R E S E A R C H
Open Access
Approximate selection theorems with
n-connectedness
Hoonjoo Kim
**Correspondence:
[email protected] Department of Mathematics Education, Sehan University, YoungAm-gun, Chunnam 526-702, Korea
Abstract
We establish new approximate selection theorems for almost lower semicontinuous multimaps withn-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
Keywords: lower semicontinuity; almost lower semicontinuity; quasi-lower semicontinuity; approximate selection theorems;D-spaces;LC-metric spaces; topological semilattice with path-connected intervals
1 Introduction
Since Michael [] constructed continuous-approximate selections for the lower semi-continuous 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 [] in . And Ben-El-Mechaiekh and Oudadess [] generalized the theorem in [] to a class of lower semicontinuous multimaps with nonconvex values inLC-metric spaces, which have generalized convex metric structures introduced by Horvath [].
Using the concept of n-connectedness, Kim [] introduced anLD-metric space and extended the result in [] toLD-metric spaces which are more general thanLC-metric spaces.
On the other hand, inLC-spaces, Wu and Li [] obtained the approximate selection the-orems for quasi-lower semicontinuous multimaps which were generalized by the author and Lee [] to almost lower semicontinuous multimaps inC-spaces.
In this paper, we establish a new approximate selection theorem for almost lower semi-continuous multimaps withD-set values except on a set of topological dimension less than or equal to zero inLD-spaces. The corollary of this gives a correct and simple proof for the result in [].
We also establish some approximate selection theorems for almost lower semicontin-uous multimaps inD-spaces and apply the results to topological semilattices with path connected intervals. Our results unify and extend the approximate selection theorems in [–, –].
2 Preliminaries
Amultimap(or simply amap)F:XYis a function from a setXinto the power set ofY; that is, a function with thevalues F(x)⊂Yforx∈X. ForA⊂X, letF(A) :={F(x)|x∈A}. Throughout this paper, we assume that multimaps have nonempty values otherwise ex-plicitly stated or obvious from the context. LetXdenote the set of all nonempty finite subsets of a setX.
LetXbe a topological space. AC-structureonXis given by a map:XXsuch that
() for allA∈ X,A=(A)is nonempty and contractible; and
() for allA,B∈ X,A⊂BimpliesA⊂B.
A pair (X,) is then called a C-spaceby Horvath [] and anH-spaceby Bardaro and Ceppitelli []. For examples of aC-space, see [, ]. For an (X,), a subsetCofXis said to be-convex(or aC-set) ifA∈ CimpliesA⊂C.
For a uniform spaceXwith a uniform structureU,A⊂XandU∈U, the setU(A) is defined to be{y∈X: (x,y)∈Ufor somex∈A}and ifx∈X,U(x) =U({x}).
AC-space (X,) is called anLC-spaceifXis a uniform space and there exists a base {Vi:i∈I} for the uniform structure such that for eachi∈I, {x∈X:C∩Vi(x)=∅}is
-convex wheneverC⊂Xis-convex.
A C-space (X,) is called anLC-metric spaceif Xis equipped with a metricd such that for any> , the setB(C,) ={x∈X:d(x,C) <}is-convex wheneverC⊂Xis -convex, and open balls are-convex. For details, see Horvath [].
A topological spaceXis said to ben-connectedforn≥ if every continuous mapf :
Sk→Xfork≤nhas a continuous extension overBk+, whereSkis the unit sphere and Bk+is the closed unit ball inRk+. Note that a contractible space isn-connected for every n≥.
The following is introduced by Kim []. LetXbe a topological space. AD-structureon
Xis a mapD:XXsuch that it satisfies the following conditions:
() for eachA∈ X,D(A)is nonempty andn-connected for alln≥;
() for eachA,B∈ X,A⊂BimpliesD(A)⊂D(B).
The pair (X,D) is called aD-space; a subsetCofXis said to be aD-setifD(A)⊂Cfor eachA∈ C.
AD-space (X,D) is called anLD-spaceifXis a uniform space and if there exists a base {Vi:i∈I}for the uniform structure such that for eachi∈I, the set{x∈X:C∩Vi(x)=∅} is aD-set wheneverC⊂Xis aD-set.
AD-space (X,D) is called anLD-metric spaceifXis a metric space such that for each > ,B(C,) is aD-set wheneverC⊂Xis aD-set and open balls areD-sets.
LetXbe a topological space and (Y,D) be aD-space with a uniformityU. A multimap
F:XY is called:
() lower semicontinuous(lsc) atx∈Xif for each open setWwithW∩F(x)=∅, there is a neighborhoodU(x)ofxsuch thatF(z)∩W=∅for allz∈U(x);
() quasi-lower semicontinuous(qlsc) atx∈Xif for eachV∈U, there arey∈F(x)and a neighborhoodU(x)ofxsuch thatF(z)∩V(y)=∅for allz∈U(x);
() almost lower semicontinuous(alsc) atx∈Xif for eachV∈U, there is a neighborhoodU(x)ofxsuch thatz∈U(x)V(F(z))=∅.
ForV∈U, a continuous functionf :X→Y is called aV-approximate selectionofFif for allx∈X,f(x)∈V(F(x)).
Let (Y,D) be anLD-metric space. For> ,f is called an-approximate selectionofF
if for allx∈X,f(x)∈B(F(x),).
LetXbe a topological space. IfZ⊂X, thendimXZ≤ means thatdimE≤ for every setE⊂Z which is closed inX, wheredimEdenotes the covering dimension ofE. Note that ifdimXZ≤, then any locally finite open covering ofZhas a disjoint locally finite
open refinement.
3 Approximate selection theorems onD-spaces As a main tool, we need Proposition of Kim [].
Proposition . Let X be a paracompact topological space andRbe a locally finite open
covering of X, (Y,D)be a D-space,andη:R→Y be a function.Then there exists a con-tinuous function f :X→Y such that
f(x)∈Dη(O) :O∈R,x∈O
for each x∈X.
With Proposition ., we establish theV-approximate selection theorem which is the key result of this paper.
Theorem . Let X be a paracompact topological space and Z be a closed subset of X with
dimXZ≤.Let(Y,D)be an LD-space with a uniformityU andD({y}) ={y}for all y∈Y. If F:XY is an alsc multimap such that F(x)is aD-set for all x∈X\Z,then F has a V -approximate selection for each V∈U.
Furthermore,if X is a precompact uniform space or a compact topological space,there is a subset A∈ Ysuch that f(X)⊂D(A).
Proof For each V ∈ U and x ∈ X, there is a neighborhood U(x) of x such that
z∈U(x)V(F(z))=∅, becauseFis alsc. SinceXis paracompact, the open cover{U(x) :x∈ X}ofXhas a locally finite refinement{ ˜U(x) :x∈X}. And sincedimXZ≤, the relatively
open cover{ ˜U(x)∩Z:x∈Z}ofZhas a relatively open disjoint refinement{W(x) :x∈Z}.
Zis closed inXso the collectionR={ ˜U(x)∩(W(x)∪(X\Z)) :x∈X}forms a locally finite open cover ofX.
For each O∈R, choose xo such thatO⊂ U(xo) and yo∈
z∈U(xo)V(F(z)). Define
η:R→Y byη(O) =yofor allO∈R. Thenη(O)∈z∈U(xo)V(F(z))⊂z∈OV(F(z)), so {η(O) :O∈R,x∈O} ⊂V(F(x)) for allx∈X. By Proposition ., there is a continuous functionf :X→Ysuch thatf(x)∈D({η(O) :O∈R,x∈O}).
We now show thatf(x)∈V(F(x)) for allx∈X. Ifx∈Z, there exists a uniqueO∈Rsuch thatx∈O, that is,{η(O) :O∈R,x∈O}is a singleton. So,f(x)∈D({η(O) :O∈R,x∈O}) =
{η(O)} ⊂V(F(x)). Ifx∈X\Z, sinceF(x) is aD-set,D({η(O) :O∈R,x∈O})⊂V(F(x)), that is,f(x)∈V(F(x)).
Remark IfZ=∅, then the condition ‘D({y}) ={y}for ally∈Y’ can be omitted. In that case, if (Y,D) is anLC-space with a uniformityUandFis qlsc, then Theorem . becomes [, Theorem .].
Proposition . Each singleton is aD-set in an LD-metric space(X,D),soD({x}) ={x}.
Proof For eachx∈X,{x}=>B(x,). Since all open balls and their intersection are
D-sets,{x}is aD-set. ThereforeD({x})⊂ {x},i.e.,D({x}) ={x}.
ForLD-metric spaces, Theorem . reduces to the following.
Corollary . Let X be a paracompact space, (Y,D)be an LD-metric space,and Z be a
closed subset of X withdimXZ≤.If F:XY is an alsc multimap such that F(x)is a
D-set for all x∈X\Z,then for> ,F has an-approximate selection.
ForLC-metric spaces, Corollary . reduces to the following.
Corollary . Let X be a paracompact space, (Y,)be an LC-metric space,and Z be a
closed subset of X with dimXZ≤.If F:XY is an alsc multimap such that F(x)is
-convex for all x∈X\Z,then for> ,F has an-approximate selection.
Remark Corollary . is Theorem . in [] which is a partial generalization of Lemma
in []. In the proof of Lemma in [] and Theorem . in [], for the subsetEofZ, it is claimed thatB(F(x),) is-convex wheneverx∈Eandx∈X\E, but it cannot be analo-gized from the assumption thatF(x) is-convex for allx∈/Z.
Theorem . in [] shows that ifX=Zand (Y,) is aC-space with a uniformityUand
F has aV-approximate selection for eachV∈U, thenFis qlsc. Using the same pattern of its proof, we also obtain the same result when (Y,D) is aD-space with a uniformityU. Since a qlsc map is alsc, so the inverse of Theorem . also holds.
Theorem . Let X be a paracompact topological space and Z be a closed subset of X with
dimXZ≤.Let(Y,D)be an LD-space with a uniformityU andD({y}) ={y}for all y∈Y.
And let F:XY be a multimap such that F(x)is aD-set for all x∈X\Z.Then F is alsc if and only if F has a V -approximate selection for each V∈U.
The following notion is motivated by Hadžić []. Let (X,D) be aD-space with a uni-formityU andKbe a nonempty subset ofX. We say thatK isof generalized Zima type
whenever for everyV∈U, there exists aV∈U such that for everyN∈ Kand every
D-setMofK, the following implication holds:
M∩V(z)= ∅, ∀z∈N =⇒ M∩V(u)=∅, ∀u∈D(N).
Note that anLD-space (X,D) is of generalized Zima type.
IfZ=∅, then theLD-space condition ofYcan be weakened in Theorem ..
Theorem . Let X be a paracompact topological space, (Y,D)be a D-space with a
The proof of Theorem . proceeds in the same fashion as Theorem in [], except that all-convex sets in aC-space is replaced byD-sets in aD-space.
LetXbe a topological space andYbe a uniform space with a uniformityU. The mul-timapsF,T:XY are said to betopologically separatedif for eachx∈X, there exist a neighborhoodU(x) ofxand an elementV∈Usuch thatF(U(x))∩V(T(x)) =∅.
Theorem . Let X be a compact topological space and Z be a closed subset of X with
dimXZ≤.And let(Y,D)be an LD-space with a uniformityU andD({y}) ={y}for all
y∈Y.If F,T:XY are two multimaps such that
() FandT are topologically separated; () Tis upper semicontinuous;and
() Fis an alsc multimap such thatF(x)is aD-set for allx∈X\Z.
Then,for each V∈U,F has a V -approximate selection f :X→Y such that
f(x) /∈T(x)
for all x∈X.
Using Theorem ., the proof of Theorem . proceeds in precisely the same fashion as Theorem . in [].
Particular forms . Zheng [, Theorem .]:Y is a locally convex space,Z=∅, andF
is sub-lower semicontinuous, that is, for eachx∈Xand each neighborhoodVof inY, there isz∈F(x) and a neighborhoodU(x) ofxinXsuch that for eachy∈U(x),z∈F(y)+V. Note that ifYis a topological vector space, thenFis sub-lower semicontinuous if and only ifFis qlsc; see [, Proposition .].
. Wu and Li [, Theorem .]: (Y,D) is anLC-space with a uniformityU,Z=∅, andF
is qlsc.
Proposition . Let X be a topological space and Y be a metric space. If a multimap
F:XY is alsc at x∈X,then F is qlsc at x∈X.
Proof For> , there is a neighborhoodU(x) ofxsuch that
z∈U(x)
BF(z),/=∅.
Select anyy∈z∈U(x)B(F(z),/). For eachz∈U(x), chooseyz∈F(z) such thatd(y,yz) <
/. Note thatyx∈F(x) andd(yx,yz)≤d(yx,y) +d(y,yz) <for eachz∈U(x). Henceyz∈
B(yx,)∩F(z) for allz∈U(x).
The following result is a generalization of Theorem . in [].
Theorem . Let X be a paracompact topological space, (Y,D)be an LD-metric space,
and Z be a closed subset of X withdimXZ≤.If F,T:XY are two multimaps such that
() Tis upper semicontinuous;and
() Fis an alsc multimap such thatF(x)is aD-set for allx∈X\Z.
Then for each> ,F has an-approximate selection f:X→Y such that
f(x) /∈T(x)
for all x∈X.
Proof For each fixed> and eachx∈X, by () and (), there exists a neighborhood
U(x) ofxand anη(x) > such thatη(x) <,F(U(x))∩B(T(x),η(x)) =∅, andT(U(x))⊂
B(T(x),η(x)/). Letζ(x) =η(x)/. For eachy∈U(x), we assertF(y)∩B(T(y),ζ(x)) =∅. Otherwise, there exist pointsp∈T(y) andz∈F(y) such thatd(p,z) <ζ(x). Becausey∈ U(x) andT(y)⊂B(T(x),ζ(x)), sop∈B(T(x),ζ(x)). Consequently, there is a pointb∈T(x) such thatd(p,b) <ζ(x). Henced(b,z) <η(x), and thusz∈F(y)∩B(T(x),η(x))⊂F(U(x))∩
B(T(x),η(x)) =∅. It is a contradiction.
Letδ(x) =sup{r: <r<andF(x)∩B(T(x),r) =∅}. Obviously,δ(x)≤and for eachy∈ U(x),δ(y)≥ζ(x). Now, we assert thatF(x)∩B(T(x),δ(x)) =∅. Otherwise, there exist points
y∈F(x) andz∈T(x) such thatd(y,z) <δ(x). Consequently, there is a numberr>d(y,z) such that <r<andF(x)∩B(T(x),r) =∅. Buty∈F(x)∩B(T(x),r), it is a contradiction. By Proposition .,F:XYis qlsc, so there exist a pointyx∈F(x) and an open neigh-borhoodN(x) ofxinXsuch thatN(x)⊂U(x) and
F(z)∩Byx,ζ(x)=∅ (∗)
for allz∈N(x). SinceXis paracompact, the open cover{N(x) :x∈X}has a locally finite open refinement{V(x) :x∈X}. SincedimXZ≤, the relative open cover{V(x)∩Z:x∈ Z}ofZ has a relatively open disjoint refinement{W(x) :x∈Z}.Zis closed inX, so the collectionR={V(x)∩(W(x)∪(X\Z)) :x∈X}forms a locally finite open cover ofX.
For eachO∈R, choose a pointxosuch thatO⊂V(xo) and defineη:R→Ybyη(O) =
yxo such thatyxo∈F(xo) satisfying the condition (∗) for allz∈V(xo). By Proposition ., there is a continuous functionf :X→Y such thatf(x)∈D({η(O) :O∈R,x∈O}). For eachx∈XandO∈Rsuch thatx∈O, by (∗), we haveF(x)∩B(η(O),ζ(xo))=∅andδ(x)≥ ζ(xo) becausex∈O⊂V(xo)⊂N(xo), soF(x)∩B(η(O),δ(x))=∅.
Note that forx∈Z,f(x) =η(O) =yxo since{O∈R:x∈O}is a singleton. Hencef(x)∈ {y∈Y:F(x)∩B(y,δ(x))=∅}.
For eachx∈X\Z,
f(x)∈Dη(O) :O∈R,x∈O⊂y∈Y:F(x)∩By,δ(x)=∅
sinceF(x) andB(F(x),δ(x)) areD-sets.
So,F(x)∩B(f(x),δ(x))=∅for allx∈Xand hence
F(x)∩Bf(x),=∅ and f(x) /∈T(x).
4 Approximate selection theorems on topological ordered spaces
upper bound, denoted by supA. In a partially ordered set (X,≤), two arbitrary elements
xandxdo not have to be comparable, but, in the case wherex≤x, the set [x,x] ={y∈ X:x≤y≤x}is calledan order interval.
The following is due to Horvath and Ciscar []: Let (X,≤) be a semilattice such that for eachA∈ X,(A) is defined bya∈A[a,supA]. Then
() (A)is well defined;
() A⊂(A);
() ifA⊂B, then(A)⊂(B).
A subsetE⊂Xis said to beconvexif, for any subsetA∈ E, we have(A)⊂E. IfXis a topological semilattice with path-connected intervals, then for anyA∈ Xand
n≥,(A) isn-connected by [, Lemma ], that is, (X,≤,) is aD-space. Note that({x}) ={x}for allx∈X.
In this section, we assume that (Y,≤,) is a topological semilattice with path-connected intervals. From the results of Section , we obtain the following theorems.
Theorem . Let X be a paracompact topological space,Z be a closed subset of X with
dimXZ≤,andU be a uniformity of(Y,≤,)such that for each V ∈U,the set{y∈Y : C∩V(y)=∅}is convex whenever C is a convex subset of Y.And let F⊆X×Y be a binary relation such that F(x)is convex for all x∈X\Z.Then F is alsc if and only if F has a V -approximate selection for each V∈U.
Theorem . Let X be a paracompact topological space andUbe a uniformity of(Y,≤,).
And let F⊆X×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 V∈U.
Theorem . Let X be a compact topological space and Z be a closed subset of X with
dimXZ≤.And letUbe a uniformity of(Y,≤,)such that for each V ∈U,the set{x∈X: C∩V(x)=∅}is convex whenever C⊂X is convex.If F,T⊆X×Y are two binary relations such that
() FandT are topologically separated; () Tis upper semicontinuous;and
() Fis an alsc relation such thatF(x)is convex for allx∈X\Z.
Then for each V∈U,F has a V -approximate selection f :X→Y such that
f(x) /∈T(x)
for all x∈X.
Theorem . Let X be a paracompact topological space and Z be a closed subset of X with
dimXZ≤.Assume that(Y,≤,)is a metric space and has the following properties:
(i) for any> ,the setB(C,)is convex wheneverCis a convex subset ofY;and (ii) open balls are convex.
If F,T⊆X×Y are two binary relations such that
() FandT are topologically separated; () Tis upper semicontinuous;and
Then for each> ,F has an-approximate selection f:X→Y such that
f(x) /∈T(x)
for all x∈X.
Competing interests
The author declares that she has no competing interests.
Abbreviations
alsc: almost lower semicontinuity; lsc: lower semicontinuity; qlsc: quasi-lower semicontinuity.
Acknowledgements
This paper was supported by the Sehan University Research Fund in 2013.
Received: 23 September 2012 Accepted: 26 February 2013 Published: 19 March 2013
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doi:10.1186/1029-242X-2013-113
