Krasinkiewicz spaces and parametric Krasinkiewicz maps

KRASINKI EWICZ SP A CES AN D P A R AMETRIC KRASINKI EWICZ MAPS EI ICHI MA TSUHASHI AND VESKO V A LOV Abstract. W e sa y that a metrizable spac e M is a K rasinkiewicz space if any map from a metrizable co mpactum X into M can be approximated by Krasinkiewicz maps (a map g : X → M is Krasinkie w ic z provided every co n tinuu m in X is either con tained in a fiber of g or co ntains a component of a fibe r of g ). In this pap er we establish the following prop erty of Kra s inkiewicz spaces: Let f : X → Y b e a per fect map b etw een metrizable spaces and M a Kras ink iewicz c omplete AN R -spac e. If Y is a countable union o f closed finite-dimensional subsets, then the function space C ( X, M ) with the so urce limitation top olog y con tains a dense G δ -subset o f maps g such that all re s trictions g | f − 1 ( y ), y ∈ Y , ar e Krasinkiewicz maps. The same conclusio n remains tr ue if M is homeomorphic to a closed conv ex subset of a B a nach spa ce and X is a C -space. 1. Intr oduction All spaces in the pap er are assumed to be metrizable and all maps con tin uous. Unless stated otherwise, any function space C ( X , M ) is endo w ed w ith the sour c e limitation top o l o gy . This topolo g y , known also as the fine top olo gy , w as in tro duced by Whitney [14] and has a base at a g iven f ∈ C ( X , M ) consisting of 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 tin uous functions into (0 , 1]. The sym b ol  ( f , g ) < ε means that   f ( x ) , g ( x )  < ε ( x ) for all x ∈ X . The source limitation t o p ology do esn’t dep end on the metric  [5] and has the Baire pro p ert y prov ided M is completely metrizable [9]. Ob viously , this top olo gy coincides with the unifor m con v ergence top o lo gy when X is compact. 1991 Mathematics Su bje ct Classific ation. Prima ry 54F1 5; Secondary 54F45, 54E40 . Key wor ds and phr ases. Krasinkie w ic z spaces, Kr asinkiewicz maps, contin ua, selections for set-v a lued maps, C -space s . The second author was par tially suppo rted by NSERC Gr ant 2 61914 -03. 1 2 W e say that a space M is a Kr asinkiewicz sp ac e if for any compactum X the function space C ( X , M ) con tains a dense subset o f Krasinkiewicz maps. Recall that a map g : X → M , where X is compact, is said to b e Krasinkiewicz [6] if ev ery contin uum in X is either con tained in a fib er of g or con tains a comp onen t of a fib er of g . Krasinkiewicz [4] pro v ed that ev ery 1- manifold is a Krasinkiewicz space (for the in terv al I this was established b y Levin-Lewis [6]). The first a utho r, g eneralizing the Krasinkiewicz result, pro v ed in [7] tha t all compact polyhedra, as w ell as all 1-dimensional P eano con tin ua and manifolds mo deled on a Menger cub e are Krasinkiewicz spaces. The main results in this pap er is the following theorem: Theorem 1.1. L et M b e a Kr asinkiew i c z c om plete AN R -sp ac e an d f : X → Y a p erfe ct map with Y b eing a str ongl y c o untable-dimensional sp ac e. Then the function sp ac e C ( X , M ) c ontains a dense G δ -set of maps g such that al l r estrictions g | f − 1 ( y ) , y ∈ Y , ar e Kr asinkiew icz maps. Mor e over, if in addition M i s a close d c onv e x subset of a Ban a ch sp ac e, then the sam e c onclusion r e mains true pr ovide d Y is a C -sp ac e. Recall that X is a C -space if for an y sequence { ν n } ∞ n =1 of op en cov ers of X there exists a sequence { γ n } ∞ n =1 of disjoin t op en fa milies in X suc h that each γ n refines ν n and ∪ ∞ n =1 γ n is a cov er of X . Ev ery strongly coun table-dimensional space (i.e. a space whic h is a union of countably man y close d finite-dimensional subsets), as well as eve ry countable- dimensional space (a countable union of 0- dimensional subsets) is a C -space [2] and there exists a compact C -space whic h is not countable- dimensional. Ev eryw here b elo w by a p o lyhe dr on w e mean the underlying space of a simplicial complex equipp ed with the metric top ology . A compactum is called a Bing sp ac e if eac h of its subcontin ua is hereditarily indecomp os- able. According to Corollary 3.2, each p o lyhedron is a Krasinkiewicz space. Moreo v er, it follo ws from [11] t ha t f o r any p olyhedron P with- out isolated p oints and a compactum X the space C ( X , P ) contains a dense set of Bing maps (maps g suc h that a ll fib ers g − 1 ( y ), y ∈ P , are Bing spaces). Therefore, Theorem 1.1 and [1 3, Theorem 1.1] imply the follo wing corollary: Corollary 1.2. L et P b e a c omple te p olyhe dr on without isolate d p oints and f : X → Y a p erfe ct map. The n the function sp ac e C ( X , P ) c on- tains a dense G δ -set of maps g such that al l r estrictions g | f − 1 ( y ) , y ∈ Y , ar e b oth Bing and Kr asin k iewicz map s in e ach o f the fol lo w ing c ases: ( i ) Y is str ongly c ountable-dim ensional; ( ii ) Y is a C -sp ac e an d P is a close d c onvex subset of a Banach sp a c e. Krasinkiewicz spaces and maps 3 Most part of the pa p er is dev oted to the pro of of Theorem 1.1 , giv en in Section 2. In Section 3 w e pro vide some prop erties of Krasinkiewicz spaces. F or example, w e show that a complete AN R is a Krasinkiewicz space if and only if it has an op en cov er of Krasinkiewicz subspace s. In part icular, all n -manifolds, n ≥ 1, are Krasinkiewicz spaces. 2. Proo f of Theorem 1. 1 W e fixed a metric d on X and for eve ry A ⊂ X and δ > 0 let B ( A, δ ) = { x ∈ X : d ( x, A ) < δ } . If y ∈ Y and m, n ≥ 1, then K ( m, n, y ) denotes the set of all maps g ∈ C ( X, M ) satisfying the follo wing condition: • F or eac h sub contin uum L ⊂ f − 1 ( y ) with diam g ( L ) ≥ 1 /n there exists x ∈ L suc h t hat C ( x, g | f − 1 ( y )) ⊂ B ( L, 1 /m ). Here, g | f − 1 ( y ) is the restriction of g o v er f − 1 ( y ) and C ( x, g | f − 1 ( y )) denotes the comp onent of the fiber g − 1 ( g ( x )) ∩ f − 1 ( y ) of g | f − 1 ( y ) con taining x . F or H ⊂ Y let K ( m, n, H ) b e the in tersection of all K ( m, n, y ), y ∈ H . W e also denote b y K ( H ) the set of all maps g ∈ C ( X, M ) suc h that g | f − 1 ( y ) : f − 1 ( y ) → M is a Krasinkiewicz map for eac h y ∈ H . Prop osition 2.1. K ( H ) = T m,n ∈ N K ( m, n, H ) . Pr o of. Ob viously K ( H ) ⊂ T m,n ∈ N K ( m, n, H ). So, w e need to prov e the inclusion T m,n ∈ N K ( m, n, H ) ⊂ K ( H ). Let g ∈ T m,n ∈ N K ( m, n, H ), y ∈ H and L ⊂ f − 1 ( y ) b e a sub con tin uum such that diam g ( L ) > 0. W e are g oing to prov e that there exists a sub contin uum L 2 ⊂ L 1 = L suc h that diam g ( L 2 ) > 0 and C ( x, g | f − 1 ( y )) ⊂ B ( L 1 , 1 / 2) for eac h x ∈ L 2 . Since diam g ( L 1 ) > 0, there exists n 1 ∈ N suc h that diam g ( L 1 ) ≥ 1 /n 1 . Since g ∈ K (2 , n 1 , y ), there exists a p oint x ∈ L 1 suc h tha t C ( x, g | f − 1 ( y )) ⊂ B ( L 1 , 1 / 2). Let E ⊂ L 1 b e the set of all suc h p oints. It is easy to see that: ( ♯ ) ev ery x ∈ E has a neigh b orho o d U x in L 1 with C ( z , g | f − 1 ( y )) ⊂ B ( L 1 , 1 / 2) for all z ∈ U x . Let x 0 ∈ E and D be the family of all sub contin ua D of L 1 suc h that x 0 ∈ D and C ( d, g | f − 1 ( y )) ⊂ B ( L 1 , 1 / 2) for eac h d ∈ D . Since { x 0 } ∈ D , D 6 = ∅ . Claim. Ther e exists D ∗ ∈ D such that diam g ( D ∗ ) > 0. Assume tha t g ( D ) is a singleton f o r eac h D ∈ D . Then cl( S D ) ∈ D . In fact, if d, d ′ ∈ cl( S D ) then C ( d, g | f − 1 ( y )) = C ( d ′ , g | f − 1 ( y )) (no t e that g (cl( S D )) is a singleton). Hence C ( d, g | f − 1 ( y )) ⊂ B ( L 1 , 1 / 2) for eac h d ∈ cl( S D ), and this implies cl( S D ) ∈ D . Then cl( S D ) 4 is a maximal elemen t of D . If cl( S D ) 6 = L 1 , then by ( ♯ ) there exists a prop er sub con tin uum D ′ ⊂ L 1 suc h that D ′ con tains cl( S D ) as a prop er sub con tin uum of D ′ and C ( d, g | f − 1 ( y )) ⊂ B ( L 1 , 1 / 2) for eac h d ∈ D ′ . But this contradicts the fact that cl( S D ) is a maximal el- emen t of D . So cl( S D ) = L 1 . But this is a contradiction b ecause diam g ( L 1 ) > 0 and g (cl( S D )) is a singleton. So there exists D ∗ ∈ D suc h that diam g ( D ∗ ) > 0. T his completes the pro of of claim. Let L 2 = D ∗ . T hen L 2 has the required prop erty . By induction, we can find a decreasing sequenc e { L k } ∞ k =1 of sub contin ua of L such that for any k ∈ N w e hav e ( ∗ ) diam g ( L k ) > 0; ( ∗∗ ) C ( x, g | f − 1 ( y )) ⊂ B ( L k , 1 / ( k + 1)) for eac h x ∈ L k +1 . It is easy to see that C ( x, g | f − 1 ( y )) ⊂ L for eac h x ∈ T ∞ k =1 L k . This implies g ∈ K ( H ), whic h completes the pro of.  Ob viously , if Y = S ∞ m =1 Y m , K ( Y ) = T ∞ i,m =1 K ( Y m ). Therefore, ac- cording to Prop osition 2.1, it suffices to sho w that K ( m, n, H ) is op en and dense in C ( X , M ) with resp ect to the source limitation top ology for m, n ≥ 1 and an y closed H ⊂ Y in the following cases: (i) H is finite-dimensional and M a Krasinkiewicz AN R - space; (ii) H is a C - space a nd M a Krasinkiewicz space homeomorphic to a closed conv ex subset of a Banac h space. In b oth of the ab ov e t wo cases w e follow the sc heme from the pro of of [13, Theorem 1.1]. In particular, w e need the following lemma es- tablished in [13, Lemma 2.1]. Lemma 2.2. [13] Every c omplete AN R -sp ac e M ′ admits a c omplete metric  gener a ting its top olo gy s atisfying the fol lowing c ondition: I f Z is a p ar ac omp act sp ac e, A ⊂ Z a close d set and ϕ : Z → M ′ a map, then fo r every function α : Z → (0 , 1] and every m ap g : A → M ′ with   g ( z ) , ϕ ( z )  < α ( z ) / 8 fo r al l z ∈ A , ther e exists a map ¯ g : Z → M ′ extending g such that   ¯ g ( z ) , ϕ ( z )  < α ( z ) for a l l z ∈ Z . 2.1. P ro of that K ( m, n, H ) is op en in C ( X , M ) for any m, n ≥ 1 and an y closed H ⊂ Y . In this subsection w e pro v e that all sets K ( m, n, H ) are op en in C ( X, M ), where ( M ,  ) is a complete metric (not necessarily an AN R or a Krasinkiewicz) space. Lemma 2.3. L et g ∈ K ( m, n, y ) fo r som e y ∈ Y a n d m, n ≥ 1 . Then ther e exists a neighb orho o d V y of y in Y and δ y > 0 such that y ′ ∈ V y and   g 1 ( x ) , g ( x )  < δ y for a l l x ∈ f − 1 ( y ′ ) yield s g 1 ∈ K ( m, n, y ′ ) . Pr o of. Indeed, otherwise w e can find a lo cal base { V k } k ∈ N of neigh- b orho o ds of y in Y , p oints y k ∈ V k and maps g k ∈ C ( X , M ) suc h Krasinkiewicz spaces and maps 5 that  ( g k ( x ) , g ( x )) < 1 / k for all x ∈ f − 1 ( y k ) but g k do es not b elong to K ( m, n, y k ). Consequen tly , for e v ery k there exists a con t inuum F k ⊂ f − 1 ( y k ) suc h that diam g k ( F k ) ≥ 1 /n and C ( x, g k | f − 1 ( y k )) is not a subset of B ( F k , 1 /m ) for an y x ∈ F k . Then all F k are contained in the compact set P = f − 1 ( { y k } k ∈ N ∪ { y } ) . W e ma y assume tha t { F k } k ∈ N con v erges to a con tin uum F . It follows that F ⊂ f − 1 ( y ) and diam g ( F ) ≥ 1 /n . Sin ce g ∈ K ( m, n, y ) there exists t ∈ F suc h that C ( t, g | f − 1 ( y )) ⊂ B ( F , 1 /m ). Since lim F k = F , for eac h k there exis ts t k ∈ F k with lim t k = t . W e may assume that { C ( t k , g k | f − 1 ( y k )) } k ∈ N con v erges to a con tin uum C . Note that C ⊂ C ( t, g | f − 1 ( y )). Since C ( t k , g k | f − 1 ( y k )) \ B ( F k , 1 /m ) 6 = ∅ , it is easy t o s ee that C is not con- tained in B ( F , 1 /m ). This is a contradiction.  No w, w e are in a p osition to sho w that the sets K ( m, n, H ) are op en in C ( X, M ). Prop osition 2.4. F or any close d H ⊂ Y and any m, n ≥ 1 , the set K ( m, n, H ) is op en in C ( X, M ) with r esp e ct to the sour c e lim itation top olo gy. Pr o of. Let g 0 ∈ K ( m, n, H ). Then, by Lemma 2.3, for ev ery y ∈ H there exist a neigh b orho o d V y and a p ositiv e δ y ≤ 1 suc h that g ∈ K ( m, n, y ′ ) provided g | f − 1 ( y ′ ) is δ y -closed to g 0 | f − 1 ( y ′ ). The family { V y ∩ H : y ∈ H } can b e supp osed to b e locally finite in H . Consider the set-v alued lo w er semi-con tin uous map ψ : H → (0 , 1 ], ψ ( y ) = S { (0 , δ z ] : y ∈ V z } . By [1 0, Theorem 6.2, p.116], ψ a dmits a con tin uous selection β : H → (0 , 1]. L et β : Y → (0 , 1] b e a contin uous extension of β and α = β ◦ f . It remains only to sho w that if g ∈ C ( X, M ) with  ( g 0 ( x ) , g ( x )) < α ( x ) for all x ∈ X , then g ∈ K ( m, n, y ) f or all y ∈ H . So, we ta ke suc h a g a nd fix y ∈ H . Then there exists z ∈ H 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 t o the c hoice o f V z , g ∈ K ( m, n, y ). This completes the pro o f.  2.2. K ( m, n, H ) is dense in C ( X , M ) for finite-dimensional H . In this subsection w e sho w that K ( m, n, H ) is dense in C ( X , M ) with resp ect to the source limitation top ology pr ovided H ⊂ Y is a closed finite-dimensional subset a nd M a Kr a sinkiewic z complete AN R -space. W e need to sho w that B  ( g , ε ) = { g ′ ∈ C ( X , M ) :  ( g , g ′ ) < ε } meets K ( m, n, H ) for ev ery g ∈ C ( X , M ) and ev ery con tin uo us func- tion ε : X → (0 , 1], where  is a complete metric o n M satisfying the h ypotheses of Lemma 2.2. T o this end , fix g 0 ∈ C ( X , M ) and ε ∈ C ( X , (0 , 1 / 64]). Consider the set-v alued map Φ ε : Y → C ( X, M ), 6 Φ ε ( y ) = K ( m, n, y ) ∩ B  ( g 0 , ε ), where C ( X, M ) carries the compac t op en top ology . Lemma 2.5. L et y 0 ∈ Y and Φ ε ( y 0 ) c ontain a c omp act set K . Then ther e exists a neighb orho o d V ( y 0 ) of y 0 such that K ⊂ Φ ε ( y ) for every y ∈ V ( y 0 ) . Pr o of. Supp ose there exists a sequence { y j } j ≥ 1 con v erging to y 0 in Y s uc h that K \ Φ ε ( y j ) 6 = ∅ . Let g j ∈ K \ Φ ε ( y j ), j ≥ 1, and P = f − 1 ( { y 0 } ∪ { y j } j ≥ 1 ). The restriction map π P : C ( X , M ) → C ( P , M ) is con tin uous when b oth C ( X , M ) and C ( P , M ) ar e equipp ed with the compact op en top olo gy . Moreo v er, the compact op en top olog y on C ( P , M ) coincides with the uniform conv ergence. Hence, there ex- ists a subsequence { g j k } of { g j } suc h that π P ( g j k ) con v erges to π P ( g ) in C ( P , M ) for some g ∈ K . Since g ∈ K ( m, n, y 0 ), we can apply Lemma 2.3 to find a neigh bo rho o d V of y 0 in Y and a positive δ > 0 suc h t hat y ′ ∈ V and   g ( x ) , g ′ ( x )  < δ fo r all x ∈ f − 1 ( y ′ ) implies g ′ ∈ K ( m, n, y ′ ). No w, c ho ose j k with y j k ∈ V and   g ( x ) , g j k ( x )  < δ for any x ∈ f − 1 ( y j k ). Th en g j k ∈ K ( m, n, y j k ). So, g j k ∈ Φ ε ( y j k ) whic h con tradicts the c hoice of the functions g j .  Lemma 2.6. Every Φ ε ( y ) has the fol lowin g pr op erty: If ˆ v : S k → Φ ε ( y ) is c ontinuous, wher e k ≥ 0 and S k is the k -spher e, then ˆ v c an b e ex- tende d to a c ontinuous map ˆ u : B k +1 → Φ 64 ε ( y ) . Pr o of. Let us men tion the followin g prop ert y of the function space C ( X , M ) with the compact op en top ology: F or a n y metrizable space Z a map ˆ w : Z → C ( X, M ) is contin uo us if and only if the map w : Z × X → M , w ( z , x ) = ˆ w ( z )( x ), is con tin uous. He nce, ev ery map ˆ v : S k → Φ ε ( y ) generates a con tin uous map v : S k × X → M defined by v ( z , x ) = ˆ v ( z ) ( x ) suc h t hat   v ( z , x ) , g 0 ( x )  < ε ( x ) for all ( z , x ) ∈ S k × X . Let π y : C ( X , M ) → C ( f − 1 ( y ) , M ) b e the restriction map. It is easily seen tha t π y is con tin uous and op en when b oth C ( X , M ) and C ( f − 1 ( y ) , M ) are equipp ed with the source limitation or the compact op en top ology . Since f − 1 ( y ) is compact, the source limitation, the com- pact op en and the uniform conv ergence top olo gies on C ( f − 1 ( y ) , M ) co- incide. Therefore, π y  K ( m, n, y )  is op en in C ( f − 1 ( y ) , M ) and contains the compact set π y  ˆ v ( S k )  . Conse quen tly , the distance ( in the space C ( f − 1 ( y ) , M )) b etw een π y  ˆ v ( S k )  and C ( f − 1 ( y ) , M ) \ π y  K ( m, n, y )  is p ositiv e. Denote this distance by δ 1 . Ob viously δ 2 = inf { ε ( x ) −   v ( z , x ) , g 0 ( x )  : ( z , x ) ∈ S k × f − 1 ( y ) } is po sitiv e. According to Lema 2.2, there exists a con tin uous ex- tension v 1 : B k +1 × f − 1 ( y ) → M of the map v |  S k × f − 1 ( y )  with Krasinkiewicz spaces and maps 7   v 1 ( z , x ) , g 0 ( x )  < 8 ε ( x ) for all ( z , x ) ∈ B k +1 × f − 1 ( y ). Let δ 3 = inf { 8 ε ( x ) −   v 1 ( z , x ) , g 0 ( x )  : ( z , x ) ∈ B k +1 × f − 1 ( y ) } . Since M is a Krasinkiewicz s pace, there exis ts a Kr asinkiewicz map v 2 : B k +1 × f − 1 ( y ) → M suc h t hat   v 2 ( z , x ) , v 1 ( z , x )  < δ / 8 for all ( z , x ) ∈ B k +1 × f − 1 ( y ), where δ = min { δ 1 , δ 2 , δ 3 } . Therefore, w e hav e a map ˆ v 2 : B k +1 → C ( f − 1 ( y ) , M ). The choice of δ 3 implies (1)   v 2 ( z , x ) , g 0 ( x )  < 8 ε ( x ) for a ll ( z , x ) ∈ B k +1 × f − 1 ( y ). Moreov er, v 2 b eing a Krasink iewicz map yields that a ll maps ˆ v 2 ( z ) : f − 1 ( y ) → M , z ∈ B k +1 , are also Krasinkiewicz. On the other hand, by Lemma 2.2 and (1), ev ery ˆ v 2 ( z ) can b e extended to a map from X in to M . T herefore, (2) . ˆ v 2  B k +1  ⊂ π y  K ( m, n, y )  Represen t ing the ba ll B k +1 as a cone with a ba se S k and a ve rtex z 0 , w e can consider v 2 as a ho motop y f rom S k × f − 1 ( y ) × [0 , 1] in to M betw een the maps v 2 |  S k × f − 1 ( y ) × { 0 }  and v 2 |  { z 0 } × f − 1 ( y )  . Observ e also that   v 2 ( z , x, 0) , v ( z , x )  < δ / 8 for an y ( z , x ) ∈ S k × f − 1 ( y ). Hence, the map ϕ : S k × f − 1 ( y ) × { 0 , 1 } → M , ϕ ( z , x, t ) = ( v ( z , x ) if t = 0; v 2 ( z , x, 0) if t = 1 . , is ( δ / 8)-close to v . Consequen tly , b y Lemma 2.2 , ϕ admits a contin uous extension v 3 : S k × f − 1 ( y ) × [0 , 1 ] → M suc h that   v 3 ( z , x, t ) , v ( z , x )  < δ for ev ery ( z , x, t ) ∈ S k × f − 1 ( y ) × [0 , 1]. Since δ < min { δ 1 , δ 2 } , for an y ( z , x, t ) ∈ S k × f − 1 ( y ) × [0 , 1] we ha v e (3)   v 3 ( z , x, t ) , v ( z , x )  < δ 1 , and (4)   v 3 ( z , x, t ) , g 0 ( x )  < ε ( x ) . Therefore, v 3 is a homotop y connecting the maps v and v 2 |  S k × f − 1 ( y ) × { 0 }  , while v 2 is a homotopy connec ting the maps v 2 |  S k × f − 1 ( y ) × { 0 }  and v 2 |  { z 0 } × f − 1 ( y )  . Com bining these t w o homotopies, w e obtain a map u 1 : S k × f − 1 ( y ) × [0 , 1] → M suc h that u 1 ( z , x, 0) = v ( z , x ), u 1 ( z , x, 1) = v 2 ( z 0 , x ) and   u 1 ( z , x, t ) , g 0 ( x )  < 8 ε ( x ) for all ( z , x, t ) ∈ S k × f − 1 ( y ) × [0 , 1 ]. Ob viously , u 1 can also b e considered as a map from B k +1 × f − 1 ( y ) in to M such that u 1 |  S k × f − 1 ( y )  = v and   u 1 ( z , x ) , g 0 ( x )  < 8 ε ( x ), ( z , x ) ∈ B k +1 × f − 1 ( y ). No w consider the map u 2 :  B k +1 × f − 1 ( y )  ∪  S k × X  → M with u 2 |  B k +1 × f − 1 ( y )  = u 1 8 and u 2 |  S k × X  = v . Finally , using Lemma 2.2 , w e extend u 2 to a map u : B k +1 × X → M suc h that (5)   u ( z , x ) , g 0 ( x )  < 64 ε ( x ) for any ( z , x ) ∈ B k +1 × X . Then ˆ u : B k +1 → C ( X , M ) extends the map ˆ v . M oreo v er, (2), (3) and the c hoice of δ 1 implies that ˆ u  B k +1  ⊂ K ( m, n, y ). On the other hand, (5) yields ˆ u  B k +1  ⊂ B  ( g 0 , 64 ε ). Hence, ˆ u  B k +1  ⊂ Φ 64 ε ( y ).  Next prop o sition completes the pro of of Theorem 1.1 in the case Y is strongly coun table-dimensional. Prop osition 2.7. L et H ⊂ Y b e a close d finite-dimens ional set. Then K ( m, n, H ) , m, n ≥ 1 , ar e dens e sets in C ( X , M ) with r esp e ct to the sour c e limitation t op olo gy. Pr o of. Let dim H ≤ k . D efine the set-v alued maps Φ j : H → C ( X , M ), j = 0 , .., k , Φ j ( y ) = Φ ε/ 8 2( k − j )+1 ( y ). Ob viously , Φ 0 ( y ) ⊂ Φ 1 ( y ) ⊂ ... ⊂ Φ k ( y ) = Φ ε/ 8 ( y ). According to Lemma 2.6, ev ery map from S k in to Φ j ( y ) can b e extended to a map from B k +1 in to Φ j +1 ( y ), where j = 0 , 1 , .., k − 1 and y ∈ H . Moreo v er, b y Lemma 2.5, an y Φ j ( y ) has the f ollo wing property : if K ⊂ Φ j ( y ) is c ompact, then there exists a neigh b orho o d V y of y in Y suc h that K ⊂ Φ j ( z ) f or all z ∈ V y ∩ H . So, we ma y apply [3, Theorem 3.1] to find a con tin uous selection θ : H → C ( X , M ) of Φ k . Hence, θ ( y ) ∈ Φ ε/ 8 ( y ) for all y ∈ H . No w, consider the map g : f − 1 ( H ) → M , g ( x ) = θ ( f ( x ))( x ). Using that C ( X , M ) carries the compact op en to p ology , o ne can show that g is con tin uous. Moreo v er,   g ( x ) , g 0 ( x )  < ε ( x ) / 8 for all x ∈ f − 1 ( H ). Then, b y Lemma 2.2 , g c an b e extended to a contin uous map ¯ g : X → M with   ¯ g ( x ) , g 0 ( x )  < ε ( x ), x ∈ X . It follow s fr o m the definition of g that g | f − 1 ( y ) = θ ( y ) | f − 1 ( y ) fo r ev ery y ∈ H . Since θ ( y ) ∈ K ( m, n, y ) for all y ∈ H , ¯ g ∈ K ( m, n, H ). Hence, B  ( g 0 , ε ) ∩ K ( m, n, H ) 6 = ∅ .  2.3. K ( m, n, H ) is dense in C ( X , M ) for H b eing a C -space. W e no w turn to the pro of of Theorem 1.1 in the case Y is a C -space and M a Kra sinkiewic z space homeomorphic to a closed conv ex subset M ′ of a giv en Banach space E . Supp ose M = M ′ and let  b e the metric on M inherited from the nor m of E and Ψ ε : Y → C ( X , M ) b e the set-v alued map Ψ ε ( y ) = B  ( g 0 , ε ) ∩ K ( m, n, y ), where C ( X, M ) is equipp ed ag ain with the compact op en top olo g y and B  ( g 0 , ε ) = { g ∈ C ( X, M ) :   g 0 ( x ) , g ( x )  ≤ ε ( x ) fo r all x ∈ X } . Krasinkiewicz spaces and maps 9 Lemma 2.8. Ψ ε has the fol lowing pr op erty: Every map ˆ v : S k → Ψ ε ( y ) , n ≥ 0 , c an b e extende d to a map ˆ u : B k +1 → Ψ ε ( y ) . Pr o of. All function spaces in this pro of a re equipp ed with the compact op en top ology . Let π y : C ( X , M ) → C ( f − 1 ( y ) , M ) b e the restriction map and P ( y ) = B  ( g 0 , ε, y ) \ π y  K ( m, n, y )  , where B  ( g 0 , ε, y ) is the set { g ∈ C ( f − 1 ( y ) , M ) :   g 0 ( x ) , g ( x )  ≤ ε ( x ) f o r all x ∈ f − 1 ( y ) } . Since π y  K ( m, n, y ) is op en in C ( f − 1 ( y ) , M ), P ( y ) ⊂ B  ( g 0 , ε, y ) is closed. W e are g oing to show that P ( y ) is a Z -set in B  ( g 0 , ε, y ), i.e., ev ery map ˆ w : K → B  ( g 0 , ε, y ), where K is compact, can b e approximated b y a map ˆ w 1 : K → B  ( g 0 , ε, y ) \ P ( y ) = B  ( g 0 , ε, y ) ∩ π y  K ( m, n, y )  . T o this end, fix δ > 0 and let w : K × f − 1 ( y ) → M b e the map generated b y ˆ w . So,   w ( z , x ) , g 0 ( x )  ≤ ε ( x ) for all ( z , x ) ∈ K × f − 1 ( y ). Since f − 1 ( y ) is compact, there exists λ ∈ ( 0 , 1) suc h that λ max { ε ( x ) : x ∈ f − 1 ( y ) } < δ / 2. D efine t he map w 1 : K × f − 1 ( y ) → M b y w 1 ( z , x ) = (1 − λ ) w ( z , x ) + λg 0 ( x ). Then, for all ( z , x ) ∈ K × f − 1 ( y ) we hav e   w 1 ( z , x ) , w ( z , x )  ≤ λ ε ( x ) < δ / 2 and   w 1 ( z , x ) , g 0 ( x )  ≤ ( 1 − λ ) ε ( x ) < ε ( x ) . Since M is a Krasinkiewicz space, there exists a Krasinkiewicz map w 2 : K × f − 1 ( y ) → M whic h is δ 1 -close to w 1 , where δ 1 = min { λ ε ( x ) : x ∈ f − 1 ( y ) } . Hence, for eve ry ( z , x ) ∈ K × f − 1 ( y ) w e ha v e   w 2 ( z , x ) , g 0 ( x )  ≤ ε ( x ) a nd   w 2 ( z , x ) , w ( z , x )  < δ. The la st tw o inequalities imply t ha t the map ˆ w 2 : K → C ( f − 1 ( y ) , M ) is δ -close to ˆ w and ˆ w 2 ( K ) ⊂ B  ( g 0 , ε, y ). Moreo v er, ev ery ˆ w 2 ( z ), z ∈ K , b eing a map from f − 1 ( y ) into M , can b e extended to a map from X to M b ecause M is a closed con v ex subset of E . Since w 2 is a Krasinkiewicz map, so are the maps ˆ w 2 ( z ), z ∈ K . Hence, ˆ w 2 ( K ) ⊂ π y  K ( m, n, y )  . So, P ( y ) is a Z - set in B  ( g 0 , ε, y ). Let us complete t he pro o f of the lemma. F or ev ery map ˆ v : S k → Ψ ε ( y ) the comp osition π y ◦ ˆ v is a map f rom S k in to B  ( g 0 , ε, y ) ∩ π y  K ( m, n, y )  . Since P ( y ) is a Z - set in the con v ex set B  ( g 0 , ε, y ), b y [12, Prop osition 6.3], there exists a map ˆ v 1 : B k +1 → B  ( g 0 , ε, y ) ∩ π y  K ( m, n, y )  extending π y ◦ ˆ v . Consider the map v 2 : A → M , where A =  B k +1 × f − 1 ( y )  ∪  S k × X  , defined b y v 2 |  B k +1 × f − 1 ( y )  = v 1 and v 2 |  S k × X  = v . Next, tak e a selection u : B k +1 × X → M fo r the set-v alued map φ : B k +1 × X → M , φ ( z , x ) = v 2 ( z , x ) if ( z , x ) ∈ 10 A and φ ( z , x ) =cl  B  ( g 0 ( x ) , ε ( x ))  if ( z , x ) 6∈ A . S uc h u exists b y Mic hael’s [8] conv ex-v alued selection theorem. Ob viously u extends v 2 and   u ( z , x ) , g 0 ( x )  ≤ ε ( x ) for eve ry ( z , x ) ∈ B k +1 × X . Finally , observ e that ˆ u is t he required extension of ˆ v .  W e can finish the pro of of Theorem 1.1. Prop osition 2.9. Supp ose H ⊂ Y is a close d C -sp ac e a n d M a close d c onve x subset of a B anach s p ac e E . Then the sets K ( m, n, H ) , m, n ≥ 1 , ar e dense in C ( X , M ) with r esp e ct to the sour c e limitation top olo gy. Pr o of. Consider the set-v alued map Ψ ε : H → C ( X, M ). It follow s from the pro o f of Lemma 2 .5 that if K ⊂ Ψ ε ( y 0 ) for some compactum K and y 0 ∈ H , then y 0 admits a neighborho od V ⊂ H with K ⊂ Ψ ε ( y ) for all y ∈ V . Moreov er, according to Lemma 2.8, ev ery imag e Ψ ε ( y ) is aspherical, i.e., an y map from S k in to Ψ ε ( y ), k ≥ 0, can b e extended to a map from B k +1 to Ψ ε ( y ). Then, b y the Usp enskij selec tion theorem [12, Theorem 1.3], Ψ ε admits a con tin uous selection θ : H → C ( X , M ). Rep eating the argumen ts from the pro of of Prop osition 2.7, w e obta in a map g : f − 1 ( H ) → M suc h that   g ( x ) , g 0 ( x )  ≤ ε ( x ) for eve ry x ∈ f − 1 ( H ) and g | f − 1 ( y ) = θ ( y ) | f − 1 ( y ), y ∈ H . Applying once more the Mic hael [8] con v ex-v alued selection theorem for the set-v alued map ϑ : X → M , ϑ ( x ) = g ( x ) if x ∈ f − 1 ( H ) and ϑ ( x ) = B   g 0 ( x ) , ε ( x )  if x 6∈ f − 1 ( H ), we obtain a selection ¯ g for ϑ . Ob viously , ¯ g extends g a nd ¯ g ∈ B  ( g 0 , ε ). Since θ ( y ) ∈ K ( m, n, y ) fo r all y ∈ H , w e hav e ¯ g ∈ B  ( g 0 , ε ) ∩ K ( m, n, H ). Hence, K ( m, n, H ) is dense in C ( X , M ).  3. Some proper ties of Krasinkiewicz sp aces In this section we inv estigate the class of Krasinkiewicz sp aces and, on t hat ba se, pro vide more spaces from this class. Let us start with the fo llo wing prop osition whose pro o f is straigh tforw ard. Prop osition 3.1. F or every sp ac e M we have: (1) If M is a Kr asinkiewi c z sp ac e, then so is any op en subset of M ; (2) If every c omp act set in M is c ontaine d in a Kr as i n kiewicz subset of M , then M is also a Kr asinkie w icz sp ac e. Corollary 3.2. Every p olyhe dr on is a Kr asinkie w icz sp ac e. Pr o of. Apply Pro p osition 3.1(2 ) and the fact that eac h compact p oly- hedron is a Krasinkiewicz space [7 ].  Next prop osition is an analog ue of [1 1, Theorem 4 .2 ]. Krasinkiewicz spaces and maps 11 Prop osition 3.3. Supp ose M is c omp letely metrizable a n d fo r eve ry ε > 0 ther e exist a Kr asin k iewicz sp ac e Z ε and maps r : M → Z ε and φ : Z ε → M such that φ is light and φ ◦ r is ε -close to the identity on M . Then M is a Kr asinkiewicz sp ac e. Pr o of. Let g ∈ C ( X, M ) a nd ε > 0, where X is compact. Then there exis ts a Krasinkiewicz space Z ε/ 2 and t w o maps r : M → Z ε/ 2 , φ : Z ε → M suc h that φ is ligh t and φ ◦ r is ε/ 2-close to the iden tit y on M . T ak e δ > 0 and a neighborho od U o f r ( g ( X )) in Z ε/ 2 suc h that dist ( φ ( z 1 ) , φ ( z 2 )) < ε / 2 pro vided z 1 , z 2 ∈ U and d ist ( z 1 , z 2 ) < δ . Next, c ho ose a Kr a sinkiewic z map h : X → Z ε/ 2 whic h is δ -close to r ◦ g and h ( X ) ⊂ U . Finally , g ′ = φ ◦ h is ε - close to g and, since φ is light, g ′ is a Krasinkiewicz map (see [7 , Prop osition 3.1]).  Prop osition 3.3 is of special interes t when all Z ε are subsets of M and the maps r are retractions (in suc h a case w e say that M ad- mits s mal l r etr actions to Kr asinkiewicz sp ac es ). Since ev ery compact Menger manifo ld (a manifold mo deled on the Menger cube µ n for some n ≥ 1) , as we ll as ev ery 1-dimensional P eano con tin uum, a dmits small retractions to compact p o lyhedra, it w as o bserv ed in [7, Theorem 3.2- 3.3] that any suc h a space is Krasinkiewicz. Moreov er, ev ery N¨ ob eling manifold also admits small retractions to p olyhedra, see [1]. So, b y Prop osition 3.3 , we ha v e: Corollary 3.4. Ea c h of the fol lowing ar e Kr a sinkiewicz sp ac es: 1 - dimensional Pe ano c ontinua, Menger manifold s and N¨ ob eling ma n i- folds. Prop osition 3.5. A pr o d uct of finitely many Kr asinkiewic z s p ac es is a Kr asinkiewicz sp ac e. Pr o of. W e need to pro v e the pro p osition for a pro duct o f t w o Krasinkie- wicz space s M 1 and M 2 . In this case, the pro of is redu ced to sho w that if X is a metric compactum and g i : X → M i , i = 1 , 2, a re Kra sinkiewic z maps, then the product map g = g 1 △ g 2 : X → M 1 × M 2 is a lso a Krasinkiewicz map. And tha t easily fo llo ws.  Some more example s of Krasinkiewicz spaces are pro vided b y next theorem. Theorem 3.6. A c omplete AN R -sp ac e M is a Kr asin kiewicz sp ac e if and only if it ha s an op en c ove r of Kr asinkiewicz sp ac es. Pro of. It suffices to show that M is Krasinkiewicz if each y ∈ M has a neighbor ho o d U y in M which is a Krasinkiewicz space. W e fix a compactum X and choose ε y > 0, y ∈ M , with B ( y , 3 ε y ) ⊂ U y . Let H y b e the set of all maps g : X → M satisfying next condition: 12 ( a ) If L ⊂ X is a su b con tin uum suc h that diam g ( L ) > 0 and g ( L ) ⊂ cl( B ( y , ε y )), then there exists x ∈ L with C ( x, g ) ⊂ L . No w, fo r eac h m, n ∈ N consider the set H m,n,y ⊂ C ( X , M ) of all maps g suc h that: ( b ) If L ⊂ X is con tinuum with diam g ( L ) ≥ 1 /n and g ( L ) ⊂ cl( B ( y , ε y )), then C ( x, g ) ⊂ B ( L, 1 /m ) for some x ∈ L . Claim 1 . H y = T m,n ∈ N H m,n,y . The pro of of this claim is similar to the proo f of Prop osition 2.1, so it is omitted. Claim 2 . Every H m,n,y is o p en in C ( X , M ). Let f ∈ cl( C ( X, M ) \ H m,n,y ). Then there exists a sequenc e o f maps { f i } ∞ i =1 ⊂ C ( X , M ) \ H m,n,y with lim f i = f . F or each i = 1 , 2 , . . . , there exists a subcontin uum L i ⊂ X suc h that diam f i ( L i ) ≥ 1 /n , f i ( L i ) ⊂ cl( B ( y , ε y )) and C ( x, f i ) is not con tained in B ( L i , 1 /m ) for eac h x ∈ L i . W e ma y assume that L i con v erges to a sub con tin uum L ⊂ X . It is easy to see that diam f ( L ) ≥ 1 /n and f ( L ) ⊂ cl( B ( y , ε y )). Let x ∈ L b e arbitrary . Then x is the limit of a sequence { x i } ∞ i =1 ⊂ X suc h that x i ∈ L i for each i = 1 , 2 , . . . . W e ma y assume that C ( x i , f i ) con v erges to a sub con tin uum C ⊂ X . Since each C ( x i , f i ) is not contained in B ( L i , 1 /m ), C is not con tained in B ( L, 1 / m ). Moreo v er, x ∈ C ⊂ C ( x, f ). So, f ∈ C ( X, M ) \ H m,n,y . This completes t he pro of of Claim 2. Claim 3 . Every H y is dense in C ( X, M ) . Let f ∈ C ( X , M ) and ε > 0 with ε < ε y . Since M is an AN R , there is a δ > 0 suc h that eac h map g : A → M , where A ⊂ X is closed, has a con tin uous extension ˆ g : X → M which is ε - close to f prov ided g is δ - close to f | A . Since U y is a Kra sinkiewic z space a nd f − 1 (cl( B ( y , 2 ε y )) is compact, there exists a K r a sinkiewic z map k : f − 1 (cl( B ( y , 2 ε y )) → U y suc h that k is δ - close to f | f − 1 (cl( B ( y , 2 ε y ))). Then there exists a con- tin uous extension ˆ k : X → Y of k suc h that ˆ k is ε - close to f . W e are going to sho w tha t ˆ k ∈ H y . Indeed, let L b e a subcontin uum of X suc h that diam ˆ k ( L ) > 0 and ˆ k ( L ) ⊂ cl( B ( y , ε y )). Then L ⊂ f − 1 (cl( B ( y , 2 ε y ))). Since k : f − 1 (cl( B ( y , 2 ε y )) → U y is a Krasinkiewic z map, there exists x ∈ L suc h that C ( x, k ) ⊂ L . Note that C ( x, k ) = C ( x, ˆ k ) b ecause ˆ k − 1 ( z ) = k − 1 ( z ) for eac h z ∈ cl( B ( y , ε y )). This com- pletes the pro of of Claim 3. No w, w e can complete the pro of of The orem 3.6. Let f ∈ C ( X, M ) and ε > 0. Since f ( X ) is compact, there exist finitely many p oints y 1 , y 2 , ..., y N ∈ f ( X ) suc h that f ( X ) ⊂ S N i =1 B ( y i , 2 − 1 ε y i ). Let δ 0 = min { ε, 2 − 1 ε y 1 , 2 − 1 ε y 2 , ..., 2 − 1 ε y N } . By previous claims, T N i =1 H y i is a Krasinkiewicz spaces and maps 13 dense G δ -subset of C ( X , M ). So, w e can find a map g 0 ∈ T N i =1 H y i δ 0 -close to f . It suffices to show that g 0 is a Krasinkiew icz map. T o this end, let T b e a sub contin uum of X with diam g 0 ( T ) > 0. Note that g 0 ( T ) ⊂ S N i =1 B ( y i , ε y i ). Hence, there exists a sub contin- uum T ′ ⊂ T and j ∈ { 1 , 2 , ..., N } suc h that diam g 0 ( T ′ ) > 0 and g 0 ( T ′ ) ⊂ cl( B ( y j , ε y j )). Since g 0 ∈ H y j , there exists a p oint x 0 ∈ T ′ suc h that C ( x 0 , g 0 ) ⊂ T ′ ⊂ T . This completes the pro of. Our final prop osition provides spaces whic h are not Krasinkiewicz. It implies, for example, that hereditarily indecomp o sable contin ua can not b e Krasinkiewicz spaces. Prop osition 3.7. L et Y b e a non-de gener ate c ontinuum such that some op en subset of Y c ontains no ar c. Then the pr oje ction p : Y × I → Y c an not b e appr oximate d by Kr asinkiew icz maps. Pr o of. Let U b e an open subset of Y suc h that U contains no arc. Cho ose a non-degenerate con tin uum L ⊂ U a nd let δ = diam L and ε = min { δ / 2 , dist ( L, X \ U ) } . W e claim that ev ery map q : Y × I → Y whic h is ε -close to p can not be Krasinkiew icz. Indeed, supp ose there exists suc h a Krasinkiewicz map q 0 and let t ∈ I . Then q 0 ( L × { t } ) is not a singleton, so there exists y ∈ q 0 ( L × { t } ) and a comp onent C of q − 1 0 ( y ) suc h that C ⊂ L × { t } . T ak e an y po in t z ∈ p ( C ). Then q 0 ( { z } × I ) is not a singleton. So q 0 ( { z } × I ) con tains a n arc. On the other hand, q 0 ( { z } × I ) ⊂ U . T his is a con tradiction.  Reference s [1] A. Chigo gidze, K. Ka wam ura and E. Tymchat yn, N¨ ob eling sp ac es and pseudo- interiors of Menger c omp acta , T op olo gy and Appl. 68 (1996 ), 33–65 . [2] R. Engelking , The ory of dimensions: Finite and Infinite , Heldermann V erlag, Lemgo (19 95). [3] V. Gutev, S ele ctions and appr oximations in fi nite-dimensional sp ac es , T op ol- ogy a nd Appl. 14 6/147 (20 05), 35 3 –383 . [4] J. K rasinkiewicz, On appr oximatio n of m appings into 1-manifolds , B ull. Pol- ish Acad. Sci. Ma th. 44, 4 (1996), 431–44 0. [5] N. Kr ikorian, A note c onc erning the fine top olo gy on function sp ac es , Comp os . Math. 21 (1 969), 3 43–34 8. [6] M. Levin and W. Lewis, Some mapping the or ems for extensional dimension , Israel J . Math. 133 (200 3), 6 1–76. [7] E. Matsuhashi, Kr asinkiewicz m aps fr om c omp acta to p olyhe dr a , Bull. Pol. Acad. Sci. Ma th. 5 4, 2 (2006), 1 37–14 6. [8] E. Michael, Continuous sele ct ions I , Ann. of Math. 63 (1956), 361–3 82. [9] J. Munkers, T op olo gy (Prent ice Ha ll, Eng lewoo d Cliffs, NY, 19 75). [10] D. Rep ov ˇ s and P . Semenov, Continu ous sele ctions of mult ivalue d mappings , Math. and its Appl. 455 , Kluw er, Dor dr ech t (1998). [11] J. Song and E. Tymc hat yn, F r e e sp ac es , F und. Math. 163 (2000 ), 22 9–239 . 14 [12] V. Uspe nskij, A sele ction the or em for C -sp ac es , T op olog y a nd Appl. 85, 1-3 (1998), 3 51–37 4. [13] V. V alov, Par ametric Bing and K r asi nkiewicz maps , T op olo g y and Appl., accepted. [14] H. Whitney , Differ ential manifolds , Ann. Math. 37 (1 936), 645–6 80. Dep ar tment o f Ma thema tics, F a cul ty of Engineering, Yokohama Na tional University Y okohama, 240-8501, Jap an E-mail addr ess : mateii@y nu.ac. jp Dep ar tment of C omputer S cience and Ma thema tics, Nipissing Uni- versity, 100 Coll ege Drive, P. O. Box 5002 , Nor th Ba y, ON, P1B 8L7, Canada E-mail addr ess : veskov@n ipissi ngu.ca