\documentclass[11pt]{amsart} \newtheorem{theorem}{Theorem}[section] \theoremstyle{definition} \newtheorem*{example}{Example} \begin{document} \setlength{\unitlength}{0.01in} \linethickness{0.01in} \begin{center} \begin{picture}(474,66)(0,0) \multiput(0,66)(1,0){40}{\line(0,-1){24}} \multiput(43,65)(1,-1){24}{\line(0,-1){40}} \multiput(1,39)(1,-1){40}{\line(1,0){24}} \multiput(70,2)(1,1){24}{\line(0,1){40}} \multiput(72,0)(1,1){24}{\line(1,0){40}} \multiput(97,66)(1,0){40}{\line(0,-1){40}} \put(143,66){\makebox(0,0)[tl]{\footnotesize Proceedings of the Ninth Prague Topological Symposium}} \put(143,50){\makebox(0,0)[tl]{\footnotesize Contributed papers from the symposium held in}} \put(143,34){\makebox(0,0)[tl]{\footnotesize Prague, Czech Republic, August 19--25, 2001}} \end{picture} \end{center} \vspace{0.25in} \setcounter{page}{309} \title[lsc multifunctions in quasi-uniform and vector spaces]{On lower semicontinuous multifunctions in quasi-uniform and vector spaces} \author{Andrzej Spakowski} \address{Instytut Matematyki i Informatyki, Uniwersytet Opolski,\\ Oleska 48, 45-052 Opole, Poland} \email{aspakow@uni.opole.pl} \keywords{Lower semicontinuous multifunctions (set-valued maps), cartesian product, intersection, vector sum, and convex hull of multifunctions.} \subjclass[2000]{54C60, 54E15, 46A16} \thanks{Andrzej Spakowski, {\em On lower semicontinuous multifunctions in quasi-uniform and vector spaces}, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp.~309--319, Topology Atlas, Toronto, 2002; {\tt arXiv:math.GN/0204142}} \begin{abstract} Given a cover $\mathcal{B}$ of a quasi-uniform space $Y$ we introduce a concept of lower semicontinuity for multifunctions $F:X\to 2^Y$, called $\mathcal{B}$-lsc. In this way, we get a common description of Vietoris-lsc, Hausdorff-lsc, and bounded-Hausdorff-lsc as well. Further, we examine set-theoretical and vector operations on such multifunctions. We also point out that the convex hull of Hausdorff-lsc multifunctions need not to be Hausdorff-lsc except the case where the range space is locally convex. \end{abstract} \maketitle \section{Lower semicontinuous multifunctions} The two most known concepts of lower semicontinuity for multifunctions are the lower semicontinuity in Vietoris sense (V-lsc) and the lower semicontinuity in Hausdorff sense (H-lsc). Given a set $Y$ we denote by $2^Y$ the family of all subsets of $Y$. Every map $F:X\rightarrow 2^Y$ will be called a {\it multifunction} from $X$ to $Y$. Now, let $X$ and $Y$ be two arbitrary topological spaces. We say that a multifunction $F:X \rightarrow 2^Y$ is {\it V-lsc} at a point $x_0 \in X$ provided for every open $G \subset Y$ such that $F(x_0) \cap G \neq \emptyset $ there exists a neighbourhood $U(x_0)$ of $x_0$ such that $F(x)\cap G\neq\emptyset$ for every $x\in U(x_0)$. This is the first concept of lower semicontinuity. Let $(Y,{\mathcal{U}})$ be a uniform space. Recall that every uniformity generate a topology, and a topological space is uniformizable provided it is a Tichonov space. For multifunctions from $X$ to $(Y,\mathcal{U})$ we may formulate the second concept of lower semicontinuity. Namely, a multifunction $F:X \rightarrow 2^Y$ is called {\it H-lsc at} $x_0 \in X$ if for every $W\in{\mathcal{U}}$ there exists a neighbourhood $U(x_0)$ of $x_0$ such that $$ F(x_0) \subset W(F(x))\ \text{for every}\ x\in U(x_0), $$ where $$W(F(x))=\{\ y \in Y: (z,y) \in W \textrm{ for some } z \in F(x)\}.$$ This is the second concept of lower semicontinuity. In particular, if $Y$ is a topological vector space, with its natural uniformity generated by the neighbourhoods of 0, the condition of H-lsc can be written in the equivalent form: $$ F(x_0) \subset F(x)+V\ \text{for every}\ x\in U(x_0),$$ where V is a neighbourhood of 0 and $C+D=\{ c+d:c \in C,\, d \in D\}$ is the vector sum of sets $C$ and $D$. It is known that every topological space $Y$ is {\it quasi-uniformizable} (\cite{Pervin-1962, Murdeshwar-Naimpally-1966}). This means that there is family ${\mathcal{U}}$ of subsets of $Y\times Y$ such that: \begin{enumerate} \item every $U\in{\mathcal{U}}$ contains the diagonal $\Delta$ of $Y\times Y$, \item $U,V \in {\mathcal{U}}$ implies that $U \cap V \in {\mathcal{U}}$, \item for every $U \in {\mathcal{U}}$ there exists $V \in\mathcal{U}$ such that $V \circ V \subset U$, where $V \circ V = \{ (x,y) \in Y \times Y:(x,z), (z,y) \in V \textrm{ for some } z\in Y \}$, \item $U \in\mathcal{U}$ and $U \subset V$ implies $V \in\mathcal{U}$, \item the family $\{ W(y):W \in\mathcal{U},\ y \in Y \} $ is a neighbourhood system generating the topology of $Y$, where $W(y)= \{ z \in Y:(y,z) \in W \}$. \end{enumerate} Every such family is called a {\it quasi-uniformity of the topological space} $Y$. If $Y$ is only a set, then a family $\mathcal{U}$ of subsets of $Y\times Y$ satisfying properties (1)--(4) is called a {\it quasi-uniformity on} $Y$, and the pair $(Y,\mathcal{U})$ a {\it quasi-uniform space}. A quasi-uniform space $(Y,\mathcal{U})$ is a uniform space provided $\mathcal{U}$ have the following symmetric property: $W\in\mathcal{U}$ implies $W^{-1}\in\mathcal{U}$, where $W^{-1}=\{ (z,y)\in Y\times Y:(y,z)\in W \}$. In every topological space $Y$ we have: for every $A\subset Y$ and every quasi-uniformity $\mathcal{U}$ of $Y$, $\textrm{\it cl\,}(A)=\bigcap\{ W^{-1}(A): W\in\mathcal{U} \}$. where $\textrm{\it cl\,}(A)$ denotes the closure of $A$ in $Y$. The usage of $W^{-1}$ is explained by the following facts: \begin{enumerate} \item the sets $W(y)$, where $W\in\mathcal{U}$ and $y\in Y$, form a neighbourhood system for the topology of $Y$, \item for arbitrary $A\subset Y$ we have: $y\in\textrm{\it cl\,}(A)$, the closure of $A$, if and only if for every $W\in\mathcal{U}$, $W(y)\cap A\neq\emptyset$ if and only if there exists $a\in A$ such that $a\in W(y)$, or equivalently $y\in W^{-1}(a)$. \end{enumerate} For a quasi-uniform $(Y,\mathcal{U})$ space the definition of H-lsc should be modified as follows. A multifunction $F:X \rightarrow 2^Y$ is called {\it H-lsc at} $x_0 \in X$ if for every $W \in \mathcal{U}$ there exists a neighbourhood $V$ of $x_0$ such that $$ F(x_0) \subset W^{-1}(F(x))\ \text{for every}\ x\in V. $$ Note that we have the following property: $F$ is H-lsc at $x_0$ if and only if $\textrm{\it cl\,}(F)$ is H-lsc, where $\textrm{\it cl\,}(F)$ is the closure multifunction of $F$, i.e.\ $\textrm{\it cl\,}(F)(x)=\textrm{\it cl\,}(F(x))$ for all $x\in X$. The basic relationships between V-lsc and H-lsc are well-known (see \cite{Klein-Thompson-1984}, \cite{Lechicki-1979}). Namely, if $Y$ is a topological space, $\mathcal{U}$ a quasi-uniformity of $Y$ and $F:X \rightarrow 2^Y$ a multifunction, then H-lsc of $F$ at $x_0\in X$ implies its V-lsc at $x_0$. The converse holds provided the set $F(x_0)$ is totally bounded. Recall that $A\subset Y$ is called {\it totally bounded} provided for every $W\in \mathcal{U}$ there exists a finite set $B\subset A$ such that $A\subset W^{-1}(B)$. In general, V-lsc need not imply H-lsc. \section{$\mathcal{B}$-lower semicontinuity} Penot \cite{Penot-1993} introduced a concept of bounded lower semicontinuity for multifunction from a topological space to a normed space. In \cite{Borwein-Vanderwerff-1996} a similar idea is applied to convergence of sets, in particular, to convergence of epigraphs, with respect to the families of single subsets, compact subsets, weakly compact subsets, and of bounded subsets. Following this, we define an abstract concept of lower semicontinuity to unify the description of the above mentioned semicontinuities. Let $X$ be a topological space, $(Y,\mathcal{U})$ a quasi-uniform space, $\mathcal{B}$ a {\it cover} of $Y$, i.e., a family of nonempty subsets of $Y$ such that $\bigcup\mathcal{B}=Y$. We say that $F:X\rightarrow 2^Y$ is {\it $\mathcal{B}$-lsc} at $x_0\in X$ provided for every $W\in \mathcal{U}$ and every $B\in\mathcal{B}$ there exists a neighbourhood $U(x_0)$ of $x_0$ such that $$ F(x_0)\cap B\subset W^{-1}(F(x))\ \text{for every}\ x\in U(x_0).$$ If $F(x_0)=\emptyset$, then $F$ is trivially $\mathcal{B}$-lsc at $x_0$ for arbitrary cover $\mathcal{B}$. Note also the following simple observations and remarks: \begin{enumerate} \item If $\mathcal{B}=\{ Y \}$, then $\mathcal{B}$-lsc is simply the H-lsc. \item If $\mathcal{B}$ is a cover of $Y$ and $\mathcal{B}_1$ the family of all finite unions of subsets of $\mathcal{B}$, then $\mathcal{B}$-lsc implies the $\mathcal{B}_1$-lsc. \item If $\mathcal{B}\subset\mathcal{B}_1$, then $\mathcal{B}_1$-lsc implies $\mathcal{B}$-lsc. \item If $\mathcal{B}$ is the family of all balls $B(r)=\{ y\in Y:||y||0$. $F$ is $\mathcal{B}$-usc at 0 but not V-usc at this point. \subsubsection*{} See \cite{Costantini-1994} for the bounded H-usc, i.e.\ for the $\mathcal{B}$-usc with $\mathcal{B}$ being the family of all bounded subsets of $Y$. \subsubsection*{} It is clear that $\mathcal{B}$-lsc is topologizable provided $\mathcal{B}$ is the family of all singletons or $\mathcal{B}=\{Y\}$. In general, $\mathcal{B}$-lsc is not topologizable. To show this we use the following Diagonalization Criterion (see e.g.\ \cite{Klein-Thompson-1984}): \begin{theorem} Let $T$ be a directed set and for each $t\in T$ there is another directed set $E(t)$. Then we define a new directed set $D=T\times\Pi_{t\in T}E(t)$ ordered as follows: $(t,(\alpha(t)))\leq(s,(\beta(t)))$ if and only if $t\leq s$ and $\alpha(t)\leq\beta(t)$ for each $t\in T$. Suppose that $z(t,\gamma)$, $t\in T$, $\gamma\in E(t)$, are elements of a topological space $Z$. Consider the following net: $z(t,\alpha)=z(t,\alpha(t))$, $(t,\alpha)\in D$, where $\alpha(t)$ is the $t$-coordinate of $\alpha$. If $\lim_{t\in T}\lim_{\gamma\in E(t)}z(t,\gamma)=z$ then $\lim_{(t,\alpha)\in D}z(t,\alpha(t))=z$. \end{theorem} Now, we can construct an example of a non-topologizable $\mathcal{B}$-lsc. Observe first that $\mathcal{B}$-lsc is simply continuity with respect to the following $\mathcal{B}^-$-convergence: $A_\lambda\rightarrow A_0$ whenever for every $W\in \mathcal{U}$ and every $B\in\mathcal{B}$ there exists $\lambda_0$ such that $$ A_0\cap B\subset W^{-1}(A_\lambda)\ \text{for every}\ \lambda>\lambda_0. $$ Let $T={\mathbb N}$ and for each $n\in T$ we take $E(n)={\mathbb N}$. Let $Y={\mathbb R}^2$ and $\mathcal{B}$ be the family of all straight line through the $(0,0)$. For every $n,k\in{\mathbb N}$ we denote: $A(n,k)=$ the line $\{y=(1/k)x+1/n\}$, $A(n)=$ the line $\{y=1/n\}$, $A(0)=B(0)=$ the line $\{y=0\}$. It is easy to check $\mathcal{B}^-$-convergence: $\lim_n\lim_kA(n,k)=\lim_nA(n)=A(0)$. On the other hand the convergence $\lim_{(n,\alpha)}A(n,\alpha(n))=A_0$ does not holds for $A(0)\cap B(0)=$ the line $\{y=0\}$ is not contained in any $A(n,\alpha(n))+V$, where $V$ is a neighbourhood of $(0,0)$. Thus the $\mathcal{B}^-$-convergence is not topologizable. \section{Unions and cartesian products} In this paragraph we deal with some set-theoretical operations on multifunctions, namely with unions and cartesian products (see \cite{Klein-Thompson-1984}). Operation of intersection of multifunctions will be examined separately in the next paragraph. \subsection*{Unions} Let $X$ and $Y$ be spaces and $F_i:X \rightarrow 2^Y$, $i\in I$, a family of multifunctions. The {\it union $F=\bigcup_{i\in I}F_i$ of multifunctions} $F_i$ is defined by $F(x) = \bigcup_{i\in I}F_i(x)$, $x\in X$. It is known and easy to prove that the union of an arbitrary family of V-lsc multifunctions is V-lsc. However, the union of an infinite family of H-lsc multifunctions need not to be H-lsc. For instance, define $F_n(0)=[0,n]$ and $F_n(x)=[0,1/ x]$ for $x\in (0,1]$, $n=1,2,\ldots$. Then the multifunctions $F_n$ are H-lsc at 0. But, their union $F=\bigcup_n F_n$ is not H-lsc at 0 for $F(0)=[0,+\infty)$. \begin{theorem} Let $X$ be a topological space, $(Y,\mathcal{U})$ a quasi-uniform space and $F_1,F_2:X \rightarrow 2^Y$ multifunctions $\mathcal{B}$-lsc at $x_0\in X$. Then the union multifunction $F=F_1\cup F_2$ is $\mathcal{B}$-lsc at $x_0$. \end{theorem} \begin{proof} The proof is a consequence of the formula: $$W^{-1}(F_1(x))\cup W^{-1}(F_2(x))=W^{-1}(F_1(x)\cup F_2(x)),$$ where $W\in\mathcal{U}$ and $x\in X$. \end{proof} It is clear that the above theorem holds for finitely many multifunctions, and need not hold when we consider an infinite family of multifunctions. \subsection*{Products} Now, let us describe the cartesian product of multifunctions. Let $X$ and $Y_1,Y_2$ be spaces and $F_i:X \rightarrow 2^{Y_i}$, $i=1,2$, multifunctions. The {\it product} of two multifunctions $F_1$ and $F_2$ is defined as the multifunction $F=F_1\times F_2:X\rightarrow 2^{Y_1\times Y_2}$ such that $F(x)=F_1(x)\times F_2(x)$, $x\in X$. In particular, if $F_2(x)=Y_2$ for all $x$, or $F_1(x)=Y_1$ for all $x$, we will write simply, $F_1\times Y_2$, or $Y_1\times F_1$, respectively. Analogously, we define the product $\Pi_{i\in I}F_i$ of an arbitrary family of multifunctions $F_i$, $i\in I$. It is known that the product of an arbitrary family of V-lsc (H-lsc) multifunctions is also V-lsc (H-lsc). Remark that the product of H-lsc multifunctions has more complicated nature than the product of V-lsc ones. To formulate a general theorem for $\mathcal{B}$-lsc we need to consider the product of quasi-uniform spaces. First, we describe the product of two quasi-uniform spaces. Let $(Y_i,\mathcal{U}_i)$, $i=1,2$, be quasi-uniform spaces, and $P_i:Y_1\times Y_2\rightarrow Y_i$, $i=1,2$, be the projections, i.e.\ $P_i(y_1,y_2)=y_i$, $i=1,2$. By the {\it product quasi-uniformity $\mathcal{U}=\mathcal{U}_1\times \mathcal{U}_2$ in $Y=Y_1\times Y_2$ } we mean a quasi-uniformity generated by the base consisting of sets of the form $$ [W_1,W_2]= \{ (z_1,z_2)\in Y\times Y:(P_i(z_1),P_i(z_2))\in W_i,\ i=1,2 \}, $$ where $W_i\in\mathcal{U}_i$, $i=1,2$. In other words, the set $[W_1,W_2]$ has a form: $$ \{ (s_1,y_1,s_2,y_2)\in Y_1\times Y_2\times Y_1\times Y_2: (s_1,s_2)\in W_1,\, (y_1,y_2)\in W_2\}. $$ Remark that $[W_1,W_2]=\tilde W_1\cap\tilde W_2$, where $$ \tilde W_i= \{ (z_1,z_2)\in Y\times Y:(P_i(z_1),P_i(z_2))\in W_i \},\ i=1,2. $$ The sets $\tilde W_i$, $W_i\in\mathcal{U}_i$, $i=1,2$, form a subbase of the product quasi-uniformity $\mathcal{U}_1\times\mathcal{U}_2$. In case of an arbitrary family of quasi-uniform spaces we proceed similarly as above and as in the construction of product topological structures. Let $(Y_i,\mathcal{U}_i)$, $i\in I$, be a family od quasi-uniform spaces. Denote: $ Y=\Pi_{i\in I}Y_i$, $P_i$ the projection on the i-th axis, i.e.\ $P_i(y)=y_i$, where $y_i$ is the i-th coordinate of $y\in Y$, $i\in I$. By the {\it product quasi-uniformity $\mathcal{U}=\Pi_{i\in I}\mathcal{U}_i$ in $Y$} we mean a quasi-uniformity generated by the subbase consisting of sets of the form $$ \tilde W_i= \{ (z_1,z_2)\in Y\times Y:(P_i(z_1),P_i(z_2))\in W_i \}, $$ where $W_i\in\mathcal{U}_i$, $i\in I$. Observe that if $\mathcal{B}_i$ is a cover of $Y_i$, $i\in I$, then $\Pi_{i\in I}\mathcal{B}_i$, i.e., the family of all sets of the form $\Pi_{i\in I} B_i$ with $B_i\in\mathcal{B}_i$ and $B_i=Y_i$ for all but a finite number of $i\in I$, is a cover of $\Pi_{i\in I}Y_i$. \begin{theorem}\label{produktowy} Let $(Y_i,\mathcal{U}_i)$, $i=1,2$, be quasi-uniform spaces and $A_i\subset Y_i$, $i=1,2$, be arbitrary subsets. Then \begin{enumerate} \item $[W_1,W_2](A_1\times A_2) = W_1(A_1)\times W_2(A_2)$, \item $[W_1,W_2]^{-1}(A_1\times A_2) = W_1^{-1}(A_1)\times W_2^{-1}(A_2)$. \end{enumerate} \end{theorem} \begin{proof} We have $$ \begin{array}{ll} \multicolumn{2}{l}{ [W_1,W_2](A_1\times A_2) } \\ & = \{ (s_1,s_2)\in Y_1\times Y_2: \exists\,y_1\in A_1\ \exists\,y_2\in A_2 \,(y_1,y_2,s_1,s_2)\in [W_1,W_2] \}\\ & = \{ (s_1,s_2)\in Y_1\times Y_2: \exists\,y_1\in A_1\ \exists\,y_2\in A_2 \,(y_1,s_1)\in W_1,\,(y_2,s_2)\in W_2 \}\\ & = \{ (s_1,s_2)\in Y_1\times Y_2:s_1\in W_1(A_1),\,s_2\in W_2(A_2) \}\\ & = W_1(A_1)\times W_2(A_2), \end{array} $$ which proves (1). The proof of (2) is similar. \end{proof} \begin{theorem} Let $X$ be a topological space, $(Y_i,\mathcal{U}_i)$, $i\in I$, a family of quasi-uniform spaces and $F_i:X\rightarrow 2^{Y_i}$ a multifunction $\mathcal{B}_i$-lsc at $x_0\in X$, where $\mathcal{B}_i$ is a cover of $Y_i$, $i\in I$. Then the product multifunction $\Pi_{i\in I}F_i$ is $\Pi_{i\in I}\mathcal{B}_i$-lsc at $x_0$. \end{theorem} \begin{proof} By the construction of the product of quasi-uniform spaces $(Y_i,\mathcal{U}_i)$, $i\in I$, it is sufficient to consider only the case $I=\{ 1,2\}$. In general case, the proof is similar. Let multifunctions $F_i:X\rightarrow 2^{Y_i}$, $i=1,2$, be $\mathcal{B}_i$-lsc at $x_0\in X$ and $W\in\mathcal{U}_1\times\mathcal{U}_2$ be arbitrary. There exist $W_i\in\mathcal{U}_i$, $i=1,2$, such that $[W_1,W_2]\subset W$. Now, let $B_i\in \mathcal{B}_1$. There exists a neighbourhood $U(x_0)$ of $x_0$ such that $F_i(x_0)\cap B_i\subset W_i^{-1}(F_i(x))$, $i=1,2,\ x\in U(x_0)$, and, by the Lemma \ref{produktowy}, we get $$ (F_1(x_0)\cap B_1)\times (F_2(x_0)\cap B_2)\subset [W_1,W_2]^{-1}(F_1(x)\times F_2(x)), $$ for every $x\in U(x_0)$. This shows the $\mathcal{B}_1\times\mathcal{B}_2$-lsc of $F_1\times F_2$ because $$ (F_1(x_0)\cap B_1)\times (F_2(x_0)\cap B_2) = (F_1(x_0)\times F_2(x_0))\cap (B_1\times B_2). $$ \end{proof} \subsection*{Remark} The converse theorem also holds. Namely, if a product multifunction $\Pi_{i\in I}F_i$ is $\Pi_{i\in I}\mathcal{B}_i$-lsc at $x_0$, then for every $i\in I$ the multifunction $F_i$ is $\mathcal{B}_i$-lsc at $x_0$. \section{Intersections} In optimization theory the lower semicontinuity properties of intersections of multifunctions play an important role \cite{Penot-1993}. The most wanted theorems are ones with no boundedness conditions on the values of intersecting multifunctions. Here we formulate a theorem of this kind. Let $Y$ be a normed space. If we assume that the considered multifunctions are {\it boundedly H-lsc}, i.e., $\mathcal{B}$-lsc with $\mathcal{B}$ being the family of all balls $B(r)$, $r>0$, then we may formulate a theorem on intersection, without boundedness conditions on $F(x_0)$. For this we need a lemma from \cite{Lechicki-Spakowski-1985} on interiority properties of convex, bounded, and with the nonempty interior subsets of a normed space. \begin{theorem}\label{interior-property} Let $Y$ be a normed space and $A\subset Y$ be convex, bounded, and with the nonempty interior. Then for every $\varepsilon >0$ there exist a set $C\subset int(A)$ and $\delta >0$ such that $C+B(\delta )\subset A \subset C+B(\varepsilon )$. \end{theorem} We need also the following well-known (see \cite{Rabinovich-1967, Urbanski-1976}) and very useful {\it law of cancellation}: \begin{theorem} Let $A$, $B$ and $C$ be subsets of a topological vector space $Y$. Assume that $B$ is bounded, and $C$ is nonempty, closed and convex. Then $A+B\subset \textrm{\it cl\,}(C+B)$ implies $A\subset C$. In particular, $A+B\subset C+B$ implies $A\subset C$, and $\textrm{\it cl\,}(A+B)\subset \textrm{\it cl\,}(C+B)$ implies $A\subset C$. \end{theorem} Let $X$ be a topological space and $Y$ a topological vector space. A multifunction $F$ from $X$ to $Y$ is called {\it locally convex-valued} ({\it locally closed-valued}) {\it at} $x_0\in X$ if there is a neighbourhood $U$ of $x_0$ such that $F(x)$ is convex (closed) for every $x\in U$. \begin{theorem} Let $X$ be a topological space, $Y$ a normed space, $\mathcal{B}$ the family of balls $B(r)\subset Y$, $00$ such that \begin{equation}\label{star0} \textrm{\it int\,}(F(x_0)\cap B(r))\neq\emptyset. \end{equation} Let $\varepsilon >0$ be arbitrary. By the Lemma \ref{interior-property} there exist a subset $C\subset F(x_0)\cap B(r)$ and $\delta >0$ such that $C+B(\delta )\subset F(x_0)\cap B(r)\subset C+B(\varepsilon )$. In virtue of the $\mathcal{B}$-lsc at $x_0$ there exists a neighbourhood $U(x_0)$ of $x_0$ such that $$ C+B(\delta )\subset F_i(x_0)\cap B(r)\subset F_i(x)+B(\delta)\ \text{for}\ x\in U(x_0)\ \text{and}\ i=1,2. $$ We can assume that the multifunctions are closed- and convex-valued on $U(x_0)$. Applying the law of cancellation, we infer that $C\subset F_1(x)\cap F_2(x)$ for every $x\in U(x_0)$. But this implies that $$ \begin{array}{lll} F_1(x_0)\cap F_2(x_0)\cap B(r)& \subset& C+B(\varepsilon )\\ & \subset& F_1(x)\cap F_2(x)+B(\varepsilon) \end{array} $$ for all $x\in U(x_0)$. This shows that the intersection $F=F_1\cap F_2$ is $\mathcal{B}$-lsc at $x_0$ and ends the proof. \end{proof} \subsection*{Remark} If $Y$ is finite dimensional we can omit in the above theorem the assumption that the multifunctions are locally closed-valued, and then proceed in a manner as in \cite{Lechicki-Spakowski-1985} using the below theorem on local interior property of $\mathcal{B}$-lower semicontinuous multifunctions. \section{Vector operations} Here we consider vector sum and convex hull operations on lower semicontinuous multifunctions with values in a topological vector space (see e.g.\ \cite{Michael-1951}). Let $X$ be a topological space, $Y$ a topological vector space and $F,G:X\rightarrow 2^Y$. We define two multifunctions: $$ \begin{array}{lll} (F+G)(x)& =& F(x)+G(x)\\ & =& \{ a+b:a\in F(x),\ b\in G(x) \},\ x\in X, \end{array} $$ called the {\it vector sum} of $F$ and $G$, and $\textrm{\it conv\,}(F)(x)=\textrm{\it conv\,}(F(x))$ the convex hull of $F(x)$, $x\in X$, called the {\it convex hull} of $F$. \subsection*{Vector sum} It is known and easy to prove that the vector sum of two H-lsc multifunctions is H-lsc. We state some further results and show that, in general, the vector sum of two $\mathcal{B}$-lsc multifunctions need not to be $\mathcal{B}$-lsc. \begin{theorem} Let $F,G:X\rightarrow 2^Y$ be V-lsc at $x_0\in X$. Then the vector sum $F+G$ is V-lsc at $x_0$. \end{theorem} \begin{proof} Let $V$ be an arbitrary neighbourhood of 0 in $Y$, $F$ and $G$ be V-lsc at $x_0$, and recall that V-lsc is equivalent to $\mathcal{B}$-lsc with $\mathcal{B}$ equals the family of all singletons of $Y$. Let $b\in Y$ be such that $(F(x_0)+G(x_0))\cap\{ b\}\neq\emptyset$. Then $b=b_1+b_2$ with $b_1\in F(x_0)$ and $b_2\in G(x_0)$. By the $\mathcal{B}$-lsc of $F$ and $G$ there exists a neighbourhood $U(x_0)$ such that $$ \begin{array}{l} F(x_0)\cap\{ b_1\}\subset F(x)+V\ \text{and}\\ G(x_0)\cap\{ b_2\}\subset G(x)+V \end{array} $$ for all $x\in U(x_0)$. This implies that $$b_1+b_2\in F(x)+G(x)+V+V$$ for $x\in U(x_0)$, or equivalently, $$(F(x_0)+G(x_0))\cap\{ b\}\subset F(x)+G(x)+V+V$$ for $x\in U(x_0)$, which shows $\mathcal{B}$-lsc of $F+G$ at $x_0$ and ends the proof. \end{proof} We say that a cover $\mathcal{B}$ of $Y$ is {\it translation invariant} if $B+c\in\mathcal{B}$ for every $B\in\mathcal{B}$ and every vector $c\in Y$. \begin{theorem} Let $\mathcal{B}$ be a translation invariant cover of $Y$, $F:X\rightarrow 2^Y$ a multifunction $\mathcal{B}$-lsc at $x_0\in X$ and $g:X\rightarrow Y$ a function continuous at $x_0$. Then the vector sum $(F+g)(x)=F(x)+g(x)$, $x\in X$, is a multifunction $\mathcal{B}$-lsc at $x_0$. \end{theorem} \begin{proof} Let $V$ be a neighbourhood of 0 in $Y$ and $B\in\mathcal{B}$. Note that $$(F(x_0)+g(x_0))\cap B=F(x_0)\cap(B-g(x_0))+g(x_0)$$ By the assumptions for all $x$ in a neighbourhood of $x_0$ we have $$\begin{array}{lll} (F(x_0)+g(x_0))\cap B& \subset& (F(x)+g(x_0)+V)\cap B\\ & \subset& (F(x)+g(x)+V+V). \end{array}$$ This shows $\mathcal{B}$-lsc of $F+g$ at $x_0$ and ends the proof. \end{proof} The following example shows that translation invariantness of $\mathcal{B}$ is not sufficient to get $\mathcal{B}$-lsc of the vector sum of a two $\mathcal{B}$-lsc multifunctions. \begin{example} Let $Y={\mathbb R}^3$, $\mathcal{B}$ consists only of the plane $\{ y=x\}$ and all of its translations. Define two multifunctions: $F(t)$ the line $\{ z=ty,\, x=0\}$ and $G(t)$ the line $\{ y=0,\, z=0\}$, $t\geq 0$. Observe that $F$ and $G$ are $\mathcal{B}$-lsc but their vector sum $F+G$ is not $\mathcal{B}$-lsc at each point. \end{example} \subsection*{Convex hull} It is known \cite{Michael-1956} and easy to proof that for every V-lsc multifunction $F$, the convex hull of $F$ is also V-lsc. We use the concept of $\mathcal{B}$-lsc to get a general result provided the space $Y$ is locally convex. In particular, we get a result for H-lsc. \begin{theorem} Let $Y$ be a locally convex space and $F:X\rightarrow 2^Y$ be $\mathcal{B}$-lsc at $x_0\in X$. Then the convex hull of $F$ is $\mathcal{B}$-lsc at $x_0$. \end{theorem} \begin{proof} Let $V\subset Y$ be a convex neighbourhood of 0 in $Y$ and $B\in\mathcal{B}$. By the $\mathcal{B}$-lsc of $F$ at $x_0$ there exists a neighbourhood $U(x_0)$ of $x_0$ such that \begin{equation}\label{star} F(x_0)\cap B\subset F(x)+V\ \text{for all}\ x\in U(x_0). \end{equation} We claim that $$ \textit{conv\,}(F(x_0))\cap B\subset\textit{conv\,}(F(x))+V\ \text{for all}\ x\in U(x_0). $$ Indeed, let $y\in\textit{conv\,}(F(x_0))\cap B$ and $x\in U(x_0)$ be arbitrary. In virtue of (\ref{star}) there exist $n\in{\mathbb N}$, $y_1,\ldots,y_n\in F(x_0)$, $z_1,\ldots,z_n\in F(x)$, $v_1,\ldots,v_n\in V$, and positive numbers $t_1,\ldots,t_n$ such that $y=t_1y_1+\cdots +t_ny_n$, $t_1+\cdots +t_n=1$, and $y_i=z_i+v_i$ for every $1\leq i\leq n$. This implies that $$ y=t_1z_1+\cdots +t_nz_n+t_1v_1+\cdots +t_nv_n\in \textit{conv\,}(F(x))+V $$ for $V$ is convex, which ends the proof. \end{proof} \subsection*{Remark} If the topological vector space $Y$ is not locally convex then the convex hull operation does not preserve H-lsc. Indeed, if $Y$ is metrizable and not locally convex then there exists a sequence $y_n\in Y$ which converges to $0$ such that the convex hull of the set $\{y_n:n=1,2,\ldots\}$ is not bounded \cite{Araki-1995}. Now, observe that the multifunction $F$ defined by: $F(0)=\{0,y_1,y_2,\ldots\}$, $F(1/n)=\{0,y_1,\ldots,y_n\}$, $n=1,2,\ldots$, is H-lsc at $0$ but the convex hull of $F$ is not. For a simple example of such sequence $y_n\in l^p$ ($0