ON TOPOLOGICAL PROPERTIES OF ARGMIN MULTIFUNCTION

Abstract

Let $X$ be a Tychonoff topological space, $C(X,\Bbb R)$ be the space of continuous real-valued functions defined on $X$ and $K(X)$ be the space of all nonempty compact subsets of $X$. The multifunction $\argm: C(X,\Bbb R) \times K(X) \to X$ is defined as follows: $\argm(f,K) = \{x \in K: f(x) = \min\{f(y): y \in K\}\}$. We present topologies on $C(X,\Bbb R) \times K(X)$ under which $\argm: C(X,\Bbb R) \times K(X) \to X$ has a closed graph. We also extend a generic optimization theorem of G. Beer from \v Cech-complete spaces to locally \v Cech-complete ones.