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Approximate selection theorems with nconnectedness
Journal of Inequalities and Applications volume 2013, Article number: 113 (2013)
Abstract
We establish new approximate selection theorems for almost lower semicontinuous multimaps with nconnectedness. Our results unify and extend the approximate selection theorems in many published works and are applied to topological semilattices with pathconnected intervals.
MSC:54C60, 54C65, 55M10.
1 Introduction
Since Michael [1] constructed continuous ϵapproximate 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 BenElMechaiekh and Oudadess [3] generalized the theorem in [2] to a class of lower semicontinuous multimaps with nonconvex values in LCmetric spaces, which have generalized convex metric structures introduced by Horvath [4].
Using the concept of nconnectedness, Kim [5] introduced an LDmetric space and extended the result in [3] to LDmetric spaces which are more general than LCmetric spaces.
On the other hand, in LCspaces, Wu and Li [6] obtained the approximate selection theorems for quasilower semicontinuous multimaps which were generalized by the author and Lee [7] to almost lower semicontinuous multimaps in Cspaces.
In this paper, we establish a new approximate selection theorem for almost lower semicontinuous multimaps with Dset values except on a set of topological dimension less than or equal to zero in LDspaces. 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 Dspaces and apply the results to topological semilattices with path connected intervals. Our results unify and extend the approximate selection theorems in [1–3, 5–9].
2 Preliminaries
A multimap (or simply a map) $F:X\u22b8Y$ is a function from a set X into the power set of Y; that is, a function with the values $F(x)\subset Y$ for $x\in X$. For $A\subset X$, let $F(A):=\bigcup \{F(x)\mid x\in A\}$. Throughout this paper, we assume that multimaps have nonempty values otherwise explicitly stated or obvious from the context. Let $\u3008X\u3009$ denote the set of all nonempty finite subsets of a set X.
Let X be a topological space. A Cstructure on X is given by a map $\mathrm{\Gamma}:\u3008X\u3009\u22b8X$ such that

(1)
for all $A\in \u3008X\u3009$, ${\mathrm{\Gamma}}_{A}=\mathrm{\Gamma}(A)$ is nonempty and contractible; and

(2)
for all $A,B\in \u3008X\u3009$, $A\subset B$ implies ${\mathrm{\Gamma}}_{A}\subset {\mathrm{\Gamma}}_{B}$.
A pair $(X,\mathrm{\Gamma})$ is then called a Cspace by Horvath [4] and an Hspace by Bardaro and Ceppitelli [10]. For examples of a Cspace, see [4, 10]. For an $(X,\mathrm{\Gamma})$, a subset C of X is said to be Γconvex (or a Cset) if $A\in \u3008C\u3009$ implies ${\mathrm{\Gamma}}_{A}\subset C$.
For a uniform space X with a uniform structure $\mathcal{U}$, $A\subset X$ and $U\in \mathcal{U}$, the set $U(A)$ is defined to be $\{y\in X:(x,y)\in U\text{for some}x\in A\}$ and if $x\in X$, $U(x)=U(\{x\})$.
A Cspace $(X,\mathrm{\Gamma})$ is called an LCspace if X is a uniform space and there exists a base $\{{V}_{i}:i\in I\}$ for the uniform structure such that for each $i\in I$, $\{x\in X:C\cap {V}_{i}(x)\ne \mathrm{\varnothing}\}$ is Γconvex whenever $C\subset X$ is Γconvex.
A Cspace $(X,\mathrm{\Gamma})$ is called an LCmetric space if X is equipped with a metric d such that for any $\u03f5>0$, the set $B(C,\u03f5)=\{x\in X:d(x,C)<\u03f5\}$ is Γconvex whenever $C\subset X$ is Γconvex, and open balls are Γconvex. For details, see Horvath [4].
A topological space X is said to be nconnected for $n\ge 0$ if every continuous map $f:{S}^{k}\to X$ for $k\le n$ 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 ${\mathbb{R}}^{k+1}$. Note that a contractible space is nconnected for every $n\ge 0$.
The following is introduced by Kim [5]. Let X be a topological space. A Dstructure on X is a map $\mathcal{D}:\u3008X\u3009\u22b8X$ such that it satisfies the following conditions:

(1)
for each $A\in \u3008X\u3009$, $\mathcal{D}(A)$ is nonempty and nconnected for all $n\ge 0$;

(2)
for each $A,B\in \u3008X\u3009$, $A\subset B$ implies $\mathcal{D}(A)\subset \mathcal{D}(B)$.
The pair $(X,\mathcal{D})$ is called a Dspace; a subset C of X is said to be a $\mathcal{D}$set if $\mathcal{D}(A)\subset C$ for each $A\in \u3008C\u3009$.
A Dspace $(X,\mathcal{D})$ is called an LDspace if X is a uniform space and if there exists a base $\{{V}_{i}:i\in I\}$ for the uniform structure such that for each $i\in I$, the set $\{x\in X:C\cap {V}_{i}(x)\ne \mathrm{\varnothing}\}$ is a $\mathcal{D}$set whenever $C\subset X$ is a $\mathcal{D}$set.
A Dspace $(X,\mathcal{D})$ is called an LDmetric space if X is a metric space such that for each $\u03f5>0$, $B(C,\u03f5)$ is a $\mathcal{D}$set whenever $C\subset X$ is a $\mathcal{D}$set and open balls are $\mathcal{D}$sets.
Let X be a topological space and $(Y,\mathcal{D})$ be a Dspace with a uniformity $\mathcal{U}$. A multimap $F:X\u22b8Y$ is called:

(1)
lower semicontinuous (lsc) at $x\in X$ if for each open set W with $W\cap F(x)\ne \mathrm{\varnothing}$, there is a neighborhood $U(x)$ of x such that $F(z)\cap W\ne \mathrm{\varnothing}$ for all $z\in U(x)$;

(2)
quasilower semicontinuous (qlsc) at $x\in X$ if for each $V\in \mathcal{U}$, there are $y\in F(x)$ and a neighborhood $U(x)$ of x such that $F(z)\cap V(y)\ne \mathrm{\varnothing}$ for all $z\in U(x)$;

(3)
almost lower semicontinuous (alsc) at $x\in X$ if for each $V\in \mathcal{U}$, there is a neighborhood $U(x)$ of x such that ${\bigcap}_{z\in U(x)}V(F(z))\ne \mathrm{\varnothing}$.
If F is lsc (qlsc, alsc, resp.) at each $x\in X$, F is called lsc (qlsc, alsc, resp.). As in [[7], Proposition 3], (1) ⟹ (2) ⟹ (3).
For $V\in \mathcal{U}$, a continuous function $f:X\to Y$ is called a Vapproximate selection of F if for all $x\in X$, $f(x)\in V(F(x))$.
Let $(Y,\mathcal{D})$ be an LDmetric space. For $\u03f5>0$, f is called an ϵapproximate selection of F if for all $x\in X$, $f(x)\in B(F(x),\u03f5)$.
Let X be a topological space. If $Z\subset X$, then ${dim}_{X}Z\le 0$ means that $dimE\le 0$ for every set $E\subset Z$ which is closed in X, where dimE denotes the covering dimension of E. Note that if ${dim}_{X}Z\le 0$, then any locally finite open covering of Z has a disjoint locally finite open refinement.
3 Approximate selection theorems on $\mathcal{D}$spaces
As a main tool, we need Proposition 1 of Kim [5].
Proposition 3.1 Let X be a paracompact topological space and ℛ be a locally finite open covering of X, $(Y,\mathcal{D})$ be a Dspace, and $\eta :\mathcal{R}\to Y$ be a function. Then there exists a continuous function $f:X\to Y$ such that
for each $x\in X$.
With Proposition 3.1, we establish the Vapproximate 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 ${dim}_{X}Z\le 0$. Let $(Y,\mathcal{D})$ be an LDspace with a uniformity $\mathcal{U}$ and $\mathcal{D}(\{y\})=\{y\}$ for all $y\in Y$. If $F:X\u22b8Y$ is an alsc multimap such that $F(x)$ is a $\mathcal{D}$set for all $x\in X\mathrm{\setminus}Z$, then F has a Vapproximate selection for each $V\in \mathcal{U}$.
Furthermore, if X is a precompact uniform space or a compact topological space, there is a subset $A\in \u3008Y\u3009$ such that $f(X)\subset \mathcal{D}(A)$.
Proof For each $V\in \mathcal{U}$ and $x\in X$, there is a neighborhood $U(x)$ of x such that ${\bigcap}_{z\in U(x)}V(F(z))\ne \mathrm{\varnothing}$, because F is alsc. Since X is paracompact, the open cover $\{U(x):x\in X\}$ of X has a locally finite refinement $\{\tilde{U}(x):x\in X\}$. And since ${dim}_{X}Z\le 0$, the relatively open cover $\{\tilde{U}(x)\cap Z:x\in Z\}$ of Z has a relatively open disjoint refinement $\{W(x):x\in Z\}$. Z is closed in X so the collection $\mathcal{R}=\{\tilde{U}(x)\cap (W(x)\cup (X\mathrm{\setminus}Z)):x\in X\}$ forms a locally finite open cover of X.
For each $O\in \mathcal{R}$, choose ${x}_{o}$ such that $O\subset U({x}_{o})$ and ${y}_{o}\in {\bigcap}_{z\in U({x}_{o})}V(F(z))$. Define $\eta :\mathcal{R}\to Y$ by $\eta (O)={y}_{o}$ for all $O\in \mathcal{R}$. Then $\eta (O)\in {\bigcap}_{z\in U({x}_{o})}V(F(z))\subset {\bigcap}_{z\in O}V(F(z))$, so $\{\eta (O):O\in \mathcal{R},x\in O\}\subset V(F(x))$ for all $x\in X$. By Proposition 3.1, there is a continuous function $f:X\to Y$ such that $f(x)\in \mathcal{D}(\{\eta (O):O\in \mathcal{R},x\in O\})$.
We now show that $f(x)\in V(F(x))$ for all $x\in X$. If $x\in Z$, there exists a unique $O\in \mathcal{R}$ such that $x\in O$, that is, $\{\eta (O):O\in \mathcal{R},x\in O\}$ is a singleton. So, $f(x)\in \mathcal{D}(\{\eta (O):O\in \mathcal{R},x\in O\})=\{\eta (O)\}\subset V(F(x))$. If $x\in X\mathrm{\setminus}Z$, since $F(x)$ is a $\mathcal{D}$set, $\mathcal{D}(\{\eta (O):O\in \mathcal{R},x\in O\})\subset V(F(x))$, that is, $f(x)\in V(F(x))$.
If X is a precompact uniform space or a compact topological space, ℛ can be chosen finite. Take $A=\{\eta (O):O\in \mathcal{R}\}$, then $A\in \u3008Y\u3009$ and $f(X)\subset \mathcal{D}(A)$. □
Remark If $Z=\mathrm{\varnothing}$, then the condition ‘$\mathcal{D}(\{y\})=\{y\}$ for all $y\in Y$’ can be omitted. In that case, if $(Y,\mathcal{D})$ is an LCspace with a uniformity $\mathcal{U}$ and F is qlsc, then Theorem 3.2 becomes [[6], Theorem 3.1].
Proposition 3.3 Each singleton is a $\mathcal{D}$set in an LDmetric space $(X,\mathcal{D})$, so $\mathcal{D}(\{x\})=\{x\}$.
Proof For each $x\in X$, $\{x\}={\bigcap}_{\u03f5>0}B(x,\u03f5)$. Since all open balls and their intersection are $\mathcal{D}$sets, $\{x\}$ is a $\mathcal{D}$set. Therefore $\mathcal{D}(\{x\})\subset \{x\}$, i.e., $\mathcal{D}(\{x\})=\{x\}$. □
For LDmetric spaces, Theorem 3.2 reduces to the following.
Corollary 3.4 Let X be a paracompact space, $(Y,\mathcal{D})$ be an LDmetric space, and Z be a closed subset of X with ${dim}_{X}Z\le 0$. If $F:X\u22b8Y$ is an alsc multimap such that $F(x)$ is a $\mathcal{D}$set for all $x\in X\mathrm{\setminus}Z$, then for $\u03f5>0$, F has an ϵapproximate selection.
For LCmetric spaces, Corollary 3.4 reduces to the following.
Corollary 3.5 Let X be a paracompact space, $(Y,\mathrm{\Gamma})$ be an LCmetric space, and Z be a closed subset of X with ${dim}_{X}Z\le 0$. If $F:X\u22b8Y$ is an alsc multimap such that $F(x)$ is Γconvex for all $x\in X\mathrm{\setminus}Z$, then for $\u03f5>0$, F has an ϵapproximate 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 $B(F(x),\u03f5)$ is Γconvex whenever $x\in E$ and $x\in X\mathrm{\setminus}E$, but it cannot be analogized from the assumption that $F(x)$ is Γconvex for all $x\notin Z$.
Theorem 3.3 in [6] shows that if $X=Z$ and $(Y,\mathrm{\Gamma})$ is a Cspace with a uniformity $\mathcal{U}$ and F has a Vapproximate selection for each $V\in \mathcal{U}$, then F is qlsc. Using the same pattern of its proof, we also obtain the same result when $(Y,\mathcal{D})$ is a Dspace with a uniformity $\mathcal{U}$. 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 ${dim}_{X}Z\le 0$. Let $(Y,\mathcal{D})$ be an LDspace with a uniformity $\mathcal{U}$ and $\mathcal{D}(\{y\})=\{y\}$ for all $y\in Y$. And let $F:X\u22b8Y$ be a multimap such that $F(x)$ is a $\mathcal{D}$set for all $x\in X\mathrm{\setminus}Z$. Then F is alsc if and only if F has a Vapproximate selection for each $V\in \mathcal{U}$.
The following notion is motivated by Hadžić [11]. Let $(X,\mathcal{D})$ be a Dspace with a uniformity $\mathcal{U}$ and K be a nonempty subset of X. We say that K is of generalized Zima type whenever for every $V\in \mathcal{U}$, there exists a ${V}_{1}\in \mathcal{U}$ such that for every $N\in \u3008K\u3009$ and every $\mathcal{D}$set M of K, the following implication holds:
Note that an LDspace $(X,\mathcal{D})$ is of generalized Zima type.
If $Z=\mathrm{\varnothing}$, then the LDspace condition of Y can be weakened in Theorem 3.6.
Theorem 3.7 Let X be a paracompact topological space, $(Y,\mathcal{D})$ be a Dspace with a uniformity $\mathcal{U}$, and $F:X\u22b8Y$ be a multimap with $\mathcal{D}$set values such that $F(X)$ is of generalized Zima type. Then F is alsc if and only if F has a Vapproximate selection for each $V\in \mathcal{U}$.
The proof of Theorem 3.7 proceeds in the same fashion as Theorem 2 in [7], except that all Γconvex sets in a Cspace is replaced by $\mathcal{D}$sets in a $\mathcal{D}$space.
Let X be a topological space and Y be a uniform space with a uniformity $\mathcal{U}$. The multimaps $F,T:X\u22b8Y$ are said to be topologically separated if for each $x\in X$, there exist a neighborhood $U(x)$ of x and an element $V\in \mathcal{U}$ such that $F(U(x))\cap V(T(x))=\mathrm{\varnothing}$.
Theorem 3.8 Let X be a compact topological space and Z be a closed subset of X with ${dim}_{X}Z\le 0$. And let $(Y,\mathcal{D})$ be an LDspace with a uniformity $\mathcal{U}$ and $\mathcal{D}(\{y\})=\{y\}$ for all $y\in Y$. If $F,T:X\u22b8Y$ are two multimaps such that

(1)
F and T are topologically separated;

(2)
T is upper semicontinuous; and

(3)
F is an alsc multimap such that $F(x)$ is a $\mathcal{D}$set for all $x\in X\mathrm{\setminus}Z$.
Then, for each $V\in \mathcal{U}$, F has a Vapproximate selection $f:X\to Y$ such that
for all $x\in X$.
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, $Z=\mathrm{\varnothing}$, and F is sublower semicontinuous, that is, for each $x\in X$ and each neighborhood V of 0 in Y, there is $z\in F(x)$ and a neighborhood $U(x)$ of x in X such that for each $y\in U(x)$, $z\in F(y)+V$. Note that if Y is a topological vector space, then F is sublower semicontinuous if and only if F is qlsc; see [[6], Proposition 1.2].

2.
Wu and Li [[6], Theorem 3.6]: $(Y,\mathcal{D})$ is an LCspace with a uniformity $\mathcal{U}$, $Z=\mathrm{\varnothing}$, and F is qlsc.
Proposition 3.9 Let X be a topological space and Y be a metric space. If a multimap $F:X\u22b8Y$ is alsc at $x\in X$, then F is qlsc at $x\in X$.
Proof For $\u03f5>0$, there is a neighborhood $U(x)$ of x such that
Select any $y\in {\bigcap}_{z\in U(x)}B(F(z),\u03f5/2)$. For each $z\in U(x)$, choose ${y}_{z}\in F(z)$ such that $d(y,{y}_{z})<\u03f5/2$. Note that ${y}_{x}\in F(x)$ and $d({y}_{x},{y}_{z})\le d({y}_{x},y)+d(y,{y}_{z})<\u03f5$ for each $z\in U(x)$. Hence ${y}_{z}\in B({y}_{x},\u03f5)\cap F(z)$ for all $z\in U(x)$. □
The following result is a generalization of Theorem 3.7 in [6].
Theorem 3.10 Let X be a paracompact topological space, $(Y,\mathcal{D})$ be an LDmetric space, and Z be a closed subset of X with ${dim}_{X}Z\le 0$. If $F,T:X\u22b8Y$ are two multimaps such that

(1)
F and T are topologically separated;

(2)
T is upper semicontinuous; and

(3)
F is an alsc multimap such that $F(x)$ is a $\mathcal{D}$set for all $x\in X\mathrm{\setminus}Z$.
Then for each $\u03f5>0$, F has an ϵapproximate selection $f:X\to Y$ such that
for all $x\in X$.
Proof For each fixed $\u03f5>0$ and each $x\in X$, by (1) and (2), there exists a neighborhood $U(x)$ of x and an $\eta (x)>0$ such that $\eta (x)<\u03f5$, $F(U(x))\cap B(T(x),\eta (x))=\mathrm{\varnothing}$, and $T(U(x))\subset B(T(x),\eta (x)/2)$. Let $\zeta (x)=\eta (x)/2$. For each $y\in U(x)$, we assert $F(y)\cap B(T(y),\zeta (x))=\mathrm{\varnothing}$. Otherwise, there exist points $p\in T(y)$ and $z\in F(y)$ such that $d(p,z)<\zeta (x)$. Because $y\in U(x)$ and $T(y)\subset B(T(x),\zeta (x))$, so $p\in B(T(x),\zeta (x))$. Consequently, there is a point $b\in T(x)$ such that $d(p,b)<\zeta (x)$. Hence $d(b,z)<\eta (x)$, and thus $z\in F(y)\cap B(T(x),\eta (x))\subset F(U(x))\cap B(T(x),\eta (x))=\mathrm{\varnothing}$. It is a contradiction.
Let $\delta (x)=sup\{r:0<r<\u03f5\text{and}F(x)\cap B(T(x),r)=\mathrm{\varnothing}\}$. Obviously, $\delta (x)\le \u03f5$ and for each $y\in U(x)$, $\delta (y)\ge \zeta (x)$. Now, we assert that $F(x)\cap B(T(x),\delta (x))=\mathrm{\varnothing}$. Otherwise, there exist points $y\in F(x)$ and $z\in T(x)$ such that $d(y,z)<\delta (x)$. Consequently, there is a number $r>d(y,z)$ such that $0<r<\u03f5$ and $F(x)\cap B(T(x),r)=\mathrm{\varnothing}$. But $y\in F(x)\cap B(T(x),r)$, it is a contradiction.
By Proposition 3.9, $F:X\u22b8Y$ is qlsc, so there exist a point ${y}_{x}\in F(x)$ and an open neighborhood $N(x)$ of x in X such that $N(x)\subset U(x)$ and
for all $z\in N(x)$. Since X is paracompact, the open cover $\{N(x):x\in X\}$ has a locally finite open refinement $\{V(x):x\in X\}$. Since ${dim}_{X}Z\le 0$, the relative open cover $\{V(x)\cap Z:x\in Z\}$ of Z has a relatively open disjoint refinement $\{W(x):x\in Z\}$. Z is closed in X, so the collection $\mathcal{R}=\{V(x)\cap (W(x)\cup (X\mathrm{\setminus}Z)):x\in X\}$ forms a locally finite open cover of X.
For each $O\in \mathcal{R}$, choose a point ${x}_{o}$ such that $O\subset V({x}_{o})$ and define $\eta :\mathcal{R}\to Y$ by $\eta (O)={y}_{{x}_{o}}$ such that ${y}_{{x}_{o}}\in F({x}_{o})$ satisfying the condition (∗) for all $z\in V({x}_{o})$. By Proposition 3.1, there is a continuous function $f:X\to Y$ such that $f(x)\in \mathcal{D}(\{\eta (O):O\in \mathcal{R},x\in O\})$. For each $x\in X$ and $O\in \mathcal{R}$ such that $x\in O$, by (∗), we have $F(x)\cap B(\eta (O),\zeta ({x}_{o}))\ne \mathrm{\varnothing}$ and $\delta (x)\ge \zeta ({x}_{o})$ because $x\in O\subset V({x}_{o})\subset N({x}_{o})$, so $F(x)\cap B(\eta (O),\delta (x))\ne \mathrm{\varnothing}$.
Note that for $x\in Z$, $f(x)=\eta (O)={y}_{{x}_{o}}$ since $\{O\in \mathcal{R}:x\in O\}$ is a singleton. Hence $f(x)\in \{y\in Y:F(x)\cap B(y,\delta (x))\ne \mathrm{\varnothing}\}$.
For each $x\in X\mathrm{\setminus}Z$,
since $F(x)$ and $B(F(x),\delta (x))$ are $\mathcal{D}$sets.
So, $F(x)\cap B(f(x),\delta (x))\ne \mathrm{\varnothing}$ for all $x\in X$ and hence
□
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 $(x,{x}^{\prime})$ of elements has a least upper bound. Any nonempty set $A\in \u3008X\u3009$ has a least upper bound, denoted by sup A. In a partially ordered set $(X,\le )$, two arbitrary elements x and ${x}^{\prime}$ do not have to be comparable, but, in the case where $x\le {x}^{\prime}$, the set $[x,{x}^{\prime}]=\{y\in X:x\le y\le {x}^{\prime}\}$ is called an order interval.
The following is due to Horvath and Ciscar [12]: Let $(X,\le )$ be a semilattice such that for each $A\in \u3008X\u3009$, $\mathrm{\Delta}(A)$ is defined by ${\bigcup}_{a\in A}[a,supA]$. Then

(1)
$$\mathrm{\Delta}(A)$$
is well defined;

(2)
$$A\subset \mathrm{\Delta}(A)$$
;

(3)
if $A\subset B$, then $\mathrm{\Delta}(A)\subset \mathrm{\Delta}(B)$.
A subset $E\subset X$ is said to be convex if, for any subset $A\in \u3008E\u3009$, we have $\mathrm{\Delta}(A)\subset E$.
If X is a topological semilattice with pathconnected intervals, then for any $A\in \u3008X\u3009$ and $n\ge 0$, $\mathrm{\Delta}(A)$ is nconnected by [[12], Lemma 1], that is, $(X,\le ,\mathrm{\Delta})$ is a Dspace.
Note that $\mathrm{\Delta}(\{x\})=\{x\}$ for all $x\in X$.
In this section, we assume that $(Y,\le ,\mathrm{\Delta})$ is a topological semilattice with pathconnected 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 ${dim}_{X}Z\le 0$, and $\mathcal{U}$ be a uniformity of $(Y,\le ,\mathrm{\Delta})$ such that for each $V\in \mathcal{U}$, the set $\{y\in Y:C\cap V(y)\ne \mathrm{\varnothing}\}$ is convex whenever C is a convex subset of Y. And let $F\subseteq X\times Y$ be a binary relation such that $F(x)$ is convex for all $x\in X\mathrm{\setminus}Z$. Then F is alsc if and only if F has a Vapproximate selection for each $V\in \mathcal{U}$.
Theorem 4.2 Let X be a paracompact topological space and $\mathcal{U}$ be a uniformity of $(Y,\le ,\mathrm{\Delta})$. And let $F\subseteq X\times Y$ 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 Vapproximate selection for each $V\in \mathcal{U}$.
Theorem 4.3 Let X be a compact topological space and Z be a closed subset of X with ${dim}_{X}Z\le 0$. And let $\mathcal{U}$ be a uniformity of $(Y,\le ,\mathrm{\Delta})$ such that for each $V\in \mathcal{U}$, the set $\{x\in X:C\cap V(x)\ne \mathrm{\varnothing}\}$ is convex whenever $C\subset X$ is convex. If $F,T\subseteq X\times Y$ are two binary relations such that

(1)
F and T are topologically separated;

(2)
T is upper semicontinuous; and

(3)
F is an alsc relation such that $F(x)$ is convex for all $x\in X\mathrm{\setminus}Z$.
Then for each $V\in \mathcal{U}$, F has a Vapproximate selection $f:X\to Y$ such that
for all $x\in X$.
Theorem 4.4 Let X be a paracompact topological space and Z be a closed subset of X with ${dim}_{X}Z\le 0$. Assume that $(Y,\le ,\mathrm{\Delta})$ is a metric space and has the following properties:

(i)
for any $\u03f5>0$, the set $B(C,\u03f5)$ is convex whenever C is a convex subset of Y; and

(ii)
open balls are convex.
If $F,T\subseteq X\times Y$ are two binary relations such that

(1)
F and T are topologically separated;

(2)
T is upper semicontinuous; and

(3)
F is an alsc relation such that $F(x)$ is convex for all $x\in X\mathrm{\setminus}Z$.
Then for each $\u03f5>0$, F has an ϵapproximate selection $f:X\to Y$ such that
for all $x\in X$.
Abbreviations
 alsc:

almost lower semicontinuity
 lsc:

lower semicontinuity
 qlsc:

quasilower semicontinuity.
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This paper was supported by the Sehan University Research Fund in 2013.
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Kim, H. Approximate selection theorems with nconnectedness. J Inequal Appl 2013, 113 (2013). https://doi.org/10.1186/1029242X2013113
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Keywords
 lower semicontinuity
 almost lower semicontinuity
 quasilower semicontinuity
 approximate selection theorems
 Dspaces
 LCmetric spaces
 topological semilattice with pathconnected intervals