CONTINUOUS SELECTIONS AND APPROXIMATIONS IN α-CONVEX METRIC SPACES
A. Kowalska
Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18, 87–100 Toru´n, Poland
Abstract
In the paper, the notion of a generalized convexity was defined and studied from the view-point of the selection and approximation theory of set-valued maps. We study the simultaneous existence of continuous relative selections and graph-approximations of lower semicontinuous and upper semicontinuous set-valued maps with α-convex values hav- ing nonempty intersection.
Keywords: generalized convexity, selections, relative selections, graph- approximations.
2000 Mathematics Subject Classification: 54C20, 54C60, 54C65.
1. Introduction
In the paper we study criteria for the existence of the so-called relative graph-approximations, selections and approximate selections of set-valued maps admitting closed and α-convex values in an α-convex metric space.
The notion of generalized convexity under consideration constitutes a slight generalization of α-convexity defined by de Blasi and Pianigiani (see [3, 4]).
Given an upper semicontinuous map ϕ : X ( Y , where X is a finite- dimensional metric space and Y is a complete α-convex metric space, a lower semicontinuous map ψ : X ( Y (both with closed and α-convex values) such that, ϕ(x)∩ ψ(x) 6= ∅, a closed set A ⊂ X and ε > 0, we establish the existence of δ > 0 such that any continuous map f : A → Y being simultaneously a selection of ψ and a δ-approximation of ϕ admits a continuous extension f∗ : X → Y such that f∗ is a selection of ψ and
an ε-approximation of ϕ. This relative generalized version of (a convex valued) result due to Ben-El-Mechaiekh and Kryszewski (see [1]) consti- tutes also an α-convex counterpart of a result from [16] and those from [14]. Moreover, we prove a result concerning the existence of the so-called strong approximations (see e.g. [7]) of α-convex upper semicontinuous maps.
As in the case of any approximation results, the theory presented here has important applications in the fixed point theory.
2. Notation and preliminaries
Let X be a metric space with a distance d. Given a ∈ X and r > 0, by BX(a, r) (resp. DX(a, r)) we denote an open (resp. closed) ball in X centered at a with the radius r; if A ⊂ X, then BX(A, r) := {x ∈ X | d(x, A) := infa∈Ad(x, a) < r}. By A we denote the closure of A, and by diam (A) := supa, b∈Ad(a, b) its diameter. If A, B ⊂ X, then D(A, B) :=
max{D+(A, B), D−(A, B)}, where D+(A, B) := supb∈Bd(b, A) and D−(A, B) := supa∈Ad(a, B), is the Hausdorff distance between A and B.
Recall (see [10]) that the covering dimension dim X ≤ p, where p ≥ −1, if any covering V admits a refinement V0 of order ≤ p, i.e., such that each collection of p + 2 elements from V0 has an empty intersection. If we write a covering, we mean an open covering. By the star of a set A ⊂ X with respect to a coveringA = {Aj}j∈J (open or closed) of X we understand the set st(A,A) :=S
{Aj | Aj∩ A 6= ∅}.
Let X and Y be metric spaces. By a set-valued map F : X ( Y we understand a function which assigns to any x ∈ X a nonempty set F (x)⊂ Y . A map F : X ( Y is lower semicontinuous (abbr. l.s.c.) (resp.
upper semicontinuous (abbr. u.s.c.)) if, for every open (resp. closed) A⊂ Y , the preimage F−1(A) := {x ∈ X|F (x) ∩ A 6= ∅} is open (resp. closed) in X. We say that F : X ( Y is D-lower semicontinuous at a point x0 ∈ X if, for any ε > 0, there exists δ > 0 such that D+(F (x), F (x0)) < ε for all x∈ BX(x0, δ). By the graph of F we understand the set
Gr (F ) :={(x, y) ∈ X × Y | y ∈ F (x)}.
Let ε > 0. We say that a function f : X → Y is a selection (resp. an ε-selection) of a map F : X ( Y if f (x)∈ F (x) (resp. f(x) ∈ BY(F (x), ε)) for all x ∈ X. A function f is an ε-graph-approximation (resp. a (µ(·), ε)- approximation, where µ : X → (0, ∞) is a continuous function) of F if
the graph Gr (f ) of f is contained in the ε-neighborhood of the graph F , i.e., f (x)∈ B(F (B(x, ε)), ε) (resp. f(x) ∈ B(F (B(x, µ(x))), ε)) for each x ∈ X.
LetU be a neighborhood of the graph of the map F in X × Y and let A be a subset of Y . A map f : A→ Y is called an U-approximation of F over A provided Gr (f )⊂ U.
Given a function f : X → [0, 1], supp f := {x ∈ X | f(x) 6= 0} is the support of f and carr f :={x ∈ X | f(x) > 0} is the carrier of f.
The following two propositions are well-known, but are included for the sake of completeness of the paper.
Proposition 2.1 (see [9]). Let X, Y be metric spaces. Let {Aβ}β∈B be a covering of X and let {gβ}β∈B be family continuous functions gβ : Aβ → Y such that, for every β, β0∈ B with Aβ∩ Aβ0 6= ∅,
gβ(x) = gβ0(x) f or every x∈ Aβ∩ Aβ0.
Then there is a unique continuous function g : X → Y such that, for all β ∈ B,
g(x) = gβ(x) f or every x∈ Aβ.
Proposition 2.2 (see [4]). Let X, Y be metric spaces. Let a map ψ : X ( Y be lower semicontinuous, a function f : X → Y continuous and ε > dY(f (x), ψ(x)) for every x∈ X. Then the map Φ : X ( Y given by
Φ(x) := ψ(x)∩ B(f(x), ε), x ∈ X, is lower semicontinuous.
Now we are going to introduce the concept of an α-convex metric space. An earlier version of this notion has been presented by F.S. de Blasi and G.
Pianigiani in [3].
Definition 2.3.By an α-convex metric space we mean a pair (Y, α), where (Y, d) is a metric space and α : Y × Y × [0, 1] → Y is a continuous function such that, for all x, y∈ Y and t ∈ [0, 1],
(i) α(x, y, 0) = x;
(ii) α(x, y, 1) = y;
(iii) α(x, x, t) = x;
and, moreover, satisfying the following Hausdorff continuity property:
(iv) for any ε > 0, there is η > 0 such that, for any x1, x2, y1, y2 ∈ Y , if d(x1, y1) < η and d(x2, y2) < η, then
D(Λα(x1, x2; [0, 1]), Λα(y1, y2; [0, 1])) < ε,
where, for any x, y∈ Y and t1, t2 ∈ [0, 1], Λα(x, y; [t1, t2]) :={α(x, y, t) | t ∈ [t1, t2]}.
In their definition, the authors in [3] require a stronger version of the above mentioned continuity and they demand that:
(∗) for all ε > 0, there is η ∈ (0, ε] such that, for any x1, x2, y1, y2 ∈ Y , if d(x1, y1) < ε and d(x2, y2) < η, then
D(Λα(x1, x2; [0, 1]), Λα(y1, y2; [0, 1])) < ε.
Our setting seems to be more natural, since – contrary to the definition of de Blasi and Pianigiani – it is just an equivalent of the usual Hausdorff continuity of the map X× X 3 (x, y) 7( Λα(x, y; [0, 1]). Consider also the following example.
Example 2.4.Consider the two-dimensional sphere S2:={x = (x1, x2, x2)
∈ R3 | kxk = 1} and denote by d the metric on S2 induced by the Eu- clidean norm k · k in R3. Let Y = {x ∈ S2 | x21 + x22 < sin2ρ}, where ρ ∈ (0,π2). It is easy to see that Y is nothing else but the set of points whose northern altitude is greater than ρ. Define α : Y × Y × [0, 1] → X for x = (x1, x2, x3), y = (y1, y2, y3)∈ Y by the formula
α((x1, y1, z1), (x2, y2, z2), t) =
x(t), y(t),p
1− x2(t)− y2(t) , where
x(t) = |B|
√A2+ B2 cos gt1,t2(t) +
r B2+ C2
A2+ B2+ C2 − A2
A2+ B2sin gt1,t2(t), y(t) = |A|
√A2+ B2 cos gt1,t2(t) +
r A2+ C2
A2+ B2+ C2 − B2
A2+ B2 sin gt1,t2(t), A = y1z2− y2z1, B = x2z1− x1z2, C = x1y2− x2y1,
gt1,t2(t) =
arctg
t
tg2πt2− π2
2π− 4ρ − tg2πt1− π2 2π− 4ρ
+ tg2πt1− π2 2π− 4ρ
+ π2 2π− 4ρ
1−2ρ π
and
t1 = arccos|B|x1+|A|y1
√A2+ B2 , t2= arccos|B|x2+|A|y2
√A2+ B2 .
It is easy to see that, for x, y ∈ Y , the set Λα(x, y; [0, 1]) is the arc of the great circle passing through x and y. It is rather obvious (but requiring tedious computation) that α satisfies properties listed in Definition 2.3. At the same time α does not have property (∗). De Blasi and Pianigiani study a similar example but they have to assume that ρ∈ (0, π4) in order to assure that property (∗) is satisfied.
It is clear that a space Y together with α satisfying property (∗) satisfies (iv).
Therefore all examples given in [3] are α-convex in the sense of Definition 2.3.
Let (Y, α) be an α-convex space. We say that A⊂ X is α-convex if, for any a1, a2 ∈ A and t ∈ [0, 1], α(a1, a2, t) ∈ A. The continuity of α implies that if A⊂ X is α-convex, then so is A.
In what follows the concept of a pseudo-barycenter plays an important role (compare [5]).
Definition 2.5. Let Y be an α-convex metric space. Let n ≥ 1 be an integer. For y1, . . . , yn ∈ Y and (λ1, . . . , λn) ∈ Σn := {(λ1, . . . , λn) | 0 ≤ λi ≤ 1, i = 1, . . . , n, λ1 + . . . + λn = 1}, the corresponding a n-pseudo- barycenter bn(y1, . . . , yn; λ1, . . . , λn) is defined as follows:
b1(y1; 1) := y1
b2(y1, y2; λ1, λ2) := α(y1, y2, λ2) and, for n≥ 3,
bn(y1, . . . , yn; λ1, . . . , λn) :=
:=
( yn, if λn= 1;
α bn−1
y1, . . . , yn−1;1−λλ1
n, . . . ,1−λλn−1
n
, yn, λn
, if λn6= 1.
We shall need the following properties of the pseudo-barycenter (compare [5]).
Property 2.6. For any n≥ 1, the map bn: Yn× Σn→ Y is continuous.
Property 2.7. Let y1, . . . , yn ∈ Y and (λ1, . . . , λn) ∈ Σn, n ≥ 2. Let {i1, . . . , ik}, 1 ≤ k ≤ n − 1, be a subset of {1, . . . , n} with i1 < i2 < . . . < ik such that, for all i∈ {i1, . . . , ik}, λi > 0 and if i∈ {1, . . . , n} \ {i1, . . . , ik}, then λi = 0. Then
bn(y1, . . . , yn; λ1, . . . , λn) = bk(yi1, . . . , yik; λi1, . . . , λik).
In what follows we shall frequently make use of the following formalism.
Assume that (J,≺) is a well-ordered set and let λ : J → [0, 1] be such that the set J0 := {j ∈ J | λ(j) > 0} is finite, i.e., J0 = {j1, . . . , jm}, where ji≺ ji+1 for all i = 1, .., m− 1. For any function y : J → Y , we define
b(y, λ) := bm(y1, . . . , ym; λ1, . . . , λm), (1)
where yi := y(ji) and λi:= λ(ji) for i = 1, . . . , m.
Obviously, by the Zermelo theorem [9], every set can be a well-ordered.
In this paper, we let each set J of indices be a well-ordered set in the sense of a relation≺.
Proposition 2.8. Let X be a metric space, Y be an α-convex metric space and assume that functions y : J → Y and λ : J × X → [0, 1] are given.
Suppose that {carr λ(j, ·)}j∈J is a locally finite covering of X (in particular, for each x∈ X, the set J(x) := {j ∈ J | λ(j, x) > 0} is finite) and, for each j∈ J, the function λ(j, ·) : X → [0, 1] is continuous. Then, for each x ∈ X, we may define
b(y, x) := b(y, λ(·, x)).
The map b(y,·) : X → Y is continuous.
Moreover, if y : J× X → Y is such that, for all j ∈ J, y(j, ·) : X → Y is continuous, then the map
X 3 x 7→ b(y(·, x), λ(·, x)) ∈ Y is also continuous.
P roof. For each j∈ J, let Uj := carr λ(j,·). Obviously Uj is open for any j ∈ J. Moreover, for any x ∈ X, let J(x) := {j ∈ J | λ(j, x) > 0} and mx := #J(x). According to formula (1),
b(y, x) = b(y, λ(·, x)) = bmx(y(j1x), . . . , y(jxmx); λ(j1x, x), . . . ., λ(jmxx, x)), (2)
where J(x) ={jx1, . . . , jmxx} and jix≺ ji+1x for all i = 1, . . . , mx− 1.
Let x0 ∈ X. There exists a neighborhood Ωx0 of x0 such that the set I(x0) := {j ∈ J | Ωx0 ∩ Uj 6= ∅} is non-empty, finite and ordered. Let nx0 := #I(x0).
Observe that, if x ∈ Ωx0, then J(x) ⊂ I(x0). By Property 2.7 and in view of (2), we have
b(y, x) = bnx0(y(j1), . . . , y(jnx0); λ(j1, x), . . . , λ(jnx0, x)), (3)
where I(x0) ={j1, . . . , jnx0} and this sequence increases (in the sense of ≺).
In view of Property 2.6 (in fact the continuity of bn(y1, . . . , yn;·) : Σn → Y is sufficient) and continuity of the function λ(j,·) : X → [0, 1], for all j ∈ J, formula (3) correctly defines a continuous map b(y,·) : Ωx0 → Y .
Now let x0, x00 ∈ X and suppose that x ∈ Ωx0 ∩ Ωx00. Then J(x) ⊂ I(x0)∩ I(x00). Hence, by (3),
b(y,·)|Ωx0(x) = bnx0(y(j1), . . . , y(jnx0); λ(j1, x), . . . , λ(jnx0, x))
= bmx(y(j1x), . . . , y(jmxx); λ(j1x, x), . . . , λ(jxmx, x))
= bnx0
0
(y(j10), . . . , y(jn0
x00
); λ(j10, x), . . . , λ(j0nx0, x)) = b(y,·)|Ωx0
0
(x).
In view of Proposition 2.1, b(y,·) is well-defined and continuous.
The proof of continuity of the function x7→ b(y(·, x), λ(·, x)) is analogous if we use the continuity of the map bn: Yn× Σn→ Y .
Proposition 2.9. Let Y be an α-convex metric space and 0≤ p < ∞ be an integer number. Then, for each ε > 0, there exists δ > 0 such that, for any 1≤ n ≤ p + 1, any y1, . . . , yn∈ Y and any non-empty α-convex set A ⊂ Y , if d(yi, A) < δ for i = 1, . . . , n, then
d(bn(y1, . . . , yn; λ1, . . . , λn), A) < ε for all (λ1, . . . , λn)∈ Σn.
P roof. Let ε > 0. Let δp+1 := η, where η > 0 corresponds to ε as in Definition 2.3. Obviously, we may take η < ε. Suppose that δi > 0, for 2 ≤ i ≤ p, is defined. By definition, there is δi−1 > 0 such that δi−1 ≤ δi
and, for all x1, x2, y1, y2 ∈ Y , if d(x1, y1), d(x2, y2) < δi−1, then D(Λα(x1, x2; [0, 1]), Λα(y1, y2; [0, 1])) < δi. In this manner the number δ := δ1 is well-defined.
Let 1≤ n ≤ p+1. Take a set A ⊂ Y and arbitrary points y1, . . . , yn∈ Y such that d(yi, A) < δ for i = 1, 2, . . . , n. We shall show that
d(bn(y1, . . . , yn; λ1, . . . , λn), A) < δn.
If n = 1, then b1(y1; 1) = y1, and hence d(b1(y1; 1), A) < δ = δ1. Let n > 1 and assume that, for any (µ1, . . . , µn)∈ Σn,
d(bn(y1, . . . , yn; µ1, . . . , µn), A) < δn. Hence there is xn∈ A such that
d(bn(y1, . . . , yn; µ1, . . . , µn), xn) < δn. (4)
Since d(yn+1, A) < δ≤ δn, there is x0n∈ A such that d(yn+1, x0n) < δn. (5)
Let (λ1, . . . , λn+1)∈ Σn+1. If λn+1 < 1, then
bn+1(y1, . . . , yn+1; λ1, . . . , λn+1) = α(bn(y1, . . . , yn; µ1, . . . , µn), yn+1, λn+1), where µi := λi(1− λn+1)−1. By (4) and (5) and the definition of δn, we see that
d(bn+1(y1, . . . , yn+1; λ1, . . . , λn+1), A)
≤ D(Λα(bn(y1, . . . , yn; µ1, . . . , µn), yn+1; [0, 1]), Λα(xn, x0n; [0, 1]) < δn+1
since λα(xn, x0n; [0, 1]) ⊂ A. If λn+1 = 1, then bn+1(y1, . . . , yn+1; λ1, . . . , λn+1) = yn+1, and then again d(bn+1(y1, . . . , yn+1; λ1, . . . , λn+1), A) < δn≤ δn+1 by (5).
Observe that if n = 2, then we can take δ := η from Definition 2.3.
3. Continuous selections and approximations
It is an important fact in this section that, for every continuous function defined on a closed subset of a finite-dimensional metric space with values in an α-convex metric space, there exists a continuous extension onto the whole domain. It means that an α-convex metric space is an ”almost”
absolute retract (in the paper [2] it is proved that an α-convex metric space defined by De Blasi and Pianigiani is an absolute retract).
Let us sketch the proof:
Let X be a metric space such that dim X = p < ∞ and Y be an α- convex metric space. Let A⊂ X be a closed set. Take a continuous function f : A→ Y .
Consider the so-called Dugundji system {Uj, aj}j∈J for X \ A (see [6]), i.e.,
(i) Uj ⊂ X \ A, aj ∈ A, j ∈ J;
(ii) {Uj}j∈J is a locally finite covering of X\ A;
(iii) if x∈ Uj, then dX(x, aj)≤ 2dX(x, A), j ∈ J.
Without loss of generality we may assume that {Uj}j∈J is such that every collection consisting of p + 2 elements Uj1, . . . , Ujp+2 of this covering has empty intersection. Let {λj}j∈J be a continuous partition of unity subordi- nate to {Uj}j∈J. Define functions λ : J × [X \ A] → [0, 1] and y : J → Y putting λ(j, x) := λj(x) and y(j) := f (aj) for j ∈ J and x ∈ X \ A.
Hence, in view of the Proposition 2.8, we have a continuous function b(y,·) : X\ A → Y .
We define the continuous extension f : X → Y of f by the formula
f (x) =
( f (x), for x∈ A;
b(y, λ(·, x)), for x∈ X \ A.
In particular, by the above assertion, it follows that, for each α-convex closed subset T of an α-convex metric space such that T ⊂ M and dim M < ∞,
there exists a retraction r : M → T as an extension of an identical function idT : T → T .
Let us start with the following result saying that any locally selectionable set-valued map with α-convex values admits a continuous selection.
Theorem 3.1. Let X be a metric space, dim X < ∞, and let Y be an α- convex space. If a set-valued map F : X ( Y has α-convex values and, for any point x∈ X, there is a neighborhood Vx and a continuous function fx: Vx → Y such that fx(y)∈ F (y) for y ∈ Vx, then F admits a continuous selection.
P roof. Let W := {Wj}j∈J be a locally finite refinement of the covering V := {Vx}x∈X and {λj}j∈J be a continuous partition of unity subordinate to W.
For all j ∈ J, let xj ∈ X be such that Wj ⊂ Vxj. Without loss of generality we may assume that, for each j ∈ J, Wj ⊂ Vxj. Since for fxj : Wj → Y there exists an extension to the whole space X, we may assume that fxj is actually defined on X. Set fj(x) := fxj(x) for x∈ X.
Let λ : J× X → [0, 1] be given by λ(j, x) = λj(x), let y : J × X → Y be given by y(j, x) = fj(x). Then all assumptions from Proposition 2.8 are satisfied, hence we may define
f (x) = b(y(·, x), λ(·, x)), x ∈ X.
Clearly, f : X → Y is well-defined and continuous. Let x ∈ X. We have that the set {j ∈ J | λ(j, x) > 0} = {j1, . . . , jm} is non-empty, finite and well- ordered. Since, for each i = 1, . . . , m, fji(x)∈ F (x) and F (x) is α-convex, we see that f (x)∈ F (x).
A theorem below shows a connection between a graph-approximation and a selection of set-values maps.
Theorem 3.2. Let X be a metric space such that dim X = p < ∞ and Y an α-convex metric space. Let ψ : X ( Y be a lower semicontinuous map with α-convex values, and ϕ : X ( Y be an upper semicontinuous map with α-convex values such that ϕ(x)∩ ψ(x) 6= ∅ for each x ∈ X. Then, for any ε > 0 and a continuous function µ : X → (0, ∞), there is a continuous map f : X → Y such that f is an ε-selection of the map ψ and a (µ(·), ε)- approximation of the map ϕ.
P roof. Let ε > 0 and µ : X → (0, ∞) be a continuous function. By Proposition 2.9, there exists 0 < δ ≤ ε such that, for every y1, . . . , yk ∈ Y , 1 ≤ k ≤ p + 1, and a non-empty α-convex set A ⊂ Y , if dY(yi, A) < δ, i = 1, . . . , k, then
dY(bk(y1, . . . , yk; λ1, . . . , λk), A) < ε for any (λ1, . . . , λk)∈ Σk.
For x∈ X, let
U (x) := µ−1(µ(x)/2,∞) ∩ BX(x, µ(x)/2)∩ {x0 ∈ X | ϕ(x0)⊂ BY(ϕ(x), δ)}.
Then, in view of the upper semicontinuity of ϕ and continuity of µ, for each x ∈ X, the set U(x) is open. Thus U := {U(x)}x∈X is a covering of X. Since dim X = p, there exists an open refinement U0 of U such that every collection consisting of p + 2 elements U10, . . . , Up+20 ofU0 has an empty intersection. Next let V := {V } be an open star-refinement of U0, i.e., for any V ∈ V, there is U0 ∈ U0 such that
st(V,V) =[
{W ∈ V | W ∩ V 6= ∅} ⊂ U0.
Without loss of generality we may assume that the covering V is such that each collection of p + 2 elements fromV has an empty intersection.
For any x ∈ X, choose zx ∈ ϕ(x) ∩ ψ(x) and consider the covering T := {TV(x)}V ∈V, x∈X of X, where
TV(x) :={x0 ∈ V | ψ(x0)∩ BY(zx, δ)6= ∅}.
Obviously, by lower continuity of ψ, the set TV(x) is open for every V ∈ V and x∈ X. Let {λj}j∈J be a partition of unity subordinated toT . Hence, for each j ∈ J, there are Vj ∈ V and xj ∈ Vj such that supp λj ⊂ TVj(xj).
Define functions λ : J × X → [0, 1] and y : J → Y putting λ(j, x) :=
λj(x) and y(j) := zj := zxj for j ∈ J and x ∈ X.
Obviously, all assumptions of Proposition 2.8 are satisfied and we may define a map f : X → Y by the formula
f (x) := b(y, λ(·, x)).
This map is well-defined and continuous.
Let x∈ X. Since, for each j ∈ J(x) := {j ∈ J | λ(j, x) > 0}, x ∈ supp λj ⊂ TVj(xj), thus ψ(x)∩ BY(zj, δ) 6= ∅. Therefore dY(zj, ψ(x)) < δ and this implies that
d(f (x), ψ(x)) < ε.
On the other hand, x∈ TVj(xj)⊂ Vj and xj ∈ Vj for all j ∈ J(x). Hence, using the fact that V is a star refinement of U0, there is U0 ∈ U0 such that x, xj ∈ U0 for all j ∈ J(x). Since U0 is a refinement of U, there is x ∈ X such that x, xj ∈ U(x) for all j ∈ J(x). Consequently, for any such j, zj ∈ ϕ(xj) and ϕ(xj) ⊂ BY(ϕ(x), δ), i.e., dY(zj, ϕ(x)) < δ. Since the value ϕ(x) is α-convex, this implies that dY(f (x), ϕ(x)) < ε. Moreover, x ∈ µ−1(µ(x)/2,∞) ∩ BX(x, µ(x)/2), then dX(x, x) < µ(x)2 < µ(x). This completes the proof.
Observe that if µ(x) = ε, for all x ∈ X, then the above function f is an ε-graph-approximation of ϕ.
Remark 3.3. Theorem 3.2 is a direct generalization of Lemma 5.1 from [1].
Below we shall show a result even stronger than Theorem 3.2. Namely, we shall show the following theorem.
Theorem 3.4.Under the assumption of Theorem 3.2, suppose that the space Y is complete and, for each x∈ X, the set ψ(x) is closed. Then, for every ε > 0 and a continuous function µ : X → (0, ∞), there exists a continuous selection f : X → Y of ψ such that f is a (µ(·), ε)-approximation of ϕ.
P roof.Let ε > 0 and let µ : X → (0, ∞) be a continuous function.
By Proposition 2.9, there exist strictly decreasing sequences (δn), (θn) such that, for any n≥ 1,
0 < 2δn< θn< 2εn,
where εn:= 2−n−1ε, and the following two conditions (C1), (C2) are satisfied.
(C1) For every α-convex A ⊂ Y , y1, . . . , yk ∈ Y and (λ1, . . . , λk) ∈ Σk, where 1 ≤ k ≤ p + 1, if dY(yi, A) < θn for all i = 1, . . . , k, then dY(bk(y1, . . . , yk; λ1, . . . , λk), A) < εn.
(C2) For every α-convex A ⊂ Y , y1, . . . , yk ∈ Y and (λ1, . . . , λk) ∈ Σk, where 1 ≤ k ≤ p + 1, if dY(yi, A) < δn for all i = 1, . . . , k, then dY(bk(y1, . . . , yk; λ1, . . . , λk), A) < θn/2.
By Theorem 3.2, there exists a continuous function g : X → Y such that, for every x∈ X,
B(g(x), θ1/2)∩ ψ(x) 6= ∅, (6)
g(x)∈ B(ϕ(B(x, µ(x))), θ1/2).
(7)
Now we shall construct a sequence (fn)∞n=1 of continuous functions fn : X → Y such that, for every x ∈ X,
dY(fn(x), ψ(x)) < θn/2, n≥ 1, (8)
dY(fn(x), fn−1(x)) < εn−1, n≥ 2.
(9)
Let f1 := g. Suppose that n≥ 2 and that continuous functions fi : X → Y , i = 1, . . . , n− 1, have been defined. In order to construct fn : X → Y , let Φn: X ( Y be given by
Φn(x) := ψ(x)∩ B(fn−1(x), θn−1/2) for x∈ X.
It is clear that Φn(x) has nonempty values and, in view of Proposition 2.2, is l.s.c. For each y∈ Y0 := ψ(X), let
Uy :={x ∈ X | Φn(x)∩ B(y, δn)6= ∅}.
Then, in view of the lower semicontinuity of Φn(x), U := {Uy}y∈Y0 is an open covering of X. Since dim X ≤ p, there exists an open refinement V := {Vj}j∈J of U such that every collection consisting of p + 2 elements V1, . . . , Vp+2 of V has an empty intersection.
Let {λj}j∈J be a partition of unity subordinate to V. As before we assume that the set J is well-ordered. For each j ∈ J, there is Vj ∈ V and y(j)∈ Y0 such that supp λj ⊂ Vj ⊂ Uy(j).
Define function λ : J × X → [0, 1] putting λ(j, x) := λj(x) for j ∈ J and x∈ X. Because the function λ satisfies assumptions of Proposition 2.8, then we may define a map fn: X → Y by the formula
fn(x) := b(y, λ(·, x)).
By Proposition 2.8, fn is well-defined and continuous.
We shall show that conditions (11), (12) hold. Indeed, let x∈ X. For each j ∈ J, if x ∈ supp λj, then x ∈ Vj ⊂ Uy(j). Hence Φn(x)∩ B(y(j), δn) 6= ∅ and therefore
dY(y(j), ψ(x)) < δn. (10)
Observe that #{j ∈ J | x ∈ supp λ(j, ·)} ≤ p + 1. By (C2), we have dY(fn(x), ψ(x)) < θn/2,
i.e., fn satisfies condition (11).
Since x∈ Uy(j), then B(fn−1(x), θn−1/2)∩ B(y(j), δn) 6= ∅ and we get that
dY(y(j), fn−1(x)) < δn+ θn−1/2 < θn/2 + θn−1/2 < θn−1. Then, by (C1),
dY(fn(x), fn−1(x)) < εn−1.
This shows condition (12) and inductively completes the construction of the sequence (fn)∞n=1.
Since, for each x∈ X, (fn(x))∞n=1 is a Cauchy sequence and the space Y is complete, (fn(x))∞n=1 converges to a point f (x)∈ Y . Because the sequence (fn)∞n=1is uniformly convergent, then the function f : X → Y is continuous.
Moreover, it is clear that f (x)∈ ψ(x) for any x ∈ X.
For every x∈ X, we have dY(f1(x), f (x)) = lim
n→∞dY(f1(x), fn+1(x))
≤ limn→∞
Xn k=1
dY(fk(x), fk+1(x)) <
X∞ k=1
εk= ε X∞ k=1
2−k−1 = ε/2.
By (7),
f (x)∈ B(ϕ(B(x, µ(x))), θ1/2 + ε/2)⊂ B(ϕ(B(x, µ(x))), ε).
The proof is complete.
Remark 3.5. Observe that if in Theorem 3.4 ψ(x) = Y , for every x ∈ X, than we have generalization of the Cellina theorem (see [7]), and if ϕ(x) = Y , for each x∈ X, than we have the Michael type continuous selection theorem (see [15]).
Now we present the relative graph-approximations theorem.
Theorem 3.6.Let X be a metric space, dim X <∞, and Y be an α-convex metric space. Let ψ : X ( Y be a D-lower semicontinuous map with α- convex values and ϕ : X ( Y an upper semicontinuous map with α-convex closed values such that ϕ(x)∩ ψ(x) 6= ∅ for each x ∈ X. Let A ⊂ X be any closed set.Then, for any ε > 0 and 0 < δ < η, where η≤ ε corresponds to ε as in Definition 2.3, we have that
(i) each δ-selection f : A→ Y of ϕ such that f is a selection of ψ extends to a map f : X → Y such that f is an ε-graph-approximation of ϕ and an ε-selection of ψ,
(ii) there exists a continuous function δ : X → (0, ∞) such that any δ(·)- graph-approximation f : A → Y of ϕ such that f is a selection of ψ extends to a map f : X → Y such that f is an ε-graph-approximation of ϕ and an ε-selection of ψ.
P roof. Let ε > 0 and 0 < δ < η, where η 6 ε corresponds to ε as in Definition 2.3.
Step 1. We shall construct a continuous function µ : X → (0, ∞) such that, for each x∈ X, there is x ∈ BX(x, η) such that
B(ϕ(B(x, µ(x))), δ)⊂ B(ϕ(x), η).
Because of ϕ is u.s.c., then we can, for each x ∈ X, choose rx ∈ (0, η) so that
ϕ(B(x, 2rx))⊂ B(ϕ(x), η − δ).
Let {λj}j∈J be a partition of unity subordinated to the covering {B(x, rx)}x∈X, that means, for each j ∈ J, there is a point xj such that supp λj ⊂ B(xj, rj), where rj := rxj, and P
j∈Jλj(x) = 1 for each x∈ X.
Put
µ(x) :=X
j∈J
λj(x)rj.
Let x∈ X, then there is j ∈ J such that λj(x) > 0 and µ(x) ≤ rj. Since λj(x) > 0, then x∈ B(xj, rj), i.e., dX(x, xj) < rj. Thus
B(x, µ(x))⊂ B(xj, 2rj), and hence
ϕ(B(x, µ(x)))⊂ ϕ(B(xj, 2rj))⊂ B(ϕ(xj), η− δ).
We have that
B(ϕ(B(x, µ(x))), δ)⊂ B(ϕ(xj), η) and dX(x, xj) < rj < η.
We finish the proof if we put x = xj. Step 2. For each (x, y)∈ X × Y , put
Uδ(x, y) := [µ−1((µ(x)/2,∞)) ∩ B(x, µ(x)/2)] × B(y, δ) and let
Uϕδ := [
(x,y)∈Gr(ϕ)
Uδ(x, y).
We shall show that there is a continuous function δ : X → (0, ∞) such that any δ(·)-graph-approximation of ϕ over A is an Uϕδ-approximation of ϕ over A.
The upper semicontinuity of ϕ implies that, for any x∈ X, there exists a number r(x) > 0 such that
B(x, r(x))× B(ϕ(B(x, 2r(x))), r(x)) ⊂ Uϕδ.
Let {λj}j∈J be a partition of unity inscribed into the covering {B(x, r(x))}x∈X. Hence, for each j∈ J, there is xj ∈ X such that supp λj ⊂ B(xj, r(xj)). Let rj := r(xj) and define
δ(x) :=X
j∈J
λj(x)rj, x∈ X.
Suppose that f : A→ Y is a δ(·)-graph-approximation of ϕ.
Take x ∈ A. There is j ∈ J such that λj(x) > 0 and δ(x) ≤ rj. Thus x∈ B(xj, rj)). Since
f (x)∈ B(ϕ(B(x, δ(x))), δ(x)),
then there is x0 ∈ B(x, δ(x)) and y0∈ ϕ(x0) such that f (x)∈ B(y0, δ(x)).
Therefore y0 ∈ ϕ(B(xj, 2rj)) and f (x) ∈ B(ϕ(B(xj, 2rj)), rj). Altogether we have
(x, f (x))∈ B(xj, rj)× B(ϕ(B(xj, 2rj)), rj)⊂ Uϕδ.
Step 3. Take any function f : A → Y such that f is a δ-selection of ϕ (adequately a δ(·)-graph-approximation of ϕ) and a selection of ψ.
We shall construct ef : X → Y a continuous extension of f such that ef is a δ-selection of ψ.
Let 0 < ξ ≤ δ be a number such that, for any x1, x2, y1, y2 ∈ Y , if d(x1, y1) < ξ and d(x2, y2) < ξ, then
D(Λα(x1, x2; [0, 1]), Λα(y1, y2; [0, 1])) < δ.
Since ψ is D-lower semicontinuous, then, for each z ∈ X, there is a neigh- bourhood Uz such that D+(ψ(x), ψ(z)) < ξ2 for all x ∈ Uz. For each (z, y)∈ Gr (ψ), let
Vξ(z, y) := Uz× B(y, ξ/2).
Obviously, Vξ(z, y) is a neighbourhood of the point (z, y). Let Uψξ := [
(x,y)∈Gr (ψ)
Vξ(x, y).
Let f∗ : X→ Y be any extension of f. Since f∗|A is a selection of ψ, then, in particular, it is anUψξ-approximation of ψ over A. Because Uψξ is an open set in X × Y , then there exists an open neighbourhood M of A such that f∗|M : M → Y is an Uψξ-approximation of ψ over M .
Let x ∈ M. We have that (x, f∗(x)) ∈ Uψξ, this means that there exist z ∈ X and y ∈ ψ(z) such that x ∈ Uz and f∗(x) ∈ B(y, ξ/2). Thus D+(ψ(x), ψ(z)) < ξ2 and dY(f∗(x), y) < ξ2. Hence y∈ ψ(z) ⊂ B(ψ(x), ξ/2) and
dY(f∗(x), ψ(x))≤ dY(f∗(x), y) + dY(y, ψ(x)) < 2ξ 2 = ξ.
Therefore f∗|M is a ξ-selection of ψ over M .
Take any ξ-selection g : X → Y of the map ψ. Such function exists, by Theorem 3.2.
Let V be an open neighbourhood of the set A such that A⊂ V ⊂ V ⊂ M and let{β, κ} be a continuous partition of unity subordinate to {M, X \ V }, that means supp β⊂ M, supp κ ⊂ X \ V and β(x) + κ(x) = 1 for all x ∈ X.
Define a continuous function ef : X → Y as follows f(x) = α(fe ∗(x), g(x), κ(x)).
Let x ∈ X. If x 6∈ M, then β(x) = 0 and κ(x) = 1, thus ef (x) = g(x). We have
f (x) = g(x)e ∈ BY(ψ(x), ξ)⊂ BY(ψ(x), δ).
If x∈ M, then f∗(x), g(x)∈ BY(ψ(x), ξ) and, by Proposition 2.9, we have f (x)e ∈ BY(ψ(x), δ).
Step 4. We shall show that ef|W : W → Y is a (µ(·), δ)-approximation of ϕ over an open neighbourhood W of the set A.
Let x ∈ A. Because ef(x) ∈ BY(ϕ(x), δ), then there is y ∈ ϕ(x) such that ef(x) ∈ BY(y, δ). Therefore (x, ef (x)) ∈ Uδ(x, y) ⊂ Uϕδ (if ef|A is a δ(·)-graph-approximation of ϕ, then it is Uϕδ-approximation, by step 2).
Since a Uϕδ is an open set in X× Y , then there exists an open neigh- bourhood W of A such that ef|W : W → Y is an Uϕδ-approximation of ϕ over W .
Let w ∈ W . Since (w, ef (w)) ∈ Uϕδ, then there is (x, y) ∈ Gr (ϕ) such that w ∈ µ−1((µ(x)/2,∞)) ∩ B(x, µ(x)/2) and ef(w) ∈ B(y, δ). Thus we have that ef(w)∈ B(y, δ) and y ∈ ϕ(x), where dX(x, w) < µ(x)2 < µ(w).
Therefore ef(w)∈ BY(ϕ(BX(w, µ(w))), δ) for all w∈ W .
Step 5. By above steps we have the continuous function ef : X → Y such that ef|A= f as well as ef|W is a (µ(·), δ)-approximation of ϕ and a δ-selection of ψ.
Take an open neighbourhood V of A such that A⊂ V ⊂ V ⊂ W . Let {β, κ} be a partition of unity inscribed into the covering {W, X \ V }, i.e., β, κ : X → [0, 1] are continuous and supp β ⊂ W, supp κ ⊂ X \ V and β(x) + κ(x) = 1 for all x∈ X.
By Theorem 3.2 there exists a continuous function g : X → Y which is a (µ(·), δ)-approximation of ϕ and a δ-selection of ψ. Define f : X → Y as follows
f (x) = α( ef (x), g(x), κ(x)).
Obviously f is continuous. Let x ∈ X. If x 6∈ W , then β(x) = 0 and κ(x) = 1, thus f(x) = g(x). We have
f(x) = g(x)∈ BY(ψ(x), δ)⊂ BY(ψ(x), ε) and, by Step 1,
f(x) = g(x)∈ BY(ϕ(BX(x, µ(x))), δ) ⊂ BY(ϕ(BX(x, η)), η)
⊂ BY(ϕ(BX(x, ε)), ε).
If x∈ W , then
f(x), g(x)e ∈ BY(ϕ(BX(x, µ(x))), δ)∩ BY(ψ(x), δ).
By Step 1, there is x∈ BX(x, η) such that
BY(ϕ(BX(x, µ(x))), δ) ⊂ BY(ϕ(x), η).
ϕ(x) is α-convex, so, by Proposition 2.9, we have
f ∈ BY(ϕ(x), ε)⊂ BY(ϕ(BX(x, ε)), ε) and also, since BY(ψ(x), δ)⊂ BY(ψ(x), η), then
f ∈ BY(ψ(x), ε).
Assumptions of Theorem 3.6 state that the map ψ is D-lower semicontinu- ous, because it was necessary in order to get the existence of an extension of a selection of the mapping to a δ-selection. But we know that D-lower semicontinuity implies lower semicontinuity and if values of the map ψ are compact, then we have a reverse implication.
Remark 3.7. Observe that Theorem 3.6 may be formulated as follows.
Let X be a metric space, dim X < ∞, and Y be an α-convex metric space. Let ψ : X ( Y be a D-lower semicontinuous map with α-convex values and ϕ : X ( Y an upper semicontinuous map with α-convex closed values such that ϕ(x)∩ ψ(x) 6= ∅ for each x ∈ X. Let A ⊂ X be any closed set. Then, for any ε, ι > 0 and 0 < δ < η, where η ≤ ε corresponds to ε as in Definition 2.3, we have that
(i) each δ-selection f : A→ Y of ϕ such that f is a selection of ψ extends to a map f : X→ Y such that f is an ε-graph-approximation of ϕ and an ι-selection of ψ,
(ii) there exists a continuous function δ : X → (0, ∞) such that any δ(·)- graph-approximation f : A → Y of ϕ such that f is a selection of ψ extends to a map f : X → Y such that f is an ε-graph-approximation of ϕ and an ι-selection of ψ.
Theorem 3.8.Let X be a metric space, dim X <∞, and Y be an α-convex complete metric space. Let ψ : X ( Y be a D-lower semicontinuous map with α-convex closed values and ϕ : X ( Y an upper semicontinuous map with α-convex closed values such that ϕ(x)∩ ψ(x) 6= ∅ for each x ∈ X. Let A ⊂ X be a non-empty closed set. Then, for any ε > 0 and 0 < δ < η, where η≤ ε2 corresponds to ε2 as in Definition 2.3, we have that
(i) each δ-selection f : A→ Y of ϕ such that f is a selection of ψ extends to a map f : X → Y such that f is an ε-graph-approximation of ϕ and a selection of ψ,
(ii) there exists a continuous function δ : X → (0, ∞) such that any δ(·)- graph-approximation f : A → Y of ϕ such that f is a selection of ψ extends to a map f : X → Y such that f is an ε-graph-approximation of ϕ and a selection of ψ.
P roof. This proof is analogous to the proof of Theorem 3.4. Let ε > 0 and 0 < δ < η, where η ≤ 2ε corresponds to 2ε. Let A⊂ X be a non-empty closed set.
By Proposition 2.9, there exist strictly decreasing sequences (δn), (θn) such that, for any n≥ 1,
0 < 2δn< θn< 2εn,
where εn := 2−n−1ε, and the following two conditions (C1), (C2) are satis- fied.
(C1) For every α-convex A ⊂ Y , y1, . . . , yk ∈ Y and (λ1, . . . , λk) ∈ Σk, where 1 ≤ k ≤ p + 1, if dY(yi, A) < θn for all i = 1, . . . , k, then dY(bk(y1, . . . , yk; λ1, . . . , λk), A) < εn.
(C2) For every α-convex A ⊂ Y , y1, . . . , yk ∈ Y and (λ1, . . . , λk) ∈ Σk, where 1 ≤ k ≤ p + 1, if dY(yi, A) < δn for all i = 1, . . . , k, then dY(bk(y1, . . . , yk; λ1, . . . , λk), A) < θn/2.
By Theorem 3.6 (in fact Remark 3.7), there exists a continuous function δ : X → (0, ∞) such that any δ(·)-graph-approximation g : A → Y of ϕ such that g is a selection of ψ extends to a map g : X → Y such that g is an ε2-graph-approximation of ϕ and a θ21-selection of ψ.
Let f : A→ Y be a δ-selection (resp. δ(·)-graph-approximation) of ϕ and a selection of ψ. By the above theorem, we have an extension g : X → Y such that g is an 2ε-graph-approximation of ϕ and a θ21-selection of ψ. Set f1 := g and construct a sequence (fn)∞n=1 of continuous functions fn: X → Y such that, for every x∈ X,
dY(fn(x), ψ(x)) < θn/2, n≥ 1, (11)
dY(fn(x), fn−1(x)) < εn−1, n≥ 2.
(12)
For this construction we use the lower semicontinuous function Φn(x) :=
( f (x), for x∈ A;
ψ(x)∩ B(fn−1(x), θn−1/2), for x∈ X \ A.
The function f := limn→∞fn is a required extension of the function f . Under the same assumptions as in the above theorem, we have the following conclusions. Let ε > 0 and A be a closed subset of X.
Corollary 3.9. Each selection f : A→ Y of ϕ and ψ over A can extend to an ε-graph-approximation of ϕ and a selection of ψ.
Corollary 3.10. If A is compact, then there exists δ > 0 such that every δ-graph-approximation f : A→ Y of ϕ such that f is a selection of ψ over A extends to a map f : X → Y such that f is an ε-graph-approximation of ϕ and a selection of ψ.
Corollary 3.11. Let 0 < δ < η, where η corresponds to ε2 as in Definition 2.3. Let f : X → Y and g : X → Y be δ-selections of ϕ and selections of ψ. Then there exists a homotopy H : X× [0, 1] → Y between f and g such that H(·, t) is an ε-graph-approximation of ϕ and a selection of ψ for every t∈ [0, 1].
P roof.Let δ > 0, f and g be as above.
Let π : X0 → X be a projection, where X0 := X × [0, 1], and let ϕ0 :=
ϕ◦ π : X0 ( Y , ψ0 := ψ◦ π : X0 ( Y . It easy to see that ϕ0 and ψ0 have properties as ϕ and ψ adequately. Moreover, X× {0, 1} is a closed set in X0 and dim X0<∞.
Now we define a function H0 : X× {0, 1} → Y as follows: H0(x, 0) = f (x), for all x ∈ X, and H0(x, 1) = g(x), for all x ∈ X. Then H0 is a δ-selection of ϕ0 and a selection of ψ0 over X × {0, 1}. Thus, by Theorem 3.8, it admits an extension H : X0 → Y being an ε-graph-approximation of ϕ0 and a selection of ψ0. It is clear that H is the required homotopy.
Recall that the Cellina theorem [7] concerns an existence of a continuous graph-approximation of an upper semicontinuous map ϕ : X ( Y . Now we shall present a stronger theorem in the spirit of the second Cellina result [7]. We shall prove that there exists a continuous function such that the Hausdorff distance of its graph to the graph of the map ϕ with α-convex values is optionally small.
But in order to proceed, we must assume additional properties of the function α:
(i’) α(x, y,·) is one-to-one for all x, y ∈ X;
(ii’) α(α(x, y, t1), α(x, y, t2)) = α(x, y, (1− t3)t1+ t2t3) for all x, y∈ X and t1, t2, t3 ∈ [0, 1].
Observe that, by (ii’), the set Λα(x1, x2; [0, 1]) ⊂ Y is α-convex for all x1, x2 ∈ Y .
We need to proof the following proposition.
Proposition 3.12. Let X, Y be α-convex metric spaces such that dim X <
∞ and let K ⊂ X be an α-convex set such that diam (K) < ε2, where ε > 0, and K contains at least two points. Let T ⊂ Y be a closed α- convex set for which there exist points y1, . . . , yn∈ Y, n ∈ N, such that T ⊂ {bn(y1, . . . , yn; λ1, . . . , λn) | (λ1, . . . , λn) ∈ Σn}. Then there is a continuous
function f : K→ T such that
D(Gr (f ), K× T ) < ε.
P roof.Let ε > 0.
Suppose that T contains only one point, T ={y0}. Then f(x) = y0 for all x∈ K. We have that
Gr (f ) = K× {y0} = K × T, and hence
D(Gr (f ), K× T ) = 0.
Suppose that T contains more then one point. Let r :{bn(y1, . . . , yn; Σn)} → T be a retraction, where
{bn(y1, . . . , yn; Σn)} := {bn(y1, . . . , yn; λ1, . . . , λn)|(λ1, . . . , λn)∈ Σn}.
If x1, x2 ∈ K, then we have Λα(x1, x2, [0, 1])⊂ K. Let ν : Λα(x1, x2, [0, 1])→ {bn(y1, . . . , yn; Σn)}
be a continuous surjection. The function ν has a form ν = ν3◦ ν2◦ ν1, where (a) ν1 : Λα(x1, x2; [0, 1])→ [0, 1] and ν1(α(x1, x2, t)) = t, for all t∈ [0, 1];
(b) ν2 : [0, 1]→ Σn is a continuous surjection; it exists because Σn is a lo- cally connected metric continuum (a continuum is a compact connected set) and hence it is a continuous image of a segment [0,1] (see [11]);
(c) ν3 : Σn→ {bn(y1, . . . , yn; Σn)} and ν3((λ1, . . . , λn)) = bn(y1, . . . , yn; λ1, . . . , λn) for all (λ1, . . . , λn)∈ Σn.
Then r◦ ν : Λα(x1, x2; [0, 1])→ T is a continuous surjection too. Let r1: K → Λα(x1, x2; [0, 1])
be a retraction.
Define f : K→ T as
f (x) := r(ν(r1(x))).
Because f (K) = T , then Gr (f )⊂ K × T and hence
supz∈Gr (f )dX×Y(z, K× T ) = 0. Take now (x, y) ∈ K × T . Since y ∈ T , then r(y) = y. There is a point x∗ ∈ Λα(x1, x2, [0, 1]) such that
ν(x∗) = y and r1(x∗) = x∗, that means y = r(ν(r1(x∗))).
Then (x∗, y)∈ Gr (f) and
dX×Y((x, y), Gr (f )) = infz∈Gr (f )dX×Y((x, y), z)≤ dX×Y((x, y), (x∗, y))
= max{dX(x, x∗), 0} ≤ ε 2, because x, x∗∈ K. Therefore
D(Gr (f ), K× T ) < ε.
Theorem 3.13. Let X, Y be α-convex metric spaces such that dim X = p < ∞ and X does not have isolated points. Let ϕ : X ( Y be an up- per semicontinuous map with closed α-convex values. Suppose that, for each x ∈ X, there exist points y1, . . . , yn ∈ Y, n ∈ N, such that ϕ(x) ⊂ {bn(y1, . . . , yn; λ1, . . . , λn) | (λ1, . . . , λn)∈ Σn}. Then, for each ε > 0, there exists a continuous function f : X → Y such that
D(Gr (f ), Gr (ϕ)) < ε.
P roof.Let ε > 0. By Proposition 2.9, there exists 0 < δ≤ ε3 such that, for any n≤ p + 1, any y1, . . . , yn ∈ Y and any non-empty α-convex set A ⊂ Y , if dY(yi, A) < δ for i = 1, . . . , n, then
dY(bn(y1, . . . , yn; λ1, . . . , λn), A) < ε 3
for all (λ1, . . . , λn) ∈ Σn. Upper semicontinuity of ϕ implies that, for each x∈ X, there exists 0 < θ = θ(x) < ε4 such that
ϕ(B(x, θ(x)))⊂ B(ϕ(x), δ).
Since dim X = p, then there exists {Uj}j∈J an open refinement of the cov- ering {B(x, θ(x)/3)}x∈X such that every subfamily consisting of p + 2 sets in{Uj}j∈J has empty intersection.
For each j ∈ J, choose xj ∈ Uj and τj0 such that D(xj, τj0) ⊂ Uj. Suppose that D(xj, τj0) has a nonempty intersection with sets Uj1, . . . , Ujn for n≤ p and jk 6= j, k = 1, . . . , n. Let
τj00:= min
1≤k≤n{dX(xj, xjk)} and τj := 1
3min{τj0, τj00}.
For every j ∈ J, let D(xj) be a closed α-convex set such that xj ∈ D(xj)⊂ D(xj, τj)
and D(xj) contains at least two points. For all j ∈ J such set D(xj) exists.
Indeed, let 0 < η ≤ τj corresponds to τj as in Definition 2.3. Take a point a ∈ B(xj, η), a 6= xj. Since dX(a, xj) < η and dX(xj, xj) = 0 < η, then D(xj, Λα(xj, a, [0, 1])) < τj and we can assume D(xj) = Λα(xj, a, [0, 1]).
Consider the family {D(xj)}j∈J. It easy to see D(xj)∩ D(xi) = ∅ for j, i∈ J, j 6= i.
Let Cj := S
i∈J,i6=jD(xi). Clearly, Cj is a closed set for j ∈ J. For all j∈ J, let Vj := Uj\ Cj. The family{Vj}j∈J is a locally finite covering of X such that, for each j ∈ J,
D(xj)⊂ Vj and D(xj)∩ Vi =∅ for i6= j.
Let{λj}j∈J be a partition of unity inscribed into the covering{Vj}j∈J. For each j ∈ J, choose ζj ∈ X such that
Vj ⊂ Uj ⊂ B(ζj, θ(ζj)/3).
Because diam D(xj) < ε6 and all assumption of Proposition 3.12 (for K = D(xj) and T = ϕ(ζj)) are true, then there exists a continuous function γj : D(xj)→ ϕ(ζj) such that
D(Gr (γj), D(xj)× ϕ(ζj)) < ε 3. Let fj : X → ϕ(ζj) be an extension of γj.
Let λ : J× X → [0, 1] be given by λ(j, x) = λj(x), let y : J × X → Y be given by y(j, x) = fj(x). Then all assumptions from Proposition 2.8 are satisfied, hence we may define
f (x) = b(y(·, x), λ(·, x)), x ∈ X.
