Approximation by light maps and parametric Lelek maps

0.37 MB·English

by  Taras Banakh

APPROXIMATION BY LIGHT MAPS AND
PARAMETRIC LELEK MAPS
8
TARAS BANAKH AND VESKO VALOV
0
0
2
Abstract. The class of metrizable spaces M with the follow-

n ing approximation property is introduced and investigated: M ∈
a AP(n,0) if for every ε > 0 and a map g: In → M there exists
J
a 0-dimensional map g′: In → M which is ε-homotopic to g. It
0
2 is shown that this class has very nice properties. For example, if
Mi ∈ AP(ni,0), i = 1,2, then M1×M2 ∈AP(n1+n2,0). More-
] over, M ∈ AP(n,0) if and only if each point of M has a local
T
base of neighborhoods U with U ∈ AP(n,0). Using the proper-
G
ties of AP(n,0)-spaces, we generalize some results of Levin and
.
h Kato-Matsuhashi concerning the existence of residual sets of n-
t dimensional Lelek maps.
a
m
[

1 1. Introduction
v
7 All spaces in the paper are assumed to be metrizable and all maps
0
continuous. By C(X,M) we denote all maps from X into M. Un-
1
3 less stated otherwise, all function spaces are endowed with the source
1. limitation topology.
0 One of the important properties of the n-dimensional cube In or the
8 Euclidean space Rn, widely exploited in dimension theory, is that any
0
: mapfromann-dimensionalcompactumX intoIn (resp.,Rn)canbeap-
v
i proximated by maps with 0-dimensional fibers. The aim of this article
X
is to introduce and investigate the class of spaces having that property.
r
a More precisely, we say that a space M has the (n,0)-approximation
property (br., M ∈ AP(n,0)) if for every ε > 0 and a map g: In → M
there exists a 0-dimensional map g′: In → M which is ε-homotopic
to g. Here, g′ is ε-homotopic to g means that there is an ε-small ho-
motopy h: In × I → M connecting g and g′. It is easily seen that
this definition doesn’t depend on the metric generating the topology
of M. If M is LCn, then M ∈ AP(n,0)) if and only if for every ε > 0
and g ∈ C(In,M) there exists a 0-dimensional map g′ which is ε-close
1991 Mathematics Subject Classification. Primary 54F45; Secondary 55M10.
Key words and phrases. dimension, n-dimensional maps, n-dimensional Lelek
maps, dendrites, Cantor n-manifolds, general position properties.
The second author was partially supported by NSERC Grant 261914-03.
1
2
to g. We show that the class of AP(n,0)-spaces is quite large and it
has very nice properties. For example, if M ∈ AP(n ,0), i = 1,2,
i i
then M × M ∈ AP(n + n ,0). We also prove that M ∈ AP(n,0)
1 2 1 2
if and only if each point of M has a local base of neighborhoods
U with U ∈ AP(n,0). Moreover, every path-connected compactum
M ∈ AP(n,0) is a (Vn)-continuum in the sense of P. Alexandroff [1].
In particular, according to [10], any such M is a Cantor n-manifold,
as well as a strong Cantor n-manifold in the since of Hadˇziivanov [9].
All complete LC0-spaces without isolated points are AP(1,0), as well
as, every manifold modeled on the n-dimensional Menger cube or the
n-dimensional N¨obeling space has the AP(n,0)-property.
The class of AP(n,0)-spaces is very natural for obtaining results
about approximation by dimensionally restricted maps. We present
such a result about n-dimensional Lelek maps. Recall that a map
g: X → M between compact spaces is said to be an n-dimensional
Lelek map [17] if the union of all non-trivial continua contained in the
fibers of g is of dimension ≤ n. For convenience, if n ≤ 0, by an n-
dimensional Lelek map we simply mean a 0-dimensional map. Every
n-dimensional Lelek mapg between compacta is at most n-dimensional
because the components of each fiber of g is at most n-dimensional (we
say that g is n-dimensional if all fibers of g are at most n-dimensional).
Lelek [15] constructed such a map from In+1 onto a dendrite. Levin
[17] improved this result by showing that the space C(X,In) of all
maps from X into In contains a dense G -subset consisting of (m−n)-
δ
dimensional Lelek maps for any compactum X with dimX = m ≤ n.
Recently Kato and Matsuhashi [12] established a version of the Levin
result with In replaced by more general class of spaces. This class
consists of complete metric ANR-spaces having a piecewise embedding
dimension ≥ n.
In the present paper we generalize the result of Kato-Matsuhashi in
twodirections. First, weestablishaparametricversionoftheirtheorem
and second, we show that this parametric version holds for AP(n,0)-
spaces (see Section 6 where it is shown that the class of AP(n,0)-
spaces contains properly the class of ANR’s with piecewise embedding
dimensions ≥ n). Here is our result about parametric Lelek maps
(a more general version is established in Section 3; moreover we also
present a version of Theorem 1.1 when M has a property weaker than
AP(n,0), see Theorem 4.5).
Theorem 1.1. Let f: X → Y be a perfect map with dim (f) ≤ m,
△
where X and Y are metric spaces. If M ∈ AP(n,0) is completely
Approximation by light maps 3
metrizable, then there exists a G -set H ⊂ C(X,M) such that ev-
δ
ery simplicially factorizable map in C(X,M) is homotopically approx-
imated by maps from H and for every g ∈ H and y ∈ Y the restriction
g|f−1(y) is an (m−n)-dimensional Lelek map.
Let us explain the notions in Theorem 1.1. A map g ∈ C(X,M)
is homotopically approximated by maps from H means that for every
function ε ∈ C(X,(0,1]) there exists g′ ∈ H which is ε-homotopic to
g. Here, the maps g and g′ are said to be ε-homotopic, if there is
a homotopy h: X × I → M connecting g and g′ such that each set
h({x}×I) has a diameter < ε(x), x ∈ X.
The function space C(X,M) appearing in this theorem is endowed
withthesourcelimitationtopologywhoseneighborhoodbaseatagiven
function f ∈ C(X,M) consists 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 over
continuous positive functions on X. The symbol ̺(f,g) < ε means
that ̺(f(x),g(x)) < ε(x) for all x ∈ X. Since X is metrizable, the
source limitation topology doesn’t depend on the metric ρ [14] and it
has the Baire property provided M is completely metrizable [20].
We say that a map g: X → M is simplicially factorizable [2] if there
exists asimplicial complex Landtwo mapsg : X → Landg : L → M
1 2
suchthatg = g ◦g . Ineachofthefollowingcasesthesetofsimplicially
2 1
factorizable maps is dense in C(X,M) (see [2, Proposition 4]): (i) M
is an ANR; (ii) dimX ≤ k and M is LCk−1; (iii) X is a C-space and
M is locally contractible.
The dimension dim (f) was defined in [2]: dim (f) of a map f :
△ △
X → Y is equal to the smallest cardinal number τ for which there is a
map g : X → Iτ such that the diagonal product f∆g : X → Y ×Iτ is
a 0-dimensional map. For any perfect map f : X → Y between metric
spaces we have: (i) dim(f) ≤ dim (f); (ii) dim (f) = dim(f) if Y is
△ △
a C-space, see [21] and [27]; (iii) dim (f) ≤ dim(f)+ 1 if the spaces
△
X,Y are compact, see [16].
Since every metric space admitting a perfect finite-dimensional map
onto a C-space isalso a C-space [11], Theorem 1.1 implies the following
(here, H is homotopically dense in C(X,M) if every g ∈ C(X,M) is
homotopically approximated by maps from H):
4
Corollary 1.2. Let f : X → Y be a perfect m-dimensional map be-
tween metric spaces with Y being a C-space. If M ∈ A(n,0) is com-
pletely metrizable, then in each of the following cases there exists a ho-
motopically dense G -subset H ⊂ C(X,M) consisting of maps g such
δ
that g|f−1(y) is an (m−n)-dimensional Lelek map for every y ∈ Y:
• M is locally contractible;
• X is finite dimensional and M ∈ LCk−1 for k = dimX.
The paper is organized as follows. In Section 2 some preliminary
results about AE(n,0)-spaces are presented. In Section 3 we establish
a generalized version of Theorem 1.1. Section 4 is devoted to almost
AP(n,0)-spaces. The final two sections contain some interesting prop-
ertiesof AP(n,0)-spaces. InSection5 weprove thatAP(n,0)-property
has a local nature, as well as, M ×M ∈ AP(n +n ,0) provided each
1 2 1 2
M ∈ AP(n ,0) is completely metrizable. It is also established in this
i i
section that every path connected compactum is a (Vn)-continuum.
Section 6 is devoted to the interplay of AP(n,0)-property and the gen-
{n,k}
eral position properties m − DD -properties. In particular, it is
shown that every point of a locally path-connected AP(n,0)-space X
is a homological Z -point in X. Another result from this section
n−1
states that every completely metrizable space possessing the disjoint
(n−1)-disks property is an AP(n,0)-space.
2. Preliminary results about AP(n,0)-spaces
In this section we established some preliminary results on AP(n,0)-
spaces which are going to be used in next sections.
Suppose M is a metric space and ε > 0. We write d (M) < ε if
n
M can be covered by an open family γ such that diamU < ε for all
U ∈ γ and ord(γ) ≤ n + 1 (the last inequality means that at most
n+1 elements of γ can have a common point). It is easily seen that
if F ⊂ M is closed and d (F) < ε, then F is covered by an open in
n
M family γ of mesh < ε and ord ≤ n + 1. We also agree to denote
by cov(M) the family of all open covers of M. Let us begin with the
following technical lemma.
Lemma 2.1. Let Z = A ∪ B be a compactum, where A and B are
closed subsets of Z. Suppose C ⊂ Z is closed such that dimC ∩A ≤ 0
and d (C ∩B) < ε. Then d (C) < ε.
0 0
Proof. Since d (C ∩ B) < ε, there exists a disjoint open family γ =
0
{W ,..,W } in B of mesh < ε. We extend every W to an open set
1 k i
W˜ in Z such that W˜ ∩ B = W and γ˜ = {W˜ : i = 1,..,k} is a
i i i i
disjoint family of mesh < ε (this can be done because B is closed
Approximation by light maps 5
in Z). Observe that C = C\ i=kW˜ is a closed subset of C ∩ A
1 i=1 i
disjoint from B. Since dimC ∩A ≤ 0, there exists a clopen set C in
2
S
C ∩ A disjoint from B and containing C . Obviously, C is clopen in
1 2
C. Moreover, dimC ≤ 0. Hence, there exists a cover ω = {V : j =
2 j
1,..,m} ∈ cov(C ) consisting of clopen subsets of C with diamV < ε
2 2 j
for every j. Then ω = ω ∪{W˜ ∩C\C } is a disjoint open cover of C
1 i 2
and mesh(ω ) < ε. So, d (C) < ε. (cid:3)
1 0
The proof of next lemma is extracted from [2, Theorem 4].
Lemma 2.2. Let G ⊂ C(X,M), where (M,̺) is a complete metric
space. Suppose (U(i) is a sequence of open subsets of C(X,M) such
i≥1
that
• for any g ∈ G, i ≥ 1 and any function η ∈ C(X,(0,1]) there
exists g ∈ B (g,η)∩U(i)∩G which is η-homotopic to g.
i ̺
Then, for any g ∈ G and ε: X → (0,1] there exists g′ ∈ ∞ U(i) and
i=1
an ε-homotopy connecting g and g′. Moreover, g′|A = g |A for some
0
T
g ∈ C(X,M) and A ⊂ X provided g |A = g |A for all i.
0 i 0
Proof. For fixed g ∈ G and ε ∈ C(X,(0,1]) let g = g and ε = ε/3.
0 0
We shall construct by induction a sequence (g : X → M) ⊂ G, a
i i≥1
sequence (ε ) of positive functions, and a sequence (H : X×[0,1] →
i i≥1 i
M) of ε -homotopies satisfying the conditions:
i≥1 i−1
• H (x,0) = g (x) and H (x,1) = g (x) for every x ∈ X;
i+1 i i+1 i+1
• g ∈ B (g ,ε )∩U(i+1)∩G;
i+1 ̺ i i
• ε ≤ ε /2;
i+1 i
• B (g ,3ε ) ⊂ U(i+1).
̺ i+1 i+1
Assumethat,forsomei,wehavealreadyconstructedmapsg ,...,g ,
1 i
positive numbers ε ,...,ε , and homotopies H ,...,H satisfying the
1 i 1 i
above conditions. According to the hypotheses, there exists a map
g ∈ B (g ,ε ) ∩ U(i + 1) ∩ G such that g is ε -homotopic to g .
i+1 ̺ i i i+1 i i
Let H : X × [0,1] → M be an ε -homotopy connecting the maps
i+1 i
g and g . Since the set U(i + 1) is open in C(X,M), there is a
i i+1
positive function ε ≤ ε /2 such that B (g ,3ε ) ⊂ U(i+1). This
i+1 i ̺ i+1 i+1
completes the inductive step.
It follows from the construction that the function sequence (g )
i i≥1
converges uniformly to some continuous function g′ : X → M. Ob-
viously, ̺(g′,g ) ≤ ∞ ε ≤ 2ε for every i. Hence, according to
i j=i j i
the choice of the sequences (ε ) and (g ), g′ ∈ U(i) for every i ≥ 1.
i i
So, g′ ∈ ∞ U(i).PMoreover, the ε -homotopies H compose an
i=1 i−1 i
T
6
ε-homotopy H : X ×[0,1] → M
1 1 1
H x,2i(t−1+ ) if t ∈ [1− ,1− ], i ≥ 1;
H(x,t) = i 2i−1 2i−1 2i

g′((cid:0)x) (cid:1) if t = 1.

connecting g = g and g′.
0
If follows from our construction that g′|A = g |A if g |A = g |A for
0 i 0
all i ≥ 1. (cid:3)
Proposition 2.3. Let M be a completely metrizable space with the
AP(n,0)-property and X be an n-dimensional compactum. Then
E = {g ∈ C(X,M) : dimg ≤ 0}
is a G -subset of C(X,M) such that every simplicially factorizable map
δ
is homotopically approximated by maps from E.
Proof. We fix a metric on X and a complete metric ̺ on M. For
every ε > 0 let C(X,M;ε) be the set all maps g ∈ C(X,M) with
d (g−1(a)) < ε for every a ∈ M. It is easily seen that C(X,M;ε) is
0
open in C(X,M) and E = ∞ C(X,M;1/i). So, E is a G -subset of
i=1 δ
C(X,M).
T
Claim. For every simplicially factorizable map h ∈ C(X,M) and
positive numbers η and δ there exists a simplicially factorizable map
h′ ∈ C(X,M;η) which is δ-homotopic to h.
To prove the claim, fix a simplicially factorizable map h ∈ C(X,M)
and positive numbers δ, η. Then there exists a simplicial complex L
and maps q : X → L, q : L → M with h = q ◦ q . We can assume
1 2 2 1
that L is finite and q (L) ⊂ ∪γ, where γ = {B (y,δ/4) : y ∈ g(X)}.
2 ̺
Because L is an ANR and dimX ≤ n, there is a polyhedron K with
dimK ≤ n and two maps f: X → K, α : K → L such that f is an
η-map and q and α ◦ f are q−1(γ)-homotopic. So, h and q ◦ α ◦ f
1 2 2
are δ/2-homotopic. Moreover, there is a cover ω ∈ cov(K) such that
diamf−1(W) < η for every W ∈ ω. Let θ be a Lebesgue number of
ω. It remains to find a map h∗: K → M which is δ/2-homotopic to
q ◦ α and h∗ ∈ C(K,M;θ) (then h′ = h∗ ◦ f would be a simplicially
2
factorizable map δ-homotopic to h and h′ ∈ C(X,M;η)).
To find such a map h∗: K → M, let {σ ,..,σ } be an enumeration
1 m
of the simplexes of K and K = j=i σ . We are going to construct
i j=1 j
by induction maps h : K → M, i = 0,..,m, satisfying the following
i
S
conditions:
• h = q ◦α;
0 2
• h |K belongs to C(K ,M;θ), 1 ≤ i ≤ m;
i i i
• h and h are (δ/2m)-homotopic, i = 0,..,m−1.
i i+1
Approximation by light maps 7
Let V = {B (y,δ/4m) : y ∈ M} ∈ cov(M) and assume that h
̺ i
has already been constructed. Since h |K ∈ C(K ,M;θ), every fiber
i i i
h−1(y)∩K of h |K , y ∈ h (K ), is covered by a finite openand disjoint
i i i i i 1
family Ω(y) in K with mesh(Ω(y)) < θ. Using that h |K is a perfect
i i i
map, we find a cover V ∈ cov(M) such that V is a star-refinement of V
i i
and if St(y,V )∩h (K ) 6= ∅ for some y ∈ M, then there is z ∈ h (K )
i i i i i
with h−1(St(y,V ))∩K ⊂ Ω(z). Since M has the AP(n,0)-property,
i i i
there exists a 0-dimensional map p : σ → M which is V -homotopic
i i+1 i
to h |σ . By the Homotopy Extension Theorem, p can be extended
i i+1 i
toamaph : K → M beingV -homotopictoh . Thenh is(δ/2m)-
i+1 i i i+1
homotopic to h . To show that h |K belongs to C(K ,M;θ), we
i i+1 i+1 i+1
observe that h−1 (y)∩K = h−1 (y)∩K ∪ h−1 (y)∩σ , y ∈ M.
i+1 i+1 i+1 i i+1 i+1
According to our construction, we have h h−1 (y)∩ K ⊂ St(y,V ).
(cid:0) i(cid:1) i+(cid:0)1 i (cid:1) i
Hence, h−1 (y)∩K iscontainedinΩ(z)forsomez ∈ h (K ). Therefore,
i+1 i (cid:0) i (cid:1)i
d h−1 (y) ∩ K < θ. Since h−1 (y) ∩ σ is 0-dimensional, Lemma
0 i+1 i i+1 i+1
2.1 implies that d (h−1 (y)∩ K ) < θ. Obviously the map h∗ = h
(cid:0) (cid:1) 0 i+1 i+1 m
is δ/2-homotopic to q ◦ α and h∗ ∈ C(K,M;θ). This completes the
2
proof of the claim.
To finish the proof of the proposition, we apply Lemma 2.2 with G
being the set of all simplicially factorizable maps from C(X,M) and
U(i) = C(X,M;1/i), i ≥ 1 (we can apply Lemma 2.2 because of the
claim). Hence, for every simplicially factorizable map g ∈ C(X,M)
and a positive number ε there exists a map g′ ∈ ∞ U(i) which is
i=1
ε-homotopic to g. Finally, we observe that ∞ U(i) consists of 0-
i=1 T (cid:3)
dimensional maps.
T
Proposition 2.4. Let M be a completely metrizable space with the
AP(n,0)-property, X a compactum and Z an F -subset of X with
σ
dimZ ≤ n − 1. Then there exists a G -subset H ⊂ C(X,M) with
δ
the following properties:
• Z is contained in the union of trivial components of the fibers
of g for all g ∈ H;
• for every simplicially factorizable map g ∈ C(X,M) and every
ε > 0 there exists g′ ∈ H which is ε-homotopic to g.
Proof. We represent Z as the union of an increasing sequence (Z )
i i≥1
with all Z being closed in Z. For every ε > 0 let
i
H(Z ,ε) = {g ∈ C(X,M) : F(g,ε)∩Z = ∅},
i i
where
F(g,ε) = ∪{C : C is a component of a fiber of g with diamC ≥ ε}.
8
It is easily seen that each H(Z ,ε) is open in C(X,M). So, the set
i
H = ∞ H(Z ,1/i) is G and Z is contained in the union of trivial
i=1 i δ
components of the fibers of g for every g ∈ H.
T
To prove the second item of our proposition, we first consider the
particular case when X is a polyhedron and Z is a (compact) subpoly-
hedron of X.
Claim 1. Suppose that, in addition to hypotheses of Proposition 2.4,
X is a polyhedron and Z is a subpolyhedron of X. Then, for every
g ∈ C(X,M) and δ > 0 there exists g ∈ H = ∞ H(Z,1/i) which is
0 i=1
δ-homotopic to g .
0
T
The proof of Claim 1 is a slight modification of the proof of [12,
Theorem 2.2]. Let g ∈ C(X,M) and δ > 0. We take an open
0
neighborhood 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 ANR, we can choose W so small that g |W is δ/2-homotopic
0
to (g ◦ r)|W. Next, denote by π: Z × I → Z the projection and
0
consider the map ϕ: W → Z × I, ϕ(x) = (r(x),α(x)). Obviously,
(g ◦ r)|W = (g ◦ π ◦ ϕ)|W. So, (g ◦ π ◦ ϕ)|W is δ/2-homotopic to
0 0 0
g |W. Because Z is a polyhedron, so is Z × I. Hence, every map
0
from C(Z×I,M) is simplicially factorizable. Moreover, dimZ×I ≤ n
and M ∈ AP(n,0). Therefore, by Proposition 2.3, there exists a 0-
dimensional map h: Z × I → M which is δ/2-homotopic to the map
g ◦π. Then (h◦ϕ)|W is δ/2-homotopicto(g ◦π◦ϕ)|W. Consequently,
0 0
(h◦ϕ)|W is δ-homotopic to g |W. By the Homotopy Extension Theo-
0
rem, there exits a map g ∈ C(X,M) such that g is δ-homotopic to g
0
and g|U = (h◦ϕ)|U, where U is an open neighborhood of Z in X with
U ⊂ W.
To finish the proof of Claim 1, it remains to show that g ∈ H. Let
C be a subcontinuum of g−1(y) for some y ∈ M and let Z ∩ C 6= ∅.
We are going to prove that C ⊂ Z. Otherwise, there would be a
subcontinuum C′ ⊂ C ∩U such 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 continuum in h−1(y). Since h is 0-dimensional, this
is a contradiction. Hence, C ⊂ Z. Using again that dim(h) ≤ 0, we
conclude that ϕ(C) is a point. On the other hand ϕ(C) = C ×{0} ⊂
Z ×I. Therefore, C should be a trivial continuum.
Now, consider the general case of Proposition 2.4.
Claim 2. Let g ∈ C(X,M) be a simplicially factorizable map and δ,
0
η positive numbers. Then for any i there exists a simplicially factoriz-
able map g ∈ H(Z ,η) which is δ-homotopic to g .
i 0
Approximation by light maps 9
Since g is simplicially factorizable, there exist a finite simplicial
0
complex L and maps q : X → L, q : L → M with g = q ◦ q . Let
1 2 0 2 1
V = {B (y,δ/4) : y ∈ M} ∈ cov(M) and W = q−1(V) ∈ cov(L). Next,
̺ 2
choose a finite cover U ∈ cov(X) of X with mesh(U) < η such that:
• at most n elements of the family γ = {U ∈ U : U ∩Z 6= ∅} can
i
have a common point;
• there exists a map h: N(U) → L such that h ◦ f is W-
U
homotopic to q , where N(U) is the nerve of U and f : X →
1 U
N(U) is the natural map.
Then q ◦ h ◦ f is δ/2-homotopic to g and the subpolyhedron K
2 U 0
of N(U) generated by the family γ is of dimension ≤ n − 1. So,
according to Claim 1, there exists a map g : N(U) → M such that
1
g is δ/2-homotopic to q ◦h and K is contained in the union of trivial
1 2
components of the fibers of g . Since allfibers of f are of diameter < η
1 U
and f (Z ) ⊂ K, the map g = g ◦f belongs to H(Z ,η). Moreover,
U i 1 U i
g is obviously simplicially factorizable and δ-homotopic to g . This
0
completes the proof of Claim 2.
Finally, the proof of Proposition 2.4 follows from Lemma 2.2 (with
G being the set of all simplicially factorizable maps from C(X,M) and
U(i) = H(Z ,1/i) for every i ≥ 1) and Claim 2. (cid:3)
i
3. Parametric Lelek maps
In this section we are going to prove Theorem 1.1. Everywhere in
this section we suppose that (M,̺) is a given complete metric space
and µ = {W : ν ∈ Λ}, µ = {G : ν ∈ Λ} are locally finite open covers
ν 1 ν
of M such that
(*) G ⊂ W and W ∈ AP(n ,0) with 0 ≤ n for every ν ∈ Λ.
ν ν ν ν ν
Obviously, Theorem 1.1 follows directly from next theorem.
Theorem 3.1. Let f: X → Y be a perfect map between metrizable
spaces with dim△(f) ≤ m. Suppose (M,̺) is a complete metric space
and µ, µ two locally finite open covers of M satisfying condition (∗).
1
Then there is a G -set H ⊂ C(X,M) such that any simplicially fac-
δ
torizable map in C(X,M) can be homotopically approximated by maps
from H and every g ∈ H has the following property: for any y ∈ Y and
ν ∈ Λ the restriction g| f−1(y)∩g−1(G ) is an (m−n )-dimensional
ν ν
Lelek map from f−1(y)∩g−1(G ) into G .
(cid:0) ν ν(cid:1)
The proof of Theorem 3.1 consists of few propositions. We can as-
sume that n ≤ m for all ν ∈ Λ. For every y ∈ Y, ε > 0, ν ∈ Λ and
ν
g ∈ C(X,M) we denote by F (g,ε,y) the union of all continua C of
ν
diameter ≥ ε such that C ⊂ f−1(y)∩g−1(z ) for some z ∈ G . Note
C C ν
10
that each F (g,ε,y) is compact as a closed subset of f−1(y). Then,
ν
for a fixed η > 0, let H(y,ε,η) be the set of all g ∈ C(X,M) such
that d F (g,ε,y) < η for every ν ∈ Λ. The last inequality means
m−nν ν
that F (g,ε,y) can be covered by an open family γ in X such that
ν
(cid:0) (cid:1)
mesh(γ) < η and no more m − n + 1 elements of γ have a com-
ν
mon point. If V ⊂ Y, then H(V,ε,η) denotes the intersection of all
H(y,ε,η), y ∈ V.
Lemma 3.2. For every y ∈ Y and every g ∈ H(y,ε,η) there exists
a neighborhood V of y in Y and δ > 0 such that if y′ ∈ V and
y y y
̺(h(x),g(x)) < δ for all x ∈ f−1(y′), then h ∈ H(y′,ε,η).
y
Proof. Since µ is locally finite and f−1(y) is compact, for every y ∈ Y
1
and g ∈ C(X,M) there exists a neighborhood O (y) of g(f−1(y)) such
g
that the family Λ (y) = {ν ∈ Λ : O (y)∩G 6= ∅} is finite.
g g ν
Assume the lemma is not true for some y ∈ Y and g ∈ H(y,ε,η).
Then there exists a local base of neighborhoods V of y, points y ∈
i i
V and functions g ∈ C(X,M) such that g |f−1(y ) is 1/i-close to
i i i i
g|f−1(y ) but g 6∈ H(y ,ε,η). It is easily seen that for some k and
i i i
all i ≥ k we have g (f−1(y )) ⊂ O (y) . Consequently, for every i ≥ k
i i g
thereexistsν(i) ∈ Λ (y)suchthatd F (g ,ε,y ) ≥ η (because
g m−nν(i) ν(i) i i
g 6∈ H(y ,ε,η)). So, each F (g ,ε,y ) doesn’t have any open cover
i i ν(i) i i (cid:0) (cid:1)
of mesh < η and order ≤ m − n + 1. Since the family Λ (y) is
ν(i) g
finite, there exists ν(0) ∈ Λ (y) with ν(i) = ν(0) for infinitely many
g
i. With out loss of generality, we may suppose that ν(i) = ν(0) for all
i ≥ k. This implies that g (f−1(y ))∩G 6= ∅, i ≥ k, where G = G .
i i ν(0)
Consequently, g(f−1(y)) ∩ G 6= ∅. Since g ∈ H(y,ε,η), F (g,ε,y)
ν(0)
can be covered by an open family γ in X of order ≤ m − n + 1
ν(0)
and mesh(γ) < η. Let U = ∪γ. To obtain a contradiction, it suffice
to show that F (g ,ε,y ) ⊂ U for some i ≥ k. Indeed, otherwise for
ν(i) i i
every i ≥ k there would exist points x ∈ F (g ,ε,y )\U, z ∈ G, and
i ν(i) i i i
a component C of f−1(y ) ∩ g−1(z ) containing x with diamC ≥ ε.
i i i i i i
Using that P = f−1({y }∞ ∪ {y}) is a compactum, we can suppose
i i=k
that {x }∞ converges to a point x ∈ f−1(y), {z }∞ converges to a
i i=k 0 i i=k
point z ∈ G and {C }∞ (considered as a sequence in the space of all
0 i i=k
closed subsets of P equipped with the Vietoris topology) converges to
a closed set C ⊂ f−1(y)∩g−1(z )). It is easily seen that C is connected
0
and diamC ≥ ε. Hence, C ⊂ F (g,ε,y) ⊂ U. So, x ∈ C ⊂ U for
ν(0) i i
some i, a contradiction. (cid:3)
Now, we are in a position to show that the sets H(Y,ε,η) are open
in C(X,M).