , M ) such that g | g − 1 ( G ν ) ∩ p − 1 ( y ) : g − 1 ( G ν ) ∩ p − 1 ( y ) → G ν is an ( m − n ν )-dimensional Lelek map for ev ery ν ∈ Λ and y ∈ B . This means that dim F ν ( g , y ) ≤ m − n ν for any y ∈ B and ν ∈ Λ, where F ν ( g , y ) = S ∞ i =1 F ν ( g , 1 /i, y ). It is easily seen that H p ( A, B ) = T i ≥ 1 H p ( A, B , 1 /i, 1 /i ). No w, let us ﬁnish the pro of of Prop osition 3 .7 . Fix g ∈ C ( N , M ) and δ ∈ C ( N , (0 , 1]). W e are going to ﬁnd h ∈ H p ( L ) whic h is δ -homot o pic to g . T o this end, let L ( i ) , i ≥ 0, denote the i -dimensional sk eleton of L and L ( − 1) = ∅ . W e put h − 1 = g a nd construct inductiv ely a sequence ( h i : N → M ) i ≥ 0 of maps suc h that • h i | p − 1 ( L ( i − 1) ) = h i − 1 | p − 1 ( L ( i − 1) ); • h i is δ 2 i +2 -homotopic to h i − 1 ; • h i ∈ H p ( L, L ( i ) ). Assuming that the map h i − 1 : N → M ha s b een constructed, con- sider the complemen t L ( i ) \ L ( i − 1) = ⊔ j ∈ J i ◦ σ j , whic h is the discrete union of op en i -dimensional simplexes. Since h i − 1 | σ j b elongs to H p ( σ j , σ ( i − 1) j ) for any simplex σ j ∈ L ( i ) , w e can a pply Lemma 3.6 to ﬁnd a map g j : p − 1 ( σ j ) → M suc h that • g j coincides with h i − 1 on the set p − 1 ( σ ( i − 1) j ); • g j is δ 2 i +2 -homotopic to h i − 1 ; • g j ∈ H p ( σ j , σ j ). Next, deﬁne a map q i : p − 1 ( L ( i ) ) → M by the form ula q i ( x ) = ( h i − 1 ( x ) if x ∈ p − 1 ( L ( i − 1) ); g j ( x ) if x ∈ p − 1 ( σ j ). It can b e sho wn that q i is δ 2 i +2 -homotopic to h i − 1 | p − 1 ( L ( i ) ). Since p − 1 ( L ( i ) ) is a subpo lyhedron o f N , w e can apply the Homotop y Exten- sion Theorem to ﬁnd a con tinuous extension h i : N → M of t he map q i whic h is δ 2 i +2 -homotopic to h i − 1 . Moreov er, h i ∈ H p ( L, L ( i ) ) b ecause h i − 1 ∈ H p ( L, L ( i − 1) ) and g j ∈ H p ( σ j , σ j ) for an y j . This completes the inductiv e step. Then the limit map h = lim i →∞ h i : N → M is well-deﬁne d, contin- uous and δ -homotopic to g (the last t w o prop erties of h hold b ecause h has this prop erties for any simplex from N and b ecause of the deﬁnition 18 of the C W -to p ology on N ). Finally , since h | p − 1 ( L ( i ) ) = h i | p − 1 ( L ( i ) ) and h i ∈ H p ( L, L ( i ) ) for ev ery i , h ∈ H p ( L ).  4. Almost AE ( n, 0) -sp a ces W e already observ ed that if M is an LC n -space, then M ∈ AP ( n, 0) if and only if M has the follow ing prop erty : • for ev ery map g ∈ C ( I n , M ) and ev ery ε > 0 there exists a 0-dimensional map g ′ ∈ C ( I n , M ) whic h is ε -close to g . An y space ha ving the ab ov e pro p ert y will b e referred as almost AP ( n, 0). Ob viously , ev ery LC n − 1 almost AP ( n, 0) - space has the AP ( n − 1 , 0)- prop erty . W e a re going to establish an ana logue of Theorem 3.1 for almost AP ( n, 0)-spaces. Lemma 4.1. Every c omplete LC n − 1 -sp ac e M admits a c omplete metric  gener ating its top olo gy and satisfying the fol lowi n g c ondition: If Z is an n -dimen sional sp ac e, A ⊂ Z its c l o se d set and h : Z → M , then for every function α : Z → (0 , 1] and every m ap g : A → M with  ( g ( z ) , h ( z )) < α ( z ) / 8 for al l z ∈ A ther e exists a map ¯ g : Z → M extending g such that  ( ¯ g ( z ) , h ( z )) < α ( z ) for al l z ∈ Z . Pr o of. W e em b ed M in a Banach space E as a closed subset. Since the Hilb ert cub e is the image of the n -dimensional Menger compactum under an n -in v ertible map [7], w e can ﬁnd a metric space E ( n ) with dim E ( n ) ≤ n and a p erfect n -in ve rtible surjection p : E ( n ) → E . Here, p is n -inv ertible means that ev ery map fro m at most n - dimensional space in to E can b e lifted to a map in to E ( n ). Since M ∈ LC n − 1 , there exist a neigh b orho o d W of M in E and a map q : p − 1 ( W ) → M extending the restriction p | p − 1 ( M ). F or ev ery op en U ⊂ M let T ( U ) = W \ p ( q − 1 ( M \ U )) . Obv io usly , T ( U ) ⊂ W is op en, T ( U ) ∩ M = U and q ( p − 1 ( T ( U ))) = U . Let T be the collection of all pairs ( U, V ) of o p en sets in M suc h that conv ( V ) ⊂ T ( U ), where conv ( V ) is the closed conv ex h ull of V in E . Now , consider the family T = { ( U, V ) : U, V ar e op en in M a nd conv ( V ) ⊂ T ( U ) } . The family T has the follo wing prop erties: (i) for any z ∈ M a nd its neigh b orho o d U in M there is a neigh b orho o d V ⊂ U of z with ( U, V ) ∈ T ; (ii) for any ( U, V ) ∈ T and op en sets U ′ , V ′ ⊂ M w e ha v e ( U ′ , V ′ ) ∈ T pro vided U ⊂ U ′ and V ′ ⊂ V . By [6, Prop osition 2.3 ], there exists a complete metric  on M suc h that for ev ery z ∈ M and r ∈ (0 , 1) the pair of op en balls ( B  ( z , r ) , ( B  ( z , r / 8)) b elongs to T . Approximation by light maps 19 Supp ose we are giv en an n -dimensional s pa ce Z , its closed sub- set A ⊂ Z and tw o maps h : Z → M and g : A → M suc h t ha t  ( g ( z ) , h ( z )) < α ( z ) / 8 for all z ∈ A , where α ∈ C ( Z , (0 , 1]). Con- sider the set-v alued map φ : Z → E , φ ( z ) = g ( z ) if z ∈ A and φ ( z ) = conv ( B  ( h ( z ) , α ( z ) / 8)) if z 6∈ A . Then φ is low er semi-con tinuous and has closed and con ve x v alues in E . So, b y the Mic hael conv ex-v alued selection theorem [19], φ has a con tin uous selection g 1 . Next, w e lift g 1 to a map g 2 : Z → E ( n ). According to the deﬁnition of T , ev- ery p − 1  conv ( B  ( h ( z ) , α ( z ) / 8))  is contained in q − 1  B  ( h ( z ) , α ( z ))  . Hence, g = q ◦ g 2 is the required extension of g .  W e also need the following lemma whose pro o f is similar to that one of Lemma 2.2. Lemma 4.2. L et G ⊂ C ( X, M ) , wher e ( M ,  ) is a c omplete metric sp ac e. Supp ose ( U ( i ) i ≥ 1 is a se quenc e of op en subsets of C ( X , M ) such that • for any g ∈ G , i ≥ 1 and any function η ∈ C ( X, (0 , 1]) ther e exists g i ∈ B  ( g , η ) ∩ U ( i ) ∩ G which is η -close to g . Then, for any g ∈ G an d ε : X → (0 , 1] ther e exists g ′ ∈ T ∞ i =1 U ( i ) which is and ε -close to g . If, in addition, al l g i c an b e cho s e n such that g i | A = g 0 | A fo r so me g 0 ∈ C ( X, M ) and A ⊂ X , then g ′ | A = g 0 | A . Next tw o prop ositions are analogues o f Prop o sition 2 .3 and Prop o- sition 2.4. Prop osition 4.3. L et M b e a c omplete LC n − 1 almost AP ( n, 0) -sp ac e. Then for every n -dimensio n al c omp actum X ther e exists a dense G δ - subset H ⊂ C ( X , M ) of 0 -dim ensional maps. Pr o of. First o f a ll, let us note that since dim X ≤ n and M is LC n − 1 , the set o f all simplicially factorizable maps f rom C ( X, M ) is dense in C ( X, M ). Analyzing the pro of of Prop osition 2.3 and using Lemma 4.2 instead of Lemma 2.2 , one can see that it suﬃces to establish the follo wing claim: Claim. If K is a ﬁnite n -dimen sional p olyhe dr on, then every map g : K → M c an b e appr oximate d by a 0 -dimensi o nal map g ′ : K → M . Since the sets C ( K, M ; η ) consisting of maps g ∈ C ( K , M ) with d 0 ( g − 1 ( g ( x )) < η , η > 0, are op en in C ( K, M ), according to Lemma 4.2, it is enough to sho w that for eve r y g ∈ C ( K, M ) and η > 0 there exists g ′ ∈ C ( K , M ; η ) whic h is η - close to g . T o this end, w e equipp ed M with a complete metric satisfying the h yp o t heses o f Lemma 4.1 and ﬁx 0 < η ≤ 1 and g ∈ C ( K, M ). Since M ∈ LC n − 1 and M is almost AP ( n, 0), M ∈ AP ( n − 1 , 0). So, there exists a 0-dimensional map 20 h : K ( n − 1) → M which is η / 16- close to g | K ( n − 1) . Here, K ( n − 1) is the ( n − 1)-dimensional sk eleton of K . By Lemma 4.1, h can b e extended to a map ¯ h ∈ C ( K , M ) with  ( ¯ h ( x ) , g ( x )) < η / 2 for all x ∈ K . Let σ j , j = 1 , .., k , b e all n -dimensional simple xes of K . Then K \ K ( n − 1) is a disjoin t union of the op en simplexes ◦ σ j . Let K i = K ( n − 1) S j = i j =1 σ j , i = 1 , .., k W e are going to construct by induction maps h i : K → M , i = 0 , .., k , satisfying the following conditions: • h 0 = ¯ h ; • h i | K i b elongs to C ( K i , M ; η ), 1 ≤ i ≤ k ; • h i and h i +1 are ( η / 2 k )-close, i = 0 , .., k − 1. Assume that h i has already b een constructed. Since h i | K i b elongs to C ( K i , M ; η ), ev ery ﬁb er h − 1 i ( y ) ∩ K i of h i | K i , y ∈ h i ( K 1 ), is co vered b y a ﬁnite op en and disjoin t family Ω( y ) in K i with mesh(Ω( y )) < η . Using that h i ( K i ) is compact, we ﬁnd 0 < δ i < η / 2 k such that if  ( y , h i ( K i )) < δ i for some y ∈ M , then there is z ∈ h i ( K i ) with h − 1 i ( B  ( y , δ i )) ∩ K i ⊂ Ω( z ). Since M is almost AP ( n, 0), there exists a 0-dimensional map p i : σ i +1 → M which is δ i / 8-close to h i | σ i +1 . By Lemma 4.1, p i can b e extended to a map h i +1 : K → M b eing δ i -close to h i . T o sho w that h i +1 | K i +1 b elongs to C ( K i +1 , M ; η ), w e observ e that h − 1 i +1 ( y ) ∩ K i +1 =  h − 1 i +1 ( y ) ∩ K i  ∪  h − 1 i +1 ( y ) ∩ σ i +1  , y ∈ M . According to our construction, w e ha ve h i  h − 1 i +1 ( y ) ∩ K i  ⊂ B  ( y , δ i ) ∩ h i ( K i ). Hence, h − 1 i +1 ( y ) ∩ K i ⊂ h − 1 i ( B  ( y , δ i )) ∩ K i ⊂ Ω( z ) for some z ∈ h i ( K i ). There- fore, d 0  h − 1 i +1 ( y ) ∩ K i  < η . Since h − 1 i +1 ( y ) ∩ σ i +1 is 0- dimensional, Lemma 2.1 implies that d 0 ( h − 1 i +1 ( y ) ∩ K i +1 ) < η . Ob viously h k ∈ C ( K, M ; η ) and h k is η / 2 -close to ¯ h . Hence, g ′ = h k is η - close t o g . This completes the pro of of the claim.  Prop osition 4.4. L et M b e a c omplete LC n − 1 almost AP ( n, 0) -sp ac e. Then for every n -dimensio nal c omp actum X an d its F σ -subset Z with dim Z ≤ n − 1 ther e exists a dense G δ -subset H ⊂ C ( X, M ) of maps g such that Z is c ontaine d in the union of trivia l c omp onents of the ﬁb ers of g . Pr o of. F ollowing the pro of of Prop osition 2.4 and using Lemma 4.2 and Prop osition 4 .3 instead of Lemma 2 .2 and Prop osition 2 .3 , r esp ective ly , it suﬃces to pro ve the following analog ue of Claim 1 from Prop osition 2.4. Claim. S upp ose X is a ﬁnite n -dimen sional p olyhe dr on and Z a subp olyhe dr on of X with dim Z ≤ n − 1 . Then for every g 0 ∈ C ( X, M ) and δ > 0 ther e ex i s ts g ∈ H = T ∞ i =1 H ( Z , 1 /i ) which is δ - c lose to g 0 . Here, H ( Z , 1 /i ) is the set of all g ∈ C ( X , M ) with F ( g , 1 / i ) ∩ Z = ∅ . W e fo llo w the pro of of Claim 1, Prop o sition 2.4 and use the same Approximation by light maps 21 notations. The ﬁrst diﬀerence is that we tak e W to b e a neigh b orho o d of Z in X suc h that g 0 | W and ( g 0 ◦ r ) | W are δ / 16- close. Then, ( g 0 ◦ π ◦ ϕ ) | W is δ / 16- close to g 0 | W . Next, w e use b y Prop osition 4.3 to c ho ose a 0-dimensional map h : Z × I → M whic h is δ / 16-close to g 0 ◦ π . So, ( h ◦ ϕ ) | W is δ / 8-close to g 0 | W . Finally , tak e a neigh b orho o d U of Z in X with U ⊂ W , and use Lemma 4.1 to ﬁnd an extension g ∈ C ( X, M ) of ( h ◦ ϕ ) | U with g b eing δ -close to g 0 . Then g ∈ H .  W e establish no w an analog ue of Theorem 3.1 for almost AP ( n, 0)- spaces. Theorem 4.5. L et f : X → Y b e a p erfe ct map b etwe en metrizable sp ac es with dim X ≤ n and M is a c ompl e te LC n − 1 -sp ac e. Supp ose µ = { W ν : ν ∈ Λ } and µ 1 = { G ν : ν ∈ Λ } ar e lo c al ly ﬁnite op en c overs of M such that G ν ⊂ W ν and e ach W ν is almost AP ( m ν , 0) for every ν ∈ Λ . T hen ther e is a dense G δ -set H ⊂ C ( X, M ) of maps g such that for any y ∈ Y and ν ∈ Λ the r estriction g |  f − 1 ( y ) ∩ g − 1 ( G ν )  is an ( n − m ν ) -dimensiona l L elek map. Pr o of. F ollowing the notations and the pro of of Theorem 3.1 , w e can assume t ha t m ν ≤ n fo r all ν . Let H ( y , ε, η ) denote the set o f all g ∈ C ( X, M ) suc h that d n − m ν  F ν ( g , ε, y )  < η for ev ery ν ∈ Λ, where y ∈ Y and ε, η > 0 a r e ﬁxed. Moreo ver, for any F ⊂ Y , let H ( F , ε, η ) b e the in tersection of all H ( y , ε, η ), y ∈ F . One can establish a lemma analogical to Lemma 3.2 . Then, as in Prop osition 3.3, w e can sho w that an y H ( F , ε, η ) is op en in C ( X, M ), where F ⊂ Y is closed. Observ e that dim △ ( f ) ≤ m b ecause X , as a space of dimension ≤ n , admits a uniformly 0-dimensional map in to I n , see [1 3 ]. More- o ve r, w e can assume that the simplicial complex K in Prop osition 3.4 is n -dimensional. So, we can apply Prop osition 3.4 in the presen t sit- uation to sho w that an y H ( Y , ε , η ) is dense in C ( X , M ) pro vided for an y p erfect P L map p : N → L with dim N ≤ n the set H p ( L ) is dense in C ( N , M ). Here , H p ( L ) is the set of all maps q : N → M suc h that q | q − 1 ( G ν ) ∩ p − 1 ( z ) is an ( n − m ν )-dimensional Lelek map for ev ery ν ∈ Λ and z ∈ L . Hence, it remains to show that H p ( L ) is dense in C ( N , M ) for an y P L -map p : N → L with dim N ≤ n . And this follo ws from the pro of of Lemma 3.5, Lemma 3.6 and Prop osition 3.7 with the only diﬀerence that no w w e replace the application of Lemma 2.2, Prop osition 2.3 a nd Prop osition 2.4 by Lemma 4.2 , Prop osition 4.3 and Prop osition 4.4, resp ective ly .  Corollary 4.6. L et M b e a c omplete LC n − 1 -sp ac e such that e ach p oint z ∈ M ha s a ne ighb orho o d w hich is alm ost AP ( n, 0) . Then, for every p erfe ct map f : X → Y b etwe en metric sp ac es with dim X ≤ n , ther e 22 is a dense G δ -set H ⊂ C ( X, M ) c onsisting of maps g such that eve ry r estriction g | f − 1 ( y ) , y ∈ Y , is a 0 -dimensi o nal map. An y manifold M mo deled o n the n -dimensional Menger cub e has the AP ( n, 0)-prop erty , see Corollary 6.5 b elow. So, Theorem 1.1 holds for suc h a space M . But Theorem 1.1 do es not pro vide any inf o rmation ab out the densit y of t he set H in C ( X , M ) except that every simplicially factorizable map in C ( X , M ) can be appro ximated b y maps fr o m H . Next prop o sition sho ws that, in this sp ecial case, the set H is dense in C ( X, M ) with r esp ect to the uniform conv ergence top o logy . Prop osition 4.7. L et f : X → Y b e a p erfe ct map b etwe en m e trizable sp ac es with dim △ ( f ) ≤ m and M b e a m a nifold mo dele d on the n - dimensional Menger cub e. Then ther e is a G δ -set H ⊂ C ( X, M ) dense in C ( X, M ) wi th r esp e ct to the uniform c onver genc e top olo gy such that for any g ∈ H and y ∈ Y the r estriction g | f − 1 ( y ) i s an ( m − n ) - dimensional L elek map. Pr o of. Let H b e t he set from the pro of of Theorem 3.1. T o sho w that H is dense in C ( X, M ) equipped with the unifo r m con v ergence top olo g y , w e used an idea from the pro o f of [12, Coro llary 2.8]. According to [4], fo r any ε > 0 there exists an n - dimensional p olyhedron P ⊂ M of piecewise embedding dimension n and maps u : M → P and v : P → M suc h that u is a retraction, v is 0-dimensional, v ◦ u is ε/ 2-close to the iden tity id M . Since ev ery AN R of piecewise em b edding dimension n has the AP ( n, 0 )-prop erty (see [1 2, Propsition 2.1]), according to Theorem 3.1, for eve ry g ∈ C ( X , M ) there is g ′ : X → P such that g ′ is δ -close to u ◦ g and g ′ | f − 1 ( y ) is an ( m − n )- dimensional Lelek map for all y ∈ Y . Here δ > 0 is c hosen suc h that d ist ( v ( x ) , v ( y )) < ε/ 2 f or an y x, y ∈ P whic h are δ - close. Then v ◦ g ′ is ε -close to g and since v is 0-dimensional, v ◦ g ′ ∈ H .  5. Proper ties of AP ( n, 0) - s p aces In this section w e inv estigate the class of AP ( n, 0)- spaces. Lemma 5.1. L et K b e a p olyhe dr on of dimension ≤ n and L ⊂ K a subp olyhe dr on. Supp ose ( X ,  ) is a c omplete metric sp ac e p ossessing the AP ( n, 0) - p r op erty and g 0 ∈ C ( K, X ) with dim g − 1 0 ( g 0 ( x )) ≤ 0 for al l x ∈ L . Then for eve ry δ > 0 ther e exists a 0 -dimensiona l map g : K → X which is δ -homotopic to g 0 and g | L = g 0 | L . Pr o of. W e already observ ed in Section 2 that all sets C ( K, X , ε ), ε > 0, consisting o f maps h ∈ C ( K, X ) with d 0 ( h − 1 ( h ( x ))) < ε for ev ery x ∈ K a re op en in C ( K , X ). Since ev ery map from C ( K , X ) is simplicially Approximation by light maps 23 factorizable, C ( K, X , ε ) are homotopically dense in C ( K, X ) according to Prop osition 2.3 . Claim 1 . F or every x ∈ L and j ≥ 1 ther e e xist ε x > 0 and a neighb orho o d U x of x in K sa tisfying the fol lowing c ondition: If h ∈ C ( K, X ) and Z ⊂ K with   g 0 ( y ) , h ( y )  < ε x for al l y ∈ Z ∪ U x , then d 0 ( h − 1 ( h ( y )) ∩ Z ) < 1 /j for any y ∈ U x . The pro o f of this claim is similar to the pro of of Lemma 3.2. Claim 2 . L e t h ∈ C ( K , X ) with h | L = g 0 | L . Then, for every η > 0 and j ≥ 1 ther e exists h j ∈ C ( K, X , 1 / j ) such that h j | L = g 0 | L and h j is η -homotopic to h . W e ﬁx j and η > 0. Cho ose ﬁnitely man y p oints x ( i ) ∈ L , i ≤ k , p o s- itiv e reals ε x ( i ) and neigh b orho o ds U ( x i ) in K satisfying the h yp otheses of Claim 1 suc h t hat L ⊂ U = S i ≤ k U ( x i ). T aking a smaller neigh b or- ho o d, if necessarily , w e can supp ose that   g 0 ( x ) , h ( x )  < η 1 / 2 for all x ∈ U , where η 1 = min { η , ε x ( i ) : i ≤ k } . Consider a tria ng ulation T of K suc h that σ ∈ T and σ ∩ K \ U 6 = ∅ imply σ ∩ L = ∅ . No w, let N b e the subp olyhedron of K give n by N = ∪{ σ ∈ T : σ ∩ K \ U 6 = ∅} . Ob- viously , N and L are disjoin t. Since X ∈ AP ( n, 0) and dim N ≤ n , b y Prop osition 2.3, there exists a 0-dimensional map g N ∈ C ( N , X ) whic h is η 1 / 2-homotopic to h | N ( we can apply Prop o sition 2.3 b ecause h | N is simplicially factorizable as a map with a p olyhedral domain). The map h ′ : N ∪ L → X , h ′ | N = g N and h ′ | L = g 0 | L , is η 1 / 2-homotopic to h | ( N ∪ L ). So, by the Homotop y Extension Theorem, h ′ admits an extension h j : K → X whic h is η 1 / 2-homotopic to h . It remains only to sho w that d 0 ( h − 1 j ( h j ( x ))) < 1 /j for a n y x ∈ K . T o this end, observ e that   g 0 ( x ) , h j ( x )  < η 1 for all x ∈ U . Because K = N ∪ U , for ev ery x ∈ K w e hav e h − 1 j ( h j ( x )) =  h − 1 j ( h j ( x )) ∩ N  ∪  h − 1 j ( h j ( x )) ∩ U  . Since h j | N = g N and g N is 0-dimensional, dim h − 1 j ( h j ( x )) ∩ N ≤ 0. On the other hand, h − 1 j ( h j ( x )) ∩ U = h − 1 j ( h j ( y )) ∩ U fo r some y ∈ U . So, there exists m ≤ k with y ∈ U x ( m ) . Since h j | U is η 1 -close to g 0 | U ,   g 0 ( z ) , h j ( z )  < ε x ( m ) for all z ∈ U . Hence, according t o the c hoice of U x ( m ) and ε x ( m ) , d 0  h − 1 j ( h j ( y )) ∩ U )  < 1 /j . Therefore, d 0  h − 1 j ( h j ( x )) ∩ U )  < 1 /j . Finally , by Lemma 2.1 , d 0  h − 1 j ( h j ( x ))  < 1 /j . This completes the pro of o f Claim 2. W e are in a p osition t o complete the pro of of Lemma 5.1. Because of Claim 2, w e can apply Lemma 2.2 (with G b eing in our case the set { h ∈ C ( K, X ) : h | L = g 0 | L } and U ( j ) = C ( K, X , 1 /j )) to obtain a map g ∈ T ∞ j =1 C ( K, X , 1 /j ) suc h t ha t g | L = g 0 | L and g is δ -homotopic to g 0 . Ob viously , g ∈ T ∞ j =1 C ( K, X , 1 /j ) yields dim g ≤ 0.  24 Prop osition 5.2. F or any sp ac e X we have: (1) If X has the AP ( n, 0) -pr op erty, then every op en subset of X also has this pr op erty. (2) If X is c omple tely metrizable , then X has the AP ( n, 0) -pr op erty if and only if it admits a c over by op en subsets w i th that pr op- erty. Pr o of. T o prov e the ﬁrst item, supp ose U is an op en subset of X ∈ AP ( n, 0), U ∈ cov ( U ) and g ∈ C ( I n , U ). Let U ′ = U ∪ { X \ g ( I n ) } . Since X ∈ AP ( n, 0), there is a 0- dimensional map g ′ ∈ C ( I n , X ) suc h that g ′ is U ′ -homotopic t o g . Then g ′ ( I n ) ⊂ U and there exists a U -homotopy h : I n → U joining g and g ′ . According to Mic hael’s theorem on lo cal prop erties [18], the second item will b e established if w e sho w that: ( i) A space has the AP ( n, 0)- prop erty provide d X is a discrete sum o f spaces with the same prop ert y; (ii) A completely metrizable space has the AP ( n, 0)- prop erty pro vided it is a union of tw o o p en subspaces with this prop erty . Condition (i) trivially follow s from the deﬁnition. Let us ch ec k condition (ii). Supp ose X is a completely metrizable space and X = X 0 ∪ X 1 is the union of tw o o p en subspaces X 1 , X 2 ⊂ X with the AP ( n, 0)-prop ert y . Fix an op en co ver U of X and a map g : I n → X and choose tw o op en sets W 1 , W 2 ⊂ X suc h that X = W 1 ∪ W 2 and W i ⊂ W i ⊂ X i for i ∈ { 1 , 2 } . Next, ﬁnd a complete metric  on X suc h that •  ( X \ W 1 , X \ W 2 ) ≥ 1; • B  ( W i , 1) ⊂ X i for i ∈ { 1 , 2 } and • eac h set of diameter < 1 in ( X,  ) lies in some U ∈ U . Let V = V 1 ∩ V 2 , where V i = B ρ ( W i , 1 / 2), i = 1 , 2, and c ho ose a triangulation T of I n suc h that for an y simplex σ ∈ T w e hav e diam g ( σ ) < 1 / 4. No w, consider t he p olyhedra K i = ∪{ σ ∈ T : g ( σ ) ∩ W i 6 = ∅} , i ∈ { 1 , 2 } and L 2 = ∪{ σ ∈ T : g ( σ ) ∩ V 2 6 = ∅} . Ob viously , g ( K i ) ⊂ V i ⊂ V i ⊂ X i , i = 1 , 2. So, g ( K 0 ) ⊂ V , where K 0 = K 1 ∩ K 2 . Moreo v er, w e ha v e K 2 ⊂ g − 1 ( V 2 ) ⊂ g − 1 ( V 2 ) ⊂ L 2 ⊂ g − 1 ( X 2 ) and K 0 ⊂ g − 1 ( V ) ⊂ L 2 . Approximation by light maps 25 Cho ose now p o sitiv e δ ≤ min {   g ( L 2 ) , X \ X 2  ,   g ( K 1 ) , X \ X 1  , 1 / 2 } suc h tha t h − 1 ( h ( K 0 )) ⊂ g − 1 ( V ) for an y h ∈ C ( I n , X ) whic h is δ - close to g . Since X 2 ∈ AP ( n, 0) and L 2 is a p olyhedron o f dimension ≤ n , b y Prop osition 2.3, there exists a 0-dimensional map g 2 : L 2 → X 2 whic h is δ -homotopic to g | L 2 . Next, by the Ho mo t op y Extension Theorem, g 2 can b e extended to a map g 2 ∈ C ( I n , X ) δ -homotopic t o g . According to the c ho ice of δ , g 2 ( L 2 ) ⊂ X 2 , g 2 ( K 1 ) ⊂ X 1 and ( g 2 ) − 1 ( g 2 ( K 0 )) ⊂ g − 1 ( V ) ⊂ L 2 . Hence, the restriction g 0 = g 2 | K 1 is a map from K 1 in to X 1 and, f or ev ery x ∈ K 0 , w e ha ve g − 1 0 ( g 0 ( x )) ⊂ g − 1 2 ( g 2 ( x )). The last inclusion implies that dim g − 1 0 ( g 0 ( x )) ≤ 0 for all x ∈ K 0 b ecause g 2 is 0-dimensional. Since X 1 ∈ AP ( n, 0), w e can apply Lemma 5.1 (fo r the p olyhedra K 0 ⊂ K 1 and the map g 0 ) to obtain a 0-dimensional map g 1 ∈ C ( K 1 , X 1 ) suc h that g 1 | K 0 = g 0 and g 1 is 1 / 2- homotopic to g 0 . F ina lly , consider the map g 12 ∈ C ( I n , X ) deﬁned by g 12 | K 1 = g 1 and g 12 | K 2 = g 2 | K 2 . Since b ot h g 1 and g 2 are 0-dimensional, so is g 12 . Moreo v er, g 12 is 1-homotopic to g b ecause δ ≤ 1 / 2. So, according to the c hoice of  , g 12 is U -homoto pic to g .  Prop osition 5.3. If X and Y ar e c omplete metric sp ac es such that X ∈ AP ( n, 0) and Y ∈ AP ( m, 0) , then X × Y ∈ AP ( n + m, 0) . Pr o of. Let  X and  Y b e complete metrics o n X and Y , resp ectiv ely . W e ﬁx g ∈ C ( I n + m , X × Y ) and consider the complete metric  = max {  X ,  Y } on X × Y . It suﬃces to show that for ev ery ε > 0 there exists a 0- dimensional map h ∈ C ( I n + m , X × Y ) whic h is ε -homotopic to g . Let g X = π X ◦ g and g Y = π Y ◦ g , where π X and π Y are the pro jec t io ns from X × Y on to X and Y , r esp ective ly . W e represen t I n + m as the union S n + m +1 i =1 A i with eac h A i b eing a 0-dimensional G δ -subset of I n + m . Then Z X = I n + m \ S n + m +1 i = n +1 A i is an F σ -subset of I n + m whic h is contained in S n i =1 A i . So, dim Z X ≤ n − 1. Since X ∈ AE ( n, 0) and ev ery map from C ( I n + m , X ) is simplicially factorizable, Prop o sition 2.4 yields the exis- tence o f a map h X ∈ C ( I n + m , X ) suc h that h X is ε -homotopic to g X and F ( h X ) = S ∞ j =1 F ( h X , 1 /j ) ⊂ I n + m \ Z X . Ob viously , F ( h X ) is an F σ -set in I n + m with F ( h X ) ⊂ S n + m +1 i = n +1 A i . Hence, Z Y = F ( h X ) \ A n + m +1 is also an F σ -set in I n + m with dim F Y ≤ m − 1. No w, since Y ∈ AE ( m, 0), w e ma y apply again Prop o sition 2.4 to obtain a map h Y ∈ C ( I n + m , Y ) whic h is ε -homotopic to g Y and F ( h Y ) = S ∞ j =1 F ( h Y , 1 /j ) ⊂ I n + m \ Z Y . Then the diago nal pro duct h = h X △ h Y : I n + m → X × Y is ε -homoto pic to g . It remains only to sho w that h is 0-dimensional. If C is a non-trivial comp onent of a ﬁb er of h , then C ⊂ F ( h X ) ∩ F ( h Y ) ⊂ 26 F ( h X ) \ Z Y ⊂ A m + n +1 . Since A m + n +1 is 0- dimensional, C s hould b e a p oint. Therefore, all comp onen ts of the ﬁb ers of h ar e trivial, i.e. dim h = 0 .  Finally , w e are going to sho w t ha t ev ery a rc-wise connected AP ( n, 0 )- compactum is a con tinuum ( V n ) in the sense of P . Alexandroﬀ. Recall that a compact metric space ( M ,  ) is a ( V n )-con tinuum [1] if f or an y pair of disjoin t closed subsets A and B of M b oth having non-empty in teriors there exists ε > 0 suc h that d n − 2 ( C ) > ε for ev ery partitio n C in M b et w een A and B . It is easily seen that this is a top olo gical prop erty , i.e., it do esn’t dep end o n the metric  . Ob viously , ev ery con tinuum ( V n ) is a Cantor n -manifold (a compactum whic h is not disconnected by an y ( n − 2)- dimensional closed subset). Moreov er, any ( V n ) contin uum has a stronger prop ert y [1 0 ]: it cannot b e decomp o sed in to a countable union of prop er closed subsets F i with dim F i ∩ F j ≤ n − 2. The compacta with the last pro p ert y are called strong Cantor n -manifolds [9]. Prop osition 5.4. Every p ath-c onne cte d c omp actum M ∈ AP ( n, 0 ) is a c ontinuum ( V n ) . Pr o of. Let A and B b e disjoin t closed subsets of M with non-empt y in terior and  b e a metric on M . Since M is path-connected, w e can c ho o se a map g : I n → M suc h that g ( I n ) ∩ I nt ( A ) 6 = ∅ 6 = g ( I n ) ∩ I nt ( B ). Then there ex ists a 0-dimensional map g 1 : I n → M whic h is so close to g that g 1 ( I n ) meets bot h I nt ( A ) a nd I nt ( B ). Th us, A 1 = g − 1 1 ( A ) and B 1 = g − 1 1 ( B ) are disjoin t closed subsets of I n with non- empt y in teriors. Since I n is a contin uum ( V n ) [1], t here exists ε > 0 suc h that d n − 2 ( C 1 ) > ε for ev ery partition C 1 ⊂ I n b et wee n A 1 and B 1 . Because g 1 is 0-dimensional, ev ery y ∈ M ha s a neighborho o d W y suc h that g − 1 1 ( W y ) splits in to a ﬁnite disjoin t family of op en subsets of I n eac h o f diameter < ε . Let δ b e the Leb esgue num ber of the co v er { W y : y ∈ M } . Then d n − 2 ( C ) ≥ δ for an y par t ition C ⊂ M b et wee n A and B . Indeed, otherwise there w ould b e a partition C and an op en family γ C in M of o rder ≤ n + 1 suc h that γ C co v ers C and diam( W ) < δ for eve r y W ∈ γ C . Hence, g − 1 1 ( γ C ) is an op en family in I n of order ≤ n + 1 and co ve r ing g − 1 1 ( C ). Moreo v er, eac h g − 1 1 ( W ), W ∈ γ C , splits into a ﬁnite disjoint op en family consisting of sets with diameter < ε . Therefore, d n − 2 ( g − 1 1 ( C )) < ε . This is a con tradiction b ecause g − 1 1 ( C ) is a partition in I n b et wee n A 1 and B 1 .  Approximation by light maps 27 6. AP ( n, 0) -sp a ce s and general pos ition proper ties The parametric general p osition prop erties w ere in tro duces in [2]. W e say that a space M has the m - D D { n,k } -prop erty , where m, n, k ≥ 0 are intege rs o r ∞ , if for an y op en co v er U of M and an y t wo maps f : I m × I n → M , g : I m × I k → M there exist maps f ′ : I m × I n → M , g ′ : I m × I k → M whic h are U -homo t o pic to f and g , resp ectiv ely , and f ′ ( { z } × I n ) × g ′ ( { z } × I k } = ∅ for all z ∈ I m . It is clear that this is exactly the we ll know n disjoin t n -disks prop ert y pro vided m = 0, n = k and M is LC n . When m = 0, w e simply write D D { n,k } instead of 0- D D { n,k } . Lemma 6.1. L et M b e c ompletely metrizable having t h e DD { n,k } - pr op erty. Supp ose X is a c omp actum and A, B ⊂ X close d disjoin t subsets wi th dim A ≤ n and dim B ≤ k . Then every simplicial ly fac- torizable map g : X → M c an b e homotopic al ly appr oxima te d by m aps g ′ ∈ C ( X, M ) such that g ′ ( A ) ∩ g ′ ( B ) = ∅ . Pr o of. Let g ∈ C ( X, M ) b e simplicially fa ctorizable and δ > 0. As in Claim 2 fr o m the pro of of Prop osition 2.4, w e can ﬁnd a ﬁnite op en co v er U of X suc h that : • at most n + 1 elemen ts of the family γ A = { U ∈ U : U ∩ A 6 = ∅} in tersect; • at most k + 1 elemen ts of the family γ B = { U ∈ U : U ∩ B 6 = ∅} in tersect; • ∪ γ A ∩ ∪ γ B = ∅ ; • there exists a map h : N ( U ) → M suc h tha t h ◦ f U is δ / 2- homotopic to g , where N ( U ) is the nerv e of U and f U : X → N ( U ) is the natural map. Let K A and K B b e the subp olyhedra of N ( U ) generated b y the families γ A and γ B , resp ectiv ely . Then dim K A ≤ n , dim K B ≤ k and K A ∩ K B = ∅ . F or any simplexes σ ∈ K A and τ ∈ K B let G ( σ , τ ) = { p ∈ C ( N ( U ) , M ) : p ( σ ) ∩ p ( τ ) = ∅} . Ob viously , eac h G ( σ, τ ) is op en in C ( N ( U ) , M ). Using M ∈ D D { n,k } and the Homotop y Extension Theorem, one can sho w that all G ( σ , τ ) are homotopically dense in C ( N ( U ) , M ) . So, by Lemma 2.2, there exists h ′ ∈ T { G ( σ, τ ) : σ ∈ K A , τ ∈ K B } whic h is δ / 2 -homotopic to h . Then g ′ = h ′ ◦ f U is δ - homotopic to g and h ′ ( A ) ∩ h ′ ( B ) = ∅ .  Prop osition 6.2. F or a sp ac e X we have : (1) If X is LC 0 , then X has the D D { 0 , 0 } -pr op erty if and on ly if X has no isolate d p oint; 28 (2) If X is c ompletely metrizable and X ∈ D D { 0 , 0 } , then X has the AP (1 , 0) -pr op erty. Pr o of. F or the ﬁrst item, see [2, Prop osiion 7 ]. T o pro of the second item, w e ﬁx coun t a bly man y closed 0 -dimensional subsets P i of I con- sisting of irrational n umbers suc h t ha t dim I \ S i ≥ 1 P i = 0. F or ev ery rational t ∈ Q and i ≥ 1 let U i ( t ) = { g ∈ C ( I , M ) : g ( t ) 6∈ g ( P i ) } . By Lemma 6.1 , each U i ( t ) is homotopically dense in C ( I , M ). On the other hand, ob viously all U i ( t ) are op en in C ( I , M ). Therefore, b y Lemma 2.2, the set U = T { U i ( t ) : t ∈ Q, i ≥ 1 } is homot o pically dense in C ( I , X ). Moreov er, for ev ery g ∈ U and x ∈ I w e ha v e the follo wing: if g − 1 ( g ( x )) ∩ Q 6 = ∅ , t hen g − 1 ( g ( x )) ⊂ I \ S i ≥ 1 P i ; if g − 1 ( g ( x )) ∩ Q = ∅ , then g − 1 ( g ( x )) ⊂ I \ Q . Hence, U consists of 0-dimensional maps.  Next corollary of Prop ositions 5.3 and 6.2 pro vides more examples of AP ( n, 0)-spaces. Corollary 6.3. If M i , i = 1 , ..n , ar e c ompletely metrizable LC 0 -sp ac es without isolate d p oints, then Q i = n i =1 M i has the AP ( n, 0) -pr op erty. Here is a c har acterization o f the D D { n,k } -prop erty . Prop osition 6.4. A c omp l e tely metrizable sp ac e M has the D D { n,k } - pr op erty, wher e n ≤ k ≤ ∞ , if and on ly if M satisﬁes the fol lowing c ondition ( n, k ) : • If X is a c omp actum and A ⊂ B ⊂ X its σ -c omp act subsets with dim A ≤ n and dim B ≤ k , then any si m plicial ly factorizable map g ∈ C ( X, M ) c an b e homotopic al ly appr oximate d by maps h ∈ C ( X , M ) such that h − 1 ( h ( x )) ∩ B = x for al l x ∈ A . Pr o of. Suppo se M ∈ D D { n,k } and let A ⊂ B ⊂ X b e t wo σ -compact subsets of a compactum X with dim A ≤ n and dim B ≤ k . Then A = S p ≥ 1 A p and B = S m ≥ 1 B m , where A p and B m are compact sets of dimension dim A p ≤ n and dim B m ≤ k . F or ev ery p, i ≥ 1 let ω i ( p ) = { A pj ( i ) : j = 1 , 2 , .., s ( p , i ) } be a family of closed subse t s of A p suc h that ω i ( p ) co v ers A p and mesh( ω i ( p )) < 1 /i . W e also c ho ose sequences { B mq ( p, i, j ) } ∞ q =1 of closed sets B mq ( p, i, j ) ⊂ B m with B m \ A pj ( i ) = S ∞ q =1 B mq ( p, i, j ), where p, i ≥ 1 and j = 1 , 2 , .., s ( p, i ). Then the sets G ( p, i, j, m, q ) = { g ∈ C ( X , M ) : g ( A pj ( i )) ∩ g ( B mq ( p, i, j )) = ∅} are op en in C ( X, M ) and the in t ersection G of all G ( p, i, j, m, q ) con- sists of maps g with g − 1 ( g ( x )) ∩ B = x for an y x ∈ A . Hence, according to Lemma 2.2, ev ery simplicially factorizable map is homot opically ap- pro ximated by maps f rom G pro vided the follow ing is true: F or any Approximation by light maps 29 simplicially factorizable map g ∈ C ( X, M ), ε > 0 and ( p, i, j, m, q ) with p, i, m, q ≥ 1 a nd 1 ≤ j ≤ s ( i, p ), there exists a simplicially fac- torizable map g ( p, i, j, m, q ) ∈ G ( p, i, j, m, q ) whic h is ε -homo t opic to g . Since M ∈ D D { n,k } and any couple A pj ( i ), B mq ( p, i, j ) consists of disjoin t closed sets in X of dimension ≤ n and ≤ k , resp ectiv ely , the last statemen t follow s from Lemma 6.1 . Hence M satisﬁes condition ( n, k ). Supp ose now that M satisﬁes condition ( n, k ). T o sho w that M ∈ D D { n,k } , let f : I n → M and g : I k → M b e tw o maps. If k < ∞ , we denote by X the disjoin t sum I n U I k and consider the map h : X → M , h | I n = f and h | I k = g . Since h is simplicially fa ctorizable (as a map with a p olyhedral do ma in) and M has the ( n, k )-prop erty , f o r ev ery ε > 0 there is a map h 1 ∈ C ( X, M ) suc h that h − 1 1 ( h 1 ( x )) = x for all x ∈ I n and h 1 is ε -ho motopic to h . Then the maps f 1 = h 1 | I n and g 1 = h 1 | I k are ε - homotopic to f and g , resp ectiv ely , and f 1 ( I n ) ∩ g 1 ( I k ) = ∅ . So, M ∈ D D { n,k } . If n < k = ∞ , w e c ho o se k (1) < ∞ suc h that n ≤ k (1) and the map g ′ = g ◦ r k (1) is ε/ 2-homo t o pic to g , where r k (1) : I ∞ → I k (1) is the retraction of I ∞ on to I k (1) ⊂ I ∞ deﬁned b y r k (1) (( x 1 , x 2 , .. )) = ( x 1 , x 2 , .., x k (1) , 0 , 0 , .. ). Then the map h : I n U I ∞ → M , h | I n = f and h | I ∞ = g ′ , is simplicially factorizable. Hence, as in the previous case, w e can use the ( n, ∞ )-prop ert y of M to show that M ∈ D D { n, ∞} . If n = k = ∞ , we homotopically approx imate b oth f and g b y maps f ′ = f ◦ r n (1) and g ′ = g ◦ r k (1) , resp ectiv ely , a nd pro ceed as in t he ﬁrst case.  Corollary 6.5. Every c omp l e tely metrizable sp ac e M ∈ D D { n − 1 ,n − 1 } has the AP ( n, 0) -pr op erty. In p articular, every manifold m o dele d on the n -dimen sional Menger c ub e or the n -dimen s i o nal N¨ ob eling sp ac e has the AP ( n, 0) -pr op erty. Pr o of. Let I n = A ∪ B , where B = I n \ A is 0 - dimensional a nd A = S i ≥ 1 A i is σ -compact with { A i } b eing a sequence of closed ( n − 1)- dimensional subsets of I n . Since C ( I n , M ) consists of simplicially fac- torizable maps, b y Prop osition 6.4, eve ry map in C ( I n , M ) is ho mo t opi- cally appro ximated b y maps g ∈ C ( I n , M ) suc h that g − 1 ( g ( x )) ∩ A = x for a ll x ∈ A . Let us show that any suc h g is 0- dimensional. In- deed, since for ev ery x ∈ I n the in tersection g − 1 ( g ( x )) ∩ A can hav e at most o ne p oin t, g − 1 ( g ( x )) = ( g − 1 ( g ( x )) ∩ A ) ∪ ( g − 1 ( g ( x )) ∩ B ) is 0-dimensional. The second part of the corollary follow s from the fact that an y n - dimensional Menger manifo ld, as w ell a s any manifold mo deled on the 30 n -dimensional N¨ ob eling space, is a complete LC n − 1 -space with the dis- join t n -disks prop ert y [4]. So, any suc h a space ha s the D D { n − 1 ,n − 1 } - prop erty .  The follo wing notio ns w ere in tro duced in [2] and [3]. Let A b e a closed subset of a space X . W e sa y that A is: • a (homotopical) Z n -set in X if f o r any an op en co v er U of X and a map f : I n → X there is a map g : I n → X suc h tha t g ( I n ) ∩ A = ∅ and g is U -near ( U -homotopic) to f ; • homological Z n -set in X if H k ( U, U \ A ) = 0 for all op en sets U ⊂ X and all k < n + 1. Here, H ∗ ( U, U \ A ) = 0 are the relativ e singular ho mology g r oups with in teger co eﬃcien ts. Eac h homotopical Z n -set in X is b ot h a Z n -set and a homological Z n -set in X . The con v erse is not alwa ys true. Theorem 6.6. L et X b e a lo c al ly p ath-c onne cte d AP ( n, 0) -sp ac e. Then every x ∈ X is a homolo gic al Z n − 1 -p oint in X . Pr o of. By [3, Corollary 8.4], it suﬃces to che ck that ( x, 0) is a ho- motopical Z n -p oint in X × [ − 1 , 1] fo r ev ery x ∈ X . Giv en a map f : I n → X × [ − 1 , 1] we are going to homotopically approx ima t e f b y a map g : I n → X × [ − 1 , 1] with ( x, 0) / ∈ g ( I n ). Let f = ( f 1 , f 2 ) where f 1 : I n → X a nd f 2 : I n → [ − 1 , 1] are the comp onen ts of f . Since X ∈ AP ( n, 0), there is a 0-dimensional map g 1 : I n → X that appro ximates f 1 . Because Z = g − 1 1 ( x ) is a 0- dimensional subset of I n and { 0 } is a Z 0 -set in [ − 1 , 1], the map f 2 can b e approximated b y a map g 2 : I n → [ − 1 , 1] suc h that 0 / ∈ f 2 ( Z ). Then for the map g = ( g 1 , g 2 ) : I n → X × [ − 1 , 1] w e hav e ( x, 0) / ∈ g ( I n ).  Prop osition 6.7. I f X ∈ AP ( n, 0) and Y ∈ ∞ - D D { 0 , 0 } , then X × Y ∈ D D { n,n } . Pr o of. Let f = ( f X , f Y ) : I n ⊕ I n → X × Y b e a giv en map. Since X ∈ AP ( n, 0), we homot opically appro ximate the ﬁrst comp onen t f X b y a 0-dimensional map g X : I n ⊕ I n → X . Next, use ∞ - D D { 0 , 0 } - prop erty of Y to homotopically approx imat e the second comp onent f Y b y a map g Y : I n ⊕ I n → Y that is injectiv e on the ﬁb ers of g X . Then the map g = ( g X , g Y ) : I n ⊕ I n → X × Y is injectiv e, witnes sing that X × Y has the D D { n,n } -prop erty .  W e sa y that a space X has the AP ( ∞ , 0)- prop erty if X ∈ AP ( n, 0) for ev ery n ≥ 1. Approximation by light maps 31 Corollary 6.8. If X, Y ar e two lo c al ly p ath-c onne cte d sp ac es b oth p os- sessing the AP ( ∞ , 0) -pr op erty and Y c ontains a dense set of hom o - topic al Z 2 -p oints, then X × Y has the D D {∞ , ∞} -pr op erty. Pr o of. By Theorem 6.6, eac h p o int of Y is a homolo gical Z ∞ -p oint. T aking into accoun t that Y has a dense set of homotopical Z 2 -p oints and applying Theorem 26(4,5) of [2], w e conclude that the space Y ∈ ∞ - D D { 0 , 0 } . Finally , b y Prop osition 6.7, X × Y ∈ D D {∞ , ∞} .  Corollary 6.9. If X , Y ar e two c omp act AR ’s b o th p o s s essing the AP ( ∞ , 0) -pr op erty, then X × Y × I is home omorp h ic to the Hilb ert cub e. Pr o of. Theorem 6.6 implies that ev ery y ∈ Y is a homolo g ical Z ∞ - p oint in Y . Then, according to [2 , Theorem 26(5)], Y ∈ DD { 1 , 1 } . Consequen tly , b y [2, Theorem 1 0(5)], eac h p oin t of Y is a homotopical Z 1 -p oint in Y . No w, a pplying the Multiplication F ormula f o r homolog- ical Z -se ts [2, Theorem 17 ( 1 )]), w e conclude that each p oint of Y × I is a homotopical Z 2 -p oint. Therefore, Corollary 6.8 yields that the pro duct X × ( Y × I ) has t he D D {∞ , ∞} -prop erty . Being a compact AR , this pro duct is homeomorphic to the Hilb ert cub e according to the T oru ´ nczy k c haracterization theorem of Q -manifolds.  In light o f the preceding corollary , it is intere sting to remark that there exists a compact space X ∈ AR ∩ AP ( ∞ , 0) whic h is not homeo- morphic to the Hilb ert cub e. T o presen t suc h an example, w e ﬁrst pro ve that the AP ( n, 0)-prop ert y is preserv ed by a sp ecial t yp e of maps. A map π : X → Y is called an elemen tary cell-lik e map if: (1) π is a ﬁne homotopy equiv alence, i.e., fo r ev ery op en cov er U of Y there exists a map s : Y → X with π ◦ s b eing U -homotopic to the iden tity of Y ; (2) the non-degeneracy set N π = { y ∈ Y : | π − 1 ( y ) | 6 = 1 } is at most coun table; (3) eac h ﬁb er π − 1 ( y ), y ∈ N π , is an arc. Prop osition 6.10. L et π : X → Y b e an elementary c el l-lik e map b etwe en metric sp ac es such that X is c omplete and every x ∈ π − 1 ( N π ) is a homotopic al Z n -p oint in X . Then Y ∈ AP ( n, 0 ) pr ovide d X ∈ AP ( n, 0) . Pr o of. Suppo se X ∈ AP ( n, 0) . T o prov e that Y ∈ AP ( n, 0), ﬁx an op en co v er γ of Y and a map f : I n → Y . Let γ 1 b e an op en co v er of Y whic h is star-reﬁnemen t of γ . Since π is a ﬁne homotop y equiv alence, there is a map s : Y → X suc h that π ◦ s is γ 1 -homotopic to the iden tity 32 map o f Y . Let D = { x i : i ≥ 1 } ⊂ f − 1 ( N π ) b e a sequence suc h that f − 1 ( y ) \ D is 0-dimensional for ev ery y ∈ N f . Then the sets W i = { g ∈ C ( I n , X ) : x i / ∈ g ( I n ) } are op en in C ( I n , X ) equipp ed with t he uniform con v erg ence top ology . Moreo v er, eac h W i is homotopically dense in C ( I n , X ) b ecause x i is a homotopical Z n -p oint in X . W e also consider the sets U i = { g ∈ C ( I n , X ) : d 0 ( g − 1 ( x )) < 1 /i for all x ∈ X } . Since X ∈ AP ( n, 0) and all maps from C ( I n , X ) a re simplicially factorizable, it follows from Prop o sition 2.3 that an y U i is op en and homotopically dense in C ( I n , X ). This easily implies that the inters ections V i = W i ∩ U i are op en and homotopically dense in C ( I n , X ). Consequ en tly , b y Lemma 2.2 (with G b eing C ( I n , X )), t here ex ists a map g ∈ T ∞ i =1 V i whic h is π − 1 ( γ 1 )-homotopic to s ◦ f . Obvious ly , g is 0-dimensional, g ( I n ) ⊂ X \ D and f 1 = π ◦ g : I n → Y is γ 1 -homotopic to π ◦ s ◦ f . Then f 1 is γ - homotopic to f b ecause π ◦ s ◦ f is γ 1 -homotopic to f . Hence, Y ∈ AP ( n, 0).  Singh [23 ] constructed an elemen tary cell-lik e map f : Q → X fro m the Hilbert cub e Q o nto a compact X ∈ AR suc h that X × I is home- omorphic to Q but X con tains no pro p er AN R -subspace of dimension ≥ 2. By the preceding prop osition, Singh’s space ha s the AP ( ∞ , 0)- prop erty . Th us we hav e: Corollary 6.11. Ther e is a c omp act X ∈ AR with X ∈ AP ( ∞ , 0) such that X × I is home omorphi c to the Hilb ert cub e but X c ontains n o pr op er AN R -subsp ac e of dim e nsion ≥ 2 . No w, we consider the spaces with piecewise em b edding dimension n . According to [12], a map h : P → M from a ﬁnite p olyhedron P is said to b e a piecewise em b edding if there is a triang ulation T of P suc h that h em b eds eac h simplex σ ∈ T and h ( P ) is an AN R . F o r a space M , the piecewise em b edding dimension ped ( M ) is the maxim um k suc h that for a ny ε > 0 and any map g : P → M from a ﬁnite p olyhedron P with dim P ≤ k there exists a piecewise em b edding g ′ : P → M whic h is ε - close to g . If y ∈ M , then ped y ( M ) is the maxim um of all ped ( U ), where U is a neigh b o r ho o d of y in M . Ob viously , ped ( M ) ≤ min { ped y ( M ) : y ∈ m } . As w e noted, b y [1 2, Propsition 2.1], ev ery complete AN R -space M with ped ( M ) ≥ n has the AP ( n, 0)-prop ert y . But there are ev en compact AN R ’s having the AP ( n, 0)-prop erty with ped ≤ n − 1 . F or example, according to Coro llary 6.3, an y pro duct M = Q i = n i =1 M i of dendrites with dense set of endp oin t s is an AP ( n, 0). On the other hand, b y [25, Theorem 3.4], ped ( M ) ≤ n − 1. Next pro p osition also sho ws that the pr o p ert y ped = n is quite restrictiv e. Approximation by light maps 33 Prop osition 6.12. If ( M ,  ) is a c omplete sp ac e and ped y ( M ) = n + 1 , then y is a Z n -p oint in M . Pr o of. Since ped y ( M ) = n + 1, there exists a neigh b orho o d U y in M with ped ( U y ) = n + 1. It is easily seen that if y is a Z n -p oint in U y , it is also a Z n -p oint in M . So, w e can supp o se that U y = M , and let ε > 0 and g ∈ C ( I n , M ). W e iden tif y I n with the set { ( x 1 , .., x n +1 ) ∈ I n +1 : x n +1 = 0 } and let π : I n +1 → I n b e the pro jection. Since ped ( M ) = n + 1, there exists a triangulation T of I n +1 and a map g 1 ∈ C ( I n +1 , M ) whic h is ε/ 2- close to g ◦ π and g 1 | σ is an em b edding for all σ ∈ T . Hence, g − 1 1 ( y ) consists of ﬁnitely man y p oin ts. Moreov er, there is δ > 0 such that for an y δ -close p oints x ′ , x ′′ in I n +1 w e ha v e  ( g 1 ( x ′ ) , g 1 ( x ′′ )) < ε/ 2. Since each x ∈ I n +1 is a Z n -p oint in I n +1 , g − 1 1 ( y ) is a Z n -set in I n +1 . Hence, there exists a map h : I n → I n +1 suc h that h is δ -close to id I n and h ( I n ) ⊂ I n +1 \ g − 1 1 ( y ). Then g 1 ◦ h ∈ C ( I n , M ) is ε -close to g and y 6∈ g 1 ◦ h ( I n ).  Finally , let us complete t he pap er with the following question: Question 1 . L et X ∈ AR b e a c omp act sp ac e such that X ∈ D D { 2 , 2 } ∩ AP ( ∞ , 0) . Is X home omorphic to the Hilb ert cub e? Reference s [1] P . Alexandro ﬀ, Die Kontinua ( V p ) - eine V erscharfung der Cantorschen Mannigfaltigke iten , Monasth. Math. 61 (1 957), 67– 76. [2] T. Banakh and V. V alo v, Par ametric gener al p osition pr op erties and emb e d- ding of n -dimensional maps into t r ivial bu nd les , preprint. [3] T. Banakh, R. Cauty and A. 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Approximation by light maps and parametric Lelek maps

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