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Approximate selection theorems with n-connectedness

Abstract

We establish new approximate selection theorems for almost lower semicontinuous multimaps with n-connectedness. Our results unify and extend the approximate selection theorems in many published works and are applied to topological semilattices with path-connected intervals.

MSC:54C60, 54C65, 55M10.

1 Introduction

Since Michael [1] constructed continuous ϵ-approximate selections for the lower semicontinuous maps with convex values in Banach spaces, the result has been improved in many ways. It was extended to lower semicontinuous maps with convex values except on a set of topological dimension less than or equal to zero by Michael and Pixley [2] in 1980. And Ben-El-Mechaiekh and Oudadess [3] generalized the theorem in [2] to a class of lower semicontinuous multimaps with nonconvex values in LC-metric spaces, which have generalized convex metric structures introduced by Horvath [4].

Using the concept of n-connectedness, Kim [5] introduced an LD-metric space and extended the result in [3] to LD-metric spaces which are more general than LC-metric spaces.

On the other hand, in LC-spaces, Wu and Li [6] obtained the approximate selection theorems for quasi-lower semicontinuous multimaps which were generalized by the author and Lee [7] to almost lower semicontinuous multimaps in C-spaces.

In this paper, we establish a new approximate selection theorem for almost lower semicontinuous multimaps with D-set values except on a set of topological dimension less than or equal to zero in LD-spaces. The corollary of this gives a correct and simple proof for the result in [8].

We also establish some approximate selection theorems for almost lower semicontinuous multimaps in D-spaces and apply the results to topological semilattices with path connected intervals. Our results unify and extend the approximate selection theorems in [13, 59].

2 Preliminaries

A multimap (or simply a map) F:XY is a function from a set X into the power set of Y; that is, a function with the values F(x)Y for xX. For AX, let F(A):={F(x)xA}. Throughout this paper, we assume that multimaps have nonempty values otherwise explicitly stated or obvious from the context. Let X denote the set of all nonempty finite subsets of a set X.

Let X be a topological space. A C-structure on X is given by a map Γ:XX such that

  1. (1)

    for all AX, Γ A =Γ(A) is nonempty and contractible; and

  2. (2)

    for all A,BX, AB implies Γ A Γ B .

A pair (X,Γ) is then called a C-space by Horvath [4] and an H-space by Bardaro and Ceppitelli [10]. For examples of a C-space, see [4, 10]. For an (X,Γ), a subset C of X is said to be Γ-convex (or a C-set) if AC implies Γ A C.

For a uniform space X with a uniform structure U, AX and UU, the set U(A) is defined to be {yX:(x,y)U for some xA} and if xX, U(x)=U({x}).

A C-space (X,Γ) is called an LC-space if X is a uniform space and there exists a base { V i :iI} for the uniform structure such that for each iI, {xX:C V i (x)} is Γ-convex whenever CX is Γ-convex.

A C-space (X,Γ) is called an LC-metric space if X is equipped with a metric d such that for any ϵ>0, the set B(C,ϵ)={xX:d(x,C)<ϵ} is Γ-convex whenever CX is Γ-convex, and open balls are Γ-convex. For details, see Horvath [4].

A topological space X is said to be n-connected for n0 if every continuous map f: S k X for kn has a continuous extension over B k + 1 , where S k is the unit sphere and B k + 1 is the closed unit ball in R k + 1 . Note that a contractible space is n-connected for every n0.

The following is introduced by Kim [5]. Let X be a topological space. A D-structure on X is a map D:XX such that it satisfies the following conditions:

  1. (1)

    for each AX, D(A) is nonempty and n-connected for all n0;

  2. (2)

    for each A,BX, AB implies D(A)D(B).

The pair (X,D) is called a D-space; a subset C of X is said to be a D-set if D(A)C for each AC.

A D-space (X,D) is called an LD-space if X is a uniform space and if there exists a base { V i :iI} for the uniform structure such that for each iI, the set {xX:C V i (x)} is a D-set whenever CX is a D-set.

A D-space (X,D) is called an LD-metric space if X is a metric space such that for each ϵ>0, B(C,ϵ) is a D-set whenever CX is a D-set and open balls are D-sets.

Let X be a topological space and (Y,D) be a D-space with a uniformity U. A multimap F:XY is called:

  1. (1)

    lower semicontinuous (lsc) at xX if for each open set W with WF(x), there is a neighborhood U(x) of x such that F(z)W for all zU(x);

  2. (2)

    quasi-lower semicontinuous (qlsc) at xX if for each VU, there are yF(x) and a neighborhood U(x) of x such that F(z)V(y) for all zU(x);

  3. (3)

    almost lower semicontinuous (alsc) at xX if for each VU, there is a neighborhood U(x) of x such that z U ( x ) V(F(z)).

If F is lsc (qlsc, alsc, resp.) at each xX, F is called lsc (qlsc, alsc, resp.). As in [[7], Proposition 3], (1) (2) (3).

For VU, a continuous function f:XY is called a V-approximate selection of F if for all xX, f(x)V(F(x)).

Let (Y,D) be an LD-metric space. For ϵ>0, f is called an ϵ-approximate selection of F if for all xX, f(x)B(F(x),ϵ).

Let X be a topological space. If ZX, then dim X Z0 means that dimE0 for every set EZ which is closed in X, where dimE denotes the covering dimension of E. Note that if dim X Z0, then any locally finite open covering of Z has a disjoint locally finite open refinement.

3 Approximate selection theorems on D-spaces

As a main tool, we need Proposition 1 of Kim [5].

Proposition 3.1 Let X be a paracompact topological space and be a locally finite open covering of X, (Y,D) be a D-space, and η:RY be a function. Then there exists a continuous function f:XY such that

f(x)D ( { η ( O ) : O R , x O } )

for each xX.

With Proposition 3.1, we establish the V-approximate selection theorem which is the key result of this paper.

Theorem 3.2 Let X be a paracompact topological space and Z be a closed subset of X with dim X Z0. Let (Y,D) be an LD-space with a uniformity U and D({y})={y} for all yY. If F:XY is an alsc multimap such that F(x) is a D-set for all xXZ, then F has a V-approximate selection for each VU.

Furthermore, if X is a precompact uniform space or a compact topological space, there is a subset AY such that f(X)D(A).

Proof For each VU and xX, there is a neighborhood U(x) of x such that z U ( x ) V(F(z)), because F is alsc. Since X is paracompact, the open cover {U(x):xX} of X has a locally finite refinement { U ˜ (x):xX}. And since dim X Z0, the relatively open cover { U ˜ (x)Z:xZ} of Z has a relatively open disjoint refinement {W(x):xZ}. Z is closed in X so the collection R={ U ˜ (x)(W(x)(XZ)):xX} forms a locally finite open cover of X.

For each OR, choose x o such that OU( x o ) and y o z U ( x o ) V(F(z)). Define η:RY by η(O)= y o for all OR. Then η(O) z U ( x o ) V(F(z)) z O V(F(z)), so {η(O):OR,xO}V(F(x)) for all xX. By Proposition 3.1, there is a continuous function f:XY such that f(x)D({η(O):OR,xO}).

We now show that f(x)V(F(x)) for all xX. If xZ, there exists a unique OR such that xO, that is, {η(O):OR,xO} is a singleton. So, f(x)D({η(O):OR,xO})={η(O)}V(F(x)). If xXZ, since F(x) is a D-set, D({η(O):OR,xO})V(F(x)), that is, f(x)V(F(x)).

If X is a precompact uniform space or a compact topological space, can be chosen finite. Take A={η(O):OR}, then AY and f(X)D(A). □

Remark If Z=, then the condition ‘D({y})={y} for all yY’ can be omitted. In that case, if (Y,D) is an LC-space with a uniformity U and F is qlsc, then Theorem 3.2 becomes [[6], Theorem 3.1].

Proposition 3.3 Each singleton is a D-set in an LD-metric space (X,D), so D({x})={x}.

Proof For each xX, {x}= ϵ > 0 B(x,ϵ). Since all open balls and their intersection are D-sets, {x} is a D-set. Therefore D({x}){x}, i.e., D({x})={x}. □

For LD-metric spaces, Theorem 3.2 reduces to the following.

Corollary 3.4 Let X be a paracompact space, (Y,D) be an LD-metric space, and Z be a closed subset of X with dim X Z0. If F:XY is an alsc multimap such that F(x) is a D-set for all xXZ, then for ϵ>0, F has an ϵ-approximate selection.

For LC-metric spaces, Corollary 3.4 reduces to the following.

Corollary 3.5 Let X be a paracompact space, (Y,Γ) be an LC-metric space, and Z be a closed subset of X with dim X Z0. If F:XY is an alsc multimap such that F(x) is Γ-convex for all xXZ, then for ϵ>0, F has an ϵ-approximate selection.

Remark Corollary 3.5 is Theorem 3.2 in [8] which is a partial generalization of Lemma 2 in [3]. In the proof of Lemma 2 in [3] and Theorem 3.2 in [8], for the subset E of Z, it is claimed that B(F(x),ϵ) is Γ-convex whenever xE and xXE, but it cannot be analogized from the assumption that F(x) is Γ-convex for all xZ.

Theorem 3.3 in [6] shows that if X=Z and (Y,Γ) is a C-space with a uniformity U and F has a V-approximate selection for each VU, then F is qlsc. Using the same pattern of its proof, we also obtain the same result when (Y,D) is a D-space with a uniformity U. Since a qlsc map is alsc, so the inverse of Theorem 3.2 also holds.

Theorem 3.6 Let X be a paracompact topological space and Z be a closed subset of X with dim X Z0. Let (Y,D) be an LD-space with a uniformity U and D({y})={y} for all yY. And let F:XY be a multimap such that F(x) is a D-set for all xXZ. Then F is alsc if and only if F has a V-approximate selection for each VU.

The following notion is motivated by Hadžić [11]. Let (X,D) be a D-space with a uniformity U and K be a nonempty subset of X. We say that K is of generalized Zima type whenever for every VU, there exists a V 1 U such that for every NK and every D-set M of K, the following implication holds:

M V 1 (z),zNMV(u),uD(N).

Note that an LD-space (X,D) is of generalized Zima type.

If Z=, then the LD-space condition of Y can be weakened in Theorem 3.6.

Theorem 3.7 Let X be a paracompact topological space, (Y,D) be a D-space with a uniformity U, and F:XY be a multimap with D-set values such that F(X) is of generalized Zima type. Then F is alsc if and only if F has a V-approximate selection for each VU.

The proof of Theorem 3.7 proceeds in the same fashion as Theorem 2 in [7], except that all Γ-convex sets in a C-space is replaced by D-sets in a D-space.

Let X be a topological space and Y be a uniform space with a uniformity U. The multimaps F,T:XY are said to be topologically separated if for each xX, there exist a neighborhood U(x) of x and an element VU such that F(U(x))V(T(x))=.

Theorem 3.8 Let X be a compact topological space and Z be a closed subset of X with dim X Z0. And let (Y,D) be an LD-space with a uniformity U and D({y})={y} for all yY. If F,T:XY are two multimaps such that

  1. (1)

    F and T are topologically separated;

  2. (2)

    T is upper semicontinuous; and

  3. (3)

    F is an alsc multimap such that F(x) is a D-set for all xXZ.

Then, for each VU, F has a V-approximate selection f:XY such that

f(x)T(x)

for all xX.

Using Theorem 3.2, the proof of Theorem 3.8 proceeds in precisely the same fashion as Theorem 3.6 in [6].

Particular forms 1. Zheng [[9], Theorem 2.2]: Y is a locally convex space, Z=, and F is sub-lower semicontinuous, that is, for each xX and each neighborhood V of 0 in Y, there is zF(x) and a neighborhood U(x) of x in X such that for each yU(x), zF(y)+V. Note that if Y is a topological vector space, then F is sub-lower semicontinuous if and only if F is qlsc; see [[6], Proposition 1.2].

  1. 2.

    Wu and Li [[6], Theorem 3.6]: (Y,D) is an LC-space with a uniformity U, Z=, and F is qlsc.

Proposition 3.9 Let X be a topological space and Y be a metric space. If a multimap F:XY is alsc at xX, then F is qlsc at xX.

Proof For ϵ>0, there is a neighborhood U(x) of x such that

z U ( x ) B ( F ( z ) , ϵ / 2 ) .

Select any y z U ( x ) B(F(z),ϵ/2). For each zU(x), choose y z F(z) such that d(y, y z )<ϵ/2. Note that y x F(x) and d( y x , y z )d( y x ,y)+d(y, y z )<ϵ for each zU(x). Hence y z B( y x ,ϵ)F(z) for all zU(x). □

The following result is a generalization of Theorem 3.7 in [6].

Theorem 3.10 Let X be a paracompact topological space, (Y,D) be an LD-metric space, and Z be a closed subset of X with dim X Z0. If F,T:XY are two multimaps such that

  1. (1)

    F and T are topologically separated;

  2. (2)

    T is upper semicontinuous; and

  3. (3)

    F is an alsc multimap such that F(x) is a D-set for all xXZ.

Then for each ϵ>0, F has an ϵ-approximate selection f:XY such that

f(x)T(x)

for all xX.

Proof For each fixed ϵ>0 and each xX, by (1) and (2), there exists a neighborhood U(x) of x and an η(x)>0 such that η(x)<ϵ, F(U(x))B(T(x),η(x))=, and T(U(x))B(T(x),η(x)/2). Let ζ(x)=η(x)/2. For each yU(x), we assert F(y)B(T(y),ζ(x))=. Otherwise, there exist points pT(y) and zF(y) such that d(p,z)<ζ(x). Because yU(x) and T(y)B(T(x),ζ(x)), so pB(T(x),ζ(x)). Consequently, there is a point bT(x) such that d(p,b)<ζ(x). Hence d(b,z)<η(x), and thus zF(y)B(T(x),η(x))F(U(x))B(T(x),η(x))=. It is a contradiction.

Let δ(x)=sup{r:0<r<ϵ and F(x)B(T(x),r)=}. Obviously, δ(x)ϵ and for each yU(x), δ(y)ζ(x). Now, we assert that F(x)B(T(x),δ(x))=. Otherwise, there exist points yF(x) and zT(x) such that d(y,z)<δ(x). Consequently, there is a number r>d(y,z) such that 0<r<ϵ and F(x)B(T(x),r)=. But yF(x)B(T(x),r), it is a contradiction.

By Proposition 3.9, F:XY is qlsc, so there exist a point y x F(x) and an open neighborhood N(x) of x in X such that N(x)U(x) and

for all zN(x). Since X is paracompact, the open cover {N(x):xX} has a locally finite open refinement {V(x):xX}. Since dim X Z0, the relative open cover {V(x)Z:xZ} of Z has a relatively open disjoint refinement {W(x):xZ}. Z is closed in X, so the collection R={V(x)(W(x)(XZ)):xX} forms a locally finite open cover of X.

For each OR, choose a point x o such that OV( x o ) and define η:RY by η(O)= y x o such that y x o F( x o ) satisfying the condition () for all zV( x o ). By Proposition 3.1, there is a continuous function f:XY such that f(x)D({η(O):OR,xO}). For each xX and OR such that xO, by (), we have F(x)B(η(O),ζ( x o )) and δ(x)ζ( x o ) because xOV( x o )N( x o ), so F(x)B(η(O),δ(x)).

Note that for xZ, f(x)=η(O)= y x o since {OR:xO} is a singleton. Hence f(x){yY:F(x)B(y,δ(x))}.

For each xXZ,

f(x)D ( { η ( O ) : O R , x O } ) { y Y : F ( x ) B ( y , δ ( x ) ) }

since F(x) and B(F(x),δ(x)) are D-sets.

So, F(x)B(f(x),δ(x)) for all xX and hence

F(x)B ( f ( x ) , ϵ ) andf(x)T(x).

 □

4 Approximate selection theorems on topological ordered spaces

A semilattice is a partially ordered set X, with the partial ordering denoted by ≤, for which any pair (x, x ) of elements has a least upper bound. Any nonempty set AX has a least upper bound, denoted by sup A. In a partially ordered set (X,), two arbitrary elements x and x do not have to be comparable, but, in the case where x x , the set [x, x ]={yX:xy x } is called an order interval.

The following is due to Horvath and Ciscar [12]: Let (X,) be a semilattice such that for each AX, Δ(A) is defined by a A [a,supA]. Then

  1. (1)
    Δ(A)

    is well defined;

  2. (2)
    AΔ(A)

    ;

  3. (3)

    if AB, then Δ(A)Δ(B).

A subset EX is said to be convex if, for any subset AE, we have Δ(A)E.

If X is a topological semilattice with path-connected intervals, then for any AX and n0, Δ(A) is n-connected by [[12], Lemma 1], that is, (X,,Δ) is a D-space.

Note that Δ({x})={x} for all xX.

In this section, we assume that (Y,,Δ) is a topological semilattice with path-connected intervals. From the results of Section 3, we obtain the following theorems.

Theorem 4.1 Let X be a paracompact topological space, Z be a closed subset of X with dim X Z0, and U be a uniformity of (Y,,Δ) such that for each VU, the set {yY:CV(y)} is convex whenever C is a convex subset of Y. And let FX×Y be a binary relation such that F(x) is convex for all xXZ. Then F is alsc if and only if F has a V-approximate selection for each VU.

Theorem 4.2 Let X be a paracompact topological space and U be a uniformity of (Y,,Δ). And let FX×Y be a binary relation with convex values and F(X) be of generalized Zima type. Then F is alsc if and only if F has a V-approximate selection for each VU.

Theorem 4.3 Let X be a compact topological space and Z be a closed subset of X with dim X Z0. And let U be a uniformity of (Y,,Δ) such that for each VU, the set {xX:CV(x)} is convex whenever CX is convex. If F,TX×Y are two binary relations such that

  1. (1)

    F and T are topologically separated;

  2. (2)

    T is upper semicontinuous; and

  3. (3)

    F is an alsc relation such that F(x) is convex for all xXZ.

Then for each VU, F has a V-approximate selection f:XY such that

f(x)T(x)

for all xX.

Theorem 4.4 Let X be a paracompact topological space and Z be a closed subset of X with dim X Z0. Assume that (Y,,Δ) is a metric space and has the following properties:

  1. (i)

    for any ϵ>0, the set B(C,ϵ) is convex whenever C is a convex subset of Y; and

  2. (ii)

    open balls are convex.

If F,TX×Y are two binary relations such that

  1. (1)

    F and T are topologically separated;

  2. (2)

    T is upper semicontinuous; and

  3. (3)

    F is an alsc relation such that F(x) is convex for all xXZ.

Then for each ϵ>0, F has an ϵ-approximate selection f:XY such that

f(x)T(x)

for all xX.

Abbreviations

alsc:

almost lower semicontinuity

lsc:

lower semicontinuity

qlsc:

quasi-lower semicontinuity.

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Acknowledgements

This paper was supported by the Sehan University Research Fund in 2013.

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Kim, H. Approximate selection theorems with n-connectedness. J Inequal Appl 2013, 113 (2013). https://doi.org/10.1186/1029-242X-2013-113

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Keywords

  • lower semicontinuity
  • almost lower semicontinuity
  • quasi-lower semicontinuity
  • approximate selection theorems
  • D-spaces
  • LC-metric spaces
  • topological semilattice with path-connected intervals