Special embeddings of finite-dimensional compacta in Euclidean spaces

SPECIAL EMBEDDINGS OF FINITE-DIMENSI ONAL COMP A CT A IN EUCLIDEAN SP A CES SEMEON BOGA TYI AND VESK O V ALOV Abstract. If g is a map fro m a space X int o R m and z 6∈ g ( X ), let P 2 , 1 ,m ( g , z ) b e the set of all lines Π 1 ⊂ R m containing z such that | g − 1 (Π 1 ) | ≥ 2. W e pro ve that fo r an y n -dimensio nal metric compactum X the functions g : X → R m , where m ≥ 2 n + 1, with dim P 2 , 1 ,m ( g , z ) ≤ 0 for all z 6∈ g ( X ) fo r m a dense G δ -subset of the function s pace C ( X , R m ). A par ametric version of the ab ove theorem is also provided. 1. I ntroductio n In this pap er w e assume that all top o lo gical spaces are metrizable and all single-v alued maps are contin uous. Ev eryw here b elo w by M m,d w e denote the space of all d -dimensional planes Π d (br., d -planes) in R m . If g is a map from a space X in to R m , q is an in teger and z 6∈ g ( X ), let P q ,d,m ( g , z ) = { Π d ∈ M m,d : | g − 1 (Π d ) | ≥ q and z ∈ Π d } . There is a metric top ology on M m,d , see [6], and we consider P q ,d,m ( g , z ) as a subs pace of M m,d with this top olo g y . One of t he results from authors’ pap er [4] stat es that if X is a metric compactum of dimension n and m ≥ 2 n + 1 , then the function space C ( X, R m ) con tains a dense G δ -subset of ma ps g suc h that the set { Π 1 ∈ M m, 1 : | g − 1 (Π d ) | ≥ 2 } is at most 2 n - dimensional. Th e next theorem pro vides more information concerning the ab ov e result: Theorem 1.1. L et X b e a metric c om p a ctum of dimension ≤ n and m ≥ 2 n + 1 . Then the maps g : X → R m such that dim P 2 , 1 ,m ( g , z ) ≤ 0 for al l z 6∈ g ( X ) form a dense G δ -subset of C ( X, R m ) . Theorem 1.1 admits a par a metric v ersion. 1991 Mathematics Subje ct Classific ation. Prima ry 5 4C10; Secondary 54F45. Key wor ds and phr ases. compact spaces, alg ebraically indepe ndent sets, general po sition, dimension, Euclidean spaces. The first author w as supp or ted b y Gra nts NSH 1562.20 08.1 and RFFI 09-01- 00741 -a. Research supp orted in part by NSERC Grant 26 1914 - 08. 1 2 Theorem 1.2. L et f : X → Y b e a p erfe ct n -dimensiona l map b etwe en metrizable sp ac es wi th dim Y = 0 , and m ≥ 2 n + 1 . T hen the maps g : X → R m such that dim P 2 , 1 ,m ( g | f − 1 ( y ) , z ) ≤ 0 for al l r estrictions g | f − 1 ( y ) , y ∈ Y , and al l z 6∈ g ( f − 1 ( y ) form a dense G δ -subset of C ( X, R m ) e quipp e d with the sour c e limitation top olo gy. F or any map g ∈ C ( X , R m ) and z 6∈ g ( X ) w e also consider the set D 2 , 1 ,m ( g , z ) consisting of p oin ts y = ( y 1 , y 2 ) ∈ ( R m ) 2 suc h that y 1 and y 2 b elong to a line Π 1 ⊂ R m with z ∈ Π 1 , and there exist t w o differen t p oints x 1 , x 2 ∈ X with g ( x i ) = y i , i = 1 , 2. Theorem 1.3 b elow follow s from the pro of of Theorem 1.2 b y consid- ering the sets D 2 , 1 ,m ( g , z ) instead of P 2 , 1 ,m ( g , z ). Theorem 1.3. L et X, Y , f and m satisfy the hyp otheses of The or em 1 . 2 . Then the maps g : X → R m such that dim D 2 , 1 ,m ( g | f − 1 ( y ) , z ) ≤ 0 for al l r estrictions g | f − 1 ( y ) , y ∈ Y , and al l z 6∈ g ( f − 1 ( y ) form a dense G δ -subset of C ( X , R m ) . Recall that for a n y metric space ( M , ρ ) the source limitation to p ol- ogy on C ( X , M ) can b e describe as follo ws: the neighborho o d base at a giv en function f ∈ C ( X, M ) consists o f the sets B ρ ( f , ǫ ) = { g ∈ C ( X, M ) : ρ ( g , f ) < ǫ } , where ǫ : X → (0 , 1] is an y contin uous p ositive functions on X . The sym b ol ρ ( f , g ) < ǫ means that ρ ( f ( x ) , g ( x )) < ǫ ( x ) for all x ∈ X . It is w ell kno w t ha t for metrizable spaces X this top olog y do esn’t dep end on the metric ρ and it has the Ba ire pro p ert y pro vided M is completely metrizable. 2. Preliminaries W e need some preliminary infor mat ion b efor e pro ving Theorem 1.1. Ev eryw here in this section w e supp ose that q , m, d are in tegers with 0 ≤ d ≤ m a nd q ≥ 1. Moreov er, the Euc lidean space R m is eq uipp ed with the standard norm || . || m . W e also supp ose t hat X is a metric compactum and Γ = { B 1 , B 2 , .., B q } is a disjoint family consisting o f q closed subsets of X . F or any g ∈ C ( X, R m ) and z 6∈ g ( X ) w e denote b y P Γ ( g , z ) the set { Π d ∈ M m,d : g − 1 (Π d ) ∩ B i 6 = ∅ for each i = 1 , .., q and z ∈ Π d } . No w, consider the o p en subset R m X of C ( X , R m ) × R m consisting of a ll pairs ( g , z ) with z 6∈ g ( X ). Define the set-v alued map Φ Γ : R m X → M m,d , Φ Γ ( g , z ) = P Γ ( g , z ). Prop osition 2.1. Φ Γ is an upp er semi-c ontinuous and close d-value d map. Em b eddings in Euclidean spaces 3 Pr o of. Supp ose ( g 0 , z 0 ) ∈ R m X . W e need to sho w that for any op en W ⊂ M m,d con taining Φ Γ ( g 0 , z 0 ) there are neigh borho o ds O ( g 0 ) ⊂ C ( X, R m ) and O ( z 0 ) ⊂ R m with O ( g 0 ) × O ( z 0 ) ⊂ R m X and Φ Γ ( g , z ) ⊂ W for all ( g , z ) ∈ O ( g 0 ) × O ( x 0 ). Assume this is not true. Then there exists a sequence { ( g k , z k ) } k ≥ 1 ∈ R m X con v erging to ( g 0 , z 0 ) a nd Π d k ∈ P Γ ( g k , z k ) with Π d k 6∈ W , k ≥ 1. F or a ny i ≤ q and k ≥ 1 t here exists a p oin t x i k ∈ B i ∩ g − 1 k (Π d k ). Since A = S i ≤ q g 0 ( B i ) ⊂ R m is compact, w e t a k e a closed ball K in R m with cen ter the origin containing A in its in terior. Because ev ery Π d ∈ P Γ ( g 0 , z 0 ) in tersects A , w e can iden tify P Γ ( g 0 , z 0 ) with { Π d ∩ K : Π d ∈ P Γ ( g 0 , z 0 ) } considered as a subspace o f exp( K ) (here exp( K ) is the h yp erspace o f all compact subset of K equipped with the Vietoris top o logy). Because { g k } k ≥ 1 con v erges in C ( X , R m ) t o g 0 , w e can assume that K con tains eac h set S i ≤ q g k ( B i ), k ≥ 1. Hence, g k ( x i k ) ∈ K ∩ Π d k for all i ≤ q and k ≥ 1. Therefore, passing to subsequenc es, we may supp o se that there exist p oints x i 0 ∈ B i i ≤ q , and a plane Π d 0 ∈ M m,d suc h that eac h sequence { x i k } k ≥ 1 , i = 1 , 2 , .., q , con v erges to x i 0 and { Π d k ∩ K } k ≥ 1 con v erges to Π d 0 ∩ K . So, lim { g 0 ( x i k ) } k ≥ 1 = g 0 ( x i 0 ), i = 1 , 2 , .., q . Then eac h { g k ( x i k ) } k ≥ 1 also con v erges to g 0 ( x i 0 ). Consequen tly , g 0 ( x i 0 ) ∈ Π d 0 for all i . Moreo v er, since z k ∈ Π d k for all k , w e also ha v e z 0 ∈ Π d 0 . Hence, Π d 0 ∈ P Γ ( g 0 , z 0 ), i.e., Π d 0 ∈ W . On the other hand, W is op en in M m,d and lim { Π d k ∩ K } k ≥ 1 = Π d 0 ∩ K implies that { Π d k } k ≥ 1 con v erges to Π d 0 in M m,d . This yields Π d k ∈ W for almo st all k , a con tradiction. The a b ov e arg umen ts also show that P Γ ( g , z ) is closed in M m,d for all ( g , z ) ∈ R m X . So, Φ Γ ,m,d is a closed-v alued map.  Let X and the inte gers q , d, m b e as ab o v e. W e c ho ose a coun table family B of closed subsets of X suc h tha t the inte riors of the elemen ts of B form a base for the top ology of X . Let also R m X ( k ) = { ( g , z ) ∈ C ( X , R m ) × R m : || z || m ≤ k and ρ m ( z , g ( X )) ≥ 1 /k , where ρ m is t he standard Euclidean metric on R m and k an inte ger. If Γ ⊂ B is a disjoint family of q elemen ts, f o r any in tegers k , s and ǫ > 0 w e consider the set H Γ ( k , s, ǫ ) of all maps g ∈ C ( X, R m ) suc h that each P Γ ( g , z ), where ( g , z ) ∈ R m X ( k ), can b e co v ered by an op en in M m,d family ω ( g , z ) satisfying the fo llowing conditions: (1) mesh( ω ( g , z ) ) < ǫ ; (2) the o rder of ω ( g , z ) is ≤ s (i.e., eac h p o in t from M m,d is con- tained in at most s + 1 elemen ts of ω ( g , z )). Prop osition 2.2. Any H Γ ( k , s, ǫ ) is op en in C ( X, R m ) . 4 Pr o of. Assume g 0 ∈ H Γ ( k , s, ǫ ). F or any ( g 0 , z ) ∈ R m X ( k ) let W ( g 0 , z ) = S { U : U ∈ ω ( g 0 , z ) } . Ob viously , w e hav e ( g 0 , z ) ∈ R m X ( k ) if a nd only if z b elongs t o the compact set B ( g 0 ) = { z ∈ R m : || z || m ≤ k and ρ m ( z , g 0 ( X )) ≥ 1 /k } . Hence, P Γ ( g 0 , z ) ⊂ W ( g 0 , z ) for ev ery z ∈ B ( g 0 ). According to Prop osition 2 .1 , for any suc h z there exists an op en neigh bo r ho o d O ( z ) ⊂ R m \ g 0 ( X ) suc h that P Γ ( g 0 , u ) ⊂ W ( g 0 , z ) for all u ∈ O ( z ). Next, shrink eac h O ( z ) to a n op en set V ( z ) suc h that z ∈ V ( z ) ⊂ V ( z ) ⊂ O ( z ). Then { V ( z ) : z ∈ B ( g 0 ) } is an op en co v er of B ( g 0 ) and we c ho o se a finite sub cov er { V ( z j ) : j = 1 , 2 , ., p } . Let η b e t he distance b et w ee n B ( g 0 ) and R m \ V , where V = S j = p j =1 V ( z j ), and A ( z ) = { j : z ∈ O ( z j ) } , z ∈ O = S j = p j =1 O ( z j ). Cho o sing smaller neigh b orho o ds V ( z j ), if ne cessarily , w e may assume that η < 1 /k . According to the c hoice of O ( z j ), w e ha v e (3) . P Γ ( g 0 , z ) ⊂ W ( g 0 , z j ) fo r a n y z ∈ O and j ∈ A ( z ) Claim 1 . L et g ∈ O ( g 0 , η ) an d ρ m ( z , g ( X )) ≥ 1 /k , wher e O ( g 0 , η ) c on sists of al l g ∈ C ( X, R m ) su ch that ρ m ( g 0 ( x ) , g ( x )) < η for al l x ∈ X . Th e n ρ m ( z , g 0 ( X )) ≥ (1 / k ) − η and z ∈ V ⊂ O . Indeed, b oth ρ m ( z , g 0 ( X )) < (1 /k ) − η and g ∈ O ( g 0 , η ) imply the existence o f x ∈ X with ρ m ( g 0 ( x ) , g ( x )) < 1 /k whic h con tradicts ρ m ( z , g ( X )) ≥ 1 /k . So, for ev ery z satisfying the hypotheses of Claim 1, we hav e ρ m ( z , g 0 ( X )) ≥ (1 / k ) − η . This yields z ∈ V ⊂ O . Eac h W ( g 0 , z j ) is the union of an op en family in M m,d of order ≤ s and mesh < ǫ . Th us , a ccording to Claim 1, it suffices to sho w the next claim. Claim 2 . Ther e exists a neighb orho o d O ( g 0 ) ⊂ O ( g 0 , η ) of g 0 sat- isfying the fol lowing c ondition: for any z ∈ V with ρ m ( z , g 0 ( X )) ≥ (1 /k ) − η ther e exists j ∈ A ( z ) such that P Γ ( g , z ) ⊂ W ( g 0 , z j ) whe n - ever g ∈ O ( g 0 ) . Supp ose the conclusion of Claim 2 is not true. Then fo r ev ery p ≥ 1 there exists a map g p ∈ O ( g 0 , η ) with ρ m ( g 0 ( x ) , g p ( x )) < 1 /p for a ll x ∈ X , a point z p ∈ V with ρ m ( z p , g 0 ( X )) ≥ (1 / k ) − η , and planes (4) Π d p ∈ P Γ ( g p , z p ) \ [ { W ( g 0 , z j ) : j ∈ A ( z p ) } . P assing to subsequences , w e ma y assume that the sequence { z p } p ≥ 1 con v erges to a p oint z 0 ∈ V and { Π d p } p ≥ 1 con v erges in M m,d to a d -plane Π d 0 . Ob vious ly , ρ m ( z 0 , g 0 ( X )) ≥ (1 / k ) − η . Since z p ∈ Π d p , w e also hav e z 0 ∈ Π d 0 . As in the pro o f of Prop osition 2 .1, w e can see that g − 1 0 (Π d 0 ) meets eac h elemen t of Γ. Consequen t ly , Π d 0 ∈ P Γ ( g 0 , z 0 ). So, by (3), Π d 0 ∈ T { W ( g 0 , z j ) : j ∈ A ( z 0 ) } . T his implies that Π d p ∈ T { W ( g 0 , z j ) : Em b eddings in Euclidean spaces 5 j ∈ A ( z 0 ) } for almost a ll p . On the other hand, since lim z p = z 0 , there exists p 0 suc h tha t A ( z 0 ) ⊂ A ( z p ) for all p ≥ p 0 . So, by (4), Π d p 6∈ S { W ( g 0 , z j ) : j ∈ A ( z 0 ) } when p ≥ p 0 , a con tradiction.  Corollary 2.3. Al l maps g ∈ C ( X, R m ) such that dim P q ,d,m ( g , z ) ≤ s for al l z 6∈ g ( X ) form a G δ -subset H X ( q , d, m, s ) of C ( X, R m ) . Pr o of. It easily seen t ha t eac h g ∈ C ( X, R m ) and z 6∈ g ( X ) w e hav e P q ,d,m ( g , z ) = S { P Γ ( g , z ) : Γ ⊂ B is disjoin t and | Γ | = q } . More- o v er, since P Γ ( g , z ) a r e closed in M m,d (b y Prop osition 2.1), we ha v e dim P q ,d,m ( g , z ) ≤ s if and only if dim P Γ ( g , z ) ≤ s for all Γ. This implies that H X ( q , d, m, s ) is the interse ction o f the se ts H Γ ( k , s, 1 /p ), where k , p ≥ 1 a re inte gers and Γ ⊂ B is a disjoin t family of q ele- men ts.  3. Pr oof of Theorems 1.1 and 1.2 Recall that a real n um ber v is called algebraically dep enden t on the real n um b ers u 1 , .., u k if v satisfies the equation p 0 ( u ) + p 1 ( u ) v + ... + p n ( u ) v n = 0, w here p 0 ( u ) , .., p n ( u ) are p olynomials in u 1 , .., u k with rational coefficien ts, not all of them 0. A finite set o f real n um b ers is a l g ebr a ic a l ly indep endent if no ne o f them dep ends alg ebraically on the others. The idea to use algebraically independen t sets for pro ving general position theorems w as originated by R ob erts in [9]. This idea w as also applied b y Berk o witz and Roy in [3]. A pro o f of t he Berko w itz- Ro y main theorem from [3] w as pro vided b y Go o dsell in [8, Theorem A.1] (see [5, Corollary 1.2] for a generalization o f the Berko witz-Roy theorem and [7] for another application of this theorem). Let us note that an y finitely many p oin ts in an Euclidean space R n whose set of co ordinates is algebraically indep enden t are in general p osition. Pr o of of The or em 1 . 1 . W e hav e to sho w that the set H X (2 , 1 , m, 0) of all maps g ∈ C ( X, R m ) suc h that dim P 2 , 1 ,m ( g , z ) ≤ 0 for a ll z 6∈ g ( X ) is dense and G δ in C ( X, R m ). According to Coro lla ry 2.3, this set is G δ . So, it remains to show it is also dense in C ( X, R m ). Fix a countable family B of closed subse ts of X suc h that the in teriors of its elemen ts is a base for X . Since H X (2 , 1 , m, 0) is the in tersec tion of the op en fa mily {H Γ ( k , 0 , 1 /p ) : Γ ⊂ B is disjoint with | Γ | = 2 and k, p ≥ 1 } (see the pro of of Corollary 2.3), it suffices to sho w that eac h H Γ ( k , 0 , ǫ ) is dens e in C ( X, R m ). Recall that H Γ ( k , 0 , ǫ ) consists of a ll maps g ∈ C ( X, R m ) suc h that P Γ ( g , z ) can b e co v ered by a disjoin t op en in M m, 1 family ω with mesh( ω ) < ǫ for ev ery map g and ev ery p oint z ∈ R m satisfying the follow ing conditions: || z | | m ≤ k and ρ m ( z , g ( X )) ≥ 1 /k . 6 T o pro v e that eac h H Γ ( k , 0 , ǫ ) is dense in C ( X , R m ), observ e that a n y map g ∈ C ( X , R m ) can b e appro ximated b y maps f = h ◦ l with l : X → K and h : K → R m , where K is a finite p o lyhedron of dimension ≤ n . Actually , K can b e suppo sed to b e a nerv e of a finite o p en cov er β of X . Moreo ver, if w e c ho ose β suc h that any its eleme n t meets at most one elem en t of Γ = { B 1 , B 2 } , then w e ha v e l ( B 1 ) ∩ l ( B 2 ) = ∅ . F urt her, taking sufficien tly small barycen tric sub division of K , w e can find disjoint subp olyhedra K i of K with l ( B i ) ⊂ K i , i = 1 , 2. Ob viously , for any z 6∈ h ( l ( X )) the se t P Γ ( h ◦ l , z ) is c on tained in P Λ ( h, z ) = { Π 1 ∈ M m, 1 : h − 1 (Π 1 ) ∩ K i 6 = ∅ , i = 1 , 2 a nd z ∈ Π 1 } , where Λ = { K 1 , K 2 } . Therefore, the density of H Γ ( k , 0 , ǫ ) in C ( X , R m ) is reduced to sho w that the maps h ∈ C ( K, R m ) suc h t ha t an y P Λ ( h, z ), z 6∈ h ( K ), a dmits a disjoin t op en cov er in M m, 1 of mesh < ǫ fo rm a dense subset of C ( K, R m ). And this follows t he next prop osition. Prop osition 3.1. L et K i , i = 1 , 2 , b e disjoint n -dimensional subp oly- he d r a of a finite p o lyhe d r on K . Then the maps h ∈ C ( K, R m ) such that for any z 6∈ h ( K ) the set { Π 1 ∈ M m, 1 : h − 1 (Π 1 ) ∩ K i 6 = ∅ , i = 1 , 2 and z ∈ Π 1 } is of dimen s ion ≤ 0 f o rm a dense subset of C ( K , R m ) . Pr o of. Let h 0 ∈ C ( K, R m ) and δ > 0. W e ta ke a sub division of K suc h that diam h 0 ( σ ) < δ / 2 for all simplexes σ . Let K (0) = { a 1 , a 2 , ..., a t } b e the v ertexes of K and v j = h 0 ( a j ), j = 1 , .., t . Then, b y [3], there are p oints b j ∈ R m suc h that the distance b et w een v j and b j is < δ / 2 for eac h j and the co ordinates of all b j , j = 1 , .., t , form an algebraically indep enden t set. D efine a map h : K → R m b y h ( a j ) = b j and h is linear on ev ery simplex of K . It is easily seen that h is δ -close to h 0 . Without loss of generalit y , w e may supp o se tha t K 1 and K 2 are tw o n -dimensional simplexes. Then eac h h ( K i ) is also a n n - dimensional simplex in R m generating a pla ne Π n i ∈ M m,n . Since the co ordinates of the points { b j : j = 1 , .., t } form an algebraically independen t set, the planes Π n 1 and Π n 2 are sk ew. Supp ose z 6∈ h ( K ). If z ∈ Π n 1 or z ∈ Π n 2 , then there is no line Π 1 ⊂ R m whic h contains z and meets b oth h ( K 1 ) and h ( K 2 ). Supp ose z 6∈ Π n 1 ∪ Π n 2 . According to [4, Corollary 3.8], t here exists a t most one line Π 1 ⊂ R m con taining z suc h that Π 1 ∩ h ( K i ) 6 = ∅ , i = 1 , 2. Hence, for an y z 6∈ h ( K ) t he set { Π 1 ∈ M m, 1 : h − 1 (Π 1 ) ∩ K i 6 = ∅ , i = 1 , 2 a nd z ∈ Π 1 } is finite.  Pr o of of The or em 1 . 2 . W e fix a metric d generating the top olo gy of X and for an y g ∈ C ( X, R m ), y ∈ Y , η > 0 and z 6∈ g ( f − 1 ( y )) let P η ( g , y , z ) b e the set of a ll Π 1 ∈ M m, 1 suc h that z ∈ Π 1 and there exist Em b eddings in Euclidean spaces 7 t w o p oin ts x 1 , x 2 ∈ g − 1 (Π 1 ) ∩ f − 1 ( y ) with d ( x 1 , x 2 ) ≥ η . Obvious ly , (5) P 2 , 1 ,m ( g | f − 1 ( y ) , z ) = ∞ [ k =1 { P 1 /k ( g , y , z ) fo r any z 6∈ g ( f − 1 ( y )) } . Claim 3 . Each P η ( g , y , z ) is close d in P 2 , 1 ,m ( g | f − 1 ( y ) , z ). The pr o of of Claim 3 follow s the argumen ts from the pro of of Prop o- sition 2.1. No w, f o r k ≥ 1 and y ∈ Y consider the set B g ( y , k ) = { z ∈ R m : | | z || m ≤ k and ρ m ( z , g ( f − 1 ( y ))) ≥ 1 /k } . Next, let P η ǫ ( y , k ) b e the set of all maps g ∈ C ( X , R m ) suc h that for eac h z ∈ B g ( y , k ) the set P η ( g , y , z ) can b e cov ered by a disjoin t op en in M m, 1 family of mesh < ǫ . If F ⊂ Y , w e consider the set P η ǫ ( F , k ) = \ y ∈ F P η ǫ ( y , k ). Ob viously the inters ection of all P η 1 /s ( Y , k ), s ≥ 1, is the set P η ( Y , k ) = { g ∈ C ( X , R m ) : dim P η ( g , y , z ) ≤ 0 , y ∈ Y , z ∈ B g ( y , k ) } . It fo llows f rom (5) that the set ∞ \ k ,s =1 P 1 /s ( Y , k ) coincides with the set P = { g ∈ C ( X, R m ) : dim P 2 , 1 ,m ( g | f − 1 ( y ) , z ) ≤ 0 , y ∈ Y , z 6∈ g ( f − 1 ( y )) . So, in order to sho w that P is dense and G δ in C ( X, R m ), it suffices to sho w that eac h P η ǫ ( Y , k ) is op en and dense in C ( X, R m ). W e are go ing first to show that an y P η ǫ ( Y , k ) is o p en in C ( X , R m ). This can b e do ne following the argumen ts fro m [4, Prop osition 5.3] using the next lemma instead of [4, Lemma 5.2]. Lemma 3.2. L et g 0 ∈ P η ǫ ( y 0 , k ) for some y 0 ∈ Y . The n ther e exists a neighb orho o d V of y 0 in Y an d δ > 0 such that g ∈ P η ǫ ( V , k ) for al l g ∈ C ( X, R m ) such that the r estrictions g | f − 1 ( V ) and g 0 | f − 1 ( V ) ar e δ -cl o se. Pr o of. Assume the conclusion of Lemma 3.2 do esn’t hold and use the argumen ts f r o m the pro of of Prop o sitions 2.1 and 2.2 to obtain a con- tradiction.  The next prop osition complete s the pro of of Theorem 1.2 . Prop osition 3.3. Any set P η ǫ ( Y , k ) is dense in C ( X , R m ) wi th r esp e ct to the sour c e limitation top olo gy. 8 Pr o of. W e mo dify the argumen t s fro m the pro of of [4, Prop osition 5.4]. Let g ∈ C ( X , R m ) and δ ∈ C ( X, (0 , 1]). W e are going to find h ∈ P η ǫ ( Y , k ) such that ρ ( g ( x ) , h ( x )) < δ ( x ) for all x ∈ X . By [1, Prop osition 4], g can b e supp osed to b e simplicially factorizable. This means that there exists a simplicial complex D and maps g D : X → D , g D : D → M with g = g D ◦ g D . F ollo wing the pro of of [2, Prop osi- tion 3.4], w e can find an op en co v er U of X , simplicial complexes N , L and maps α : X → N , β : Y → L , p : N → L , ϕ : N → R m and δ 1 : N → (0 , 1] satisfying the fo llo wing conditions, where h ′ = ϕ ◦ α : • α is an U -map and for any x 1 , x 2 ∈ X with d ( x 1 , x 2 ) ≥ η w e ha v e α ( x 1 ) 6 = α ( x 2 ); • β ◦ f = p ◦ α ; • p is a p erfect P L -map with dim p ≤ n and dim L = 0; • h ′ is ( δ / 2)-close to g ; • δ 1 ◦ α ≤ δ . So, we hav e the fo llo wing comm utativ e diagram: L Y ◗ ◗ s β X ❄ f ✲ h ′ ◗ ◗ s α N ❄ p ✚ ✚ ❃ ϕ R m Since L is a 0- dimensional simplicial complex and p is a p erfect P L - map, N is a discre te union of the finite complexes K l = p − 1 ( l ) , l ∈ L . Because dim p ≤ n , dim K l ≤ n , l ∈ L . Applying Theorem 1.1 to each complex K l , w e can find a map ϕ 1 : N → R m suc h that for any l ∈ L and z 6∈ ϕ 1 ( p − 1 ( l ) ) w e ha v e dim P 2 , 1 ,m ( ϕ 1 | p − 1 ( l ) , z ) ≤ 0 a nd ϕ 1 | p − 1 ( l ) is θ l -close to ϕ | p − 1 ( l ) , where θ l = min { δ 1 ( u ) : u ∈ p − 1 ( l ) } . Moreov er, the map h = ϕ 1 ◦ α is δ -close to g . W e claim that h ∈ P η ǫ ( Y , k ). Indeed, let y ∈ Y and z ∈ B h ( y , k ). If Π 1 ∈ P η ( h, y , z ), then there exist t w o p oin ts x i ∈ h − 1 (Π 1 ) ∩ f − 1 ( y ), i = 1 , 2, with d ( x 1 , x 2 ) ≥ η . According to the choice of the cov er U , w e hav e α ( x 1 ) 6 = α ( x 2 ). Since ϕ − 1 1 (Π 1 ) ∩ p − 1 ( β ( y )) contains the p oints α ( x i ), i = 1 , 2, w e obtain that Π 1 ∈ P 2 , 1 ,m ( ϕ 1 | p − 1 ( β ( y )) , z ). T h us, w e established the inclusion P η ( h, y , z ) ⊂ P 2 , 1 ,m ( ϕ 1 | p − 1 ( β ( y )) , z ) whic h implies dim P η ( h, y , z ) ≤ 0 for ev ery y ∈ Y and z ∈ B h ( y , k ). Conseq uen tly , h ∈ P η ǫ ( Y , k ).  Ac kno wledgmen t s. The results from this pap er we re obtained during the second author’s visit o f D epart ment of Computer Science and Mathematics (COMA), Nipissing Univ ersit y in August 20 10. He ac kno wledges COMA f or the supp ort and hospitalit y . Em b eddings in Euclidean spaces 9 Reference s [1] T. Banakh and V. V a lov, Gener al Position Pr o p erties in Fib erwise Ge ometric T op olo gy , arXiv:ma th.GT/1001 249 4. [2] T. Banakh a nd V. V alov, Appr oximation by light maps and p a r ametric L elek maps , T op olo gy a nd Appl. 157 (2010), 23 25–2 341. [3] H. Berko witz and P . Roy , Gener a l p osition and algebr ai c indep en denc e , in: L.C. Gla s er, T.B. Rushing (Eds.), Geometr ic T o p ology , Pro c e edings of the Geometry T op olog y Confer ence held at Park Cit y , UT, Springer , New Y ork , 1975, 9–15. [4] S. B ogataya, S. Bo gatyi and V. V alov, Emb e dding of finite-dimensional c om- p a cta in Euclide an sp ac es , a rXiv:math.GN/101 01892. [5] S. B ogatyi and V. V alov, R ob ert s ’ typ e emb e d dings and c onversion of the tr ansversal Tverb er g’s the or em , Mat. Sb or nik 196, 11 (200 5), 3 3–52 (in Rus- sian); translation in: Sb. Math. 196:11 -12 (2005 ), 15 8 5–16 03. [6] B. Dubrovin, S. No viko v and A. F o menko, Mo dern Ge ometry , Nauk a, Mosc ow 1979 (in Russian). [7] T. Go o dsell, S t r ong gener al p osition and Menger curves , T op olo gy and Appl. 120 (2002), 4 7–55 . [8] T. Go o dsell, Pr oje ctions of c omp acta in R n , Ph.D. Thes is , Br igham Y oung Univ ersity , Prov o, UT, 1997 . [9] J. Rob erts, A t he o r em on dimension , Duk e Math. J. 8 (194 1), 5 6 5–57 4. F acul ty o f Mechanics and Ma thema tics, Moscow St a te U niversity, Vor obevy gor y , Moscow, 119899 R ussia E-mail addr ess : bo gatyi @inbo x.ru Dep ar tment of Computer Science and Ma thema tics, Nipissing Uni- versity, 1 00 College Drive, P. O. Box 5002, Nor th Ba y, ON, P 1B 8L7, Canada E-mail addr ess : ve skov@ nipis singu.ca