Most of the minimization problems have a unique solution

Abstract

Let X be a Tychonoff topological space, $C\left(X\right)$ be the space of continuous real-valued functions defined on X and $K\left(X\right)$ be the space of all nonempty compact subsets of X. Define the multifunction argmin$:C\left(X\right)\times K\left(X\right)\to X$ as follows: argmin $(f,K)=\{x\in K:f(x)=\min \{f\left(y\right):y\in K\left\}\right\}$. Let ${\tau }_U$ be the topology of uniform convergence on $C\left(X\right)$ and ${\tau }_V$ the Vietoris topology on $K\left(X\right)$. We prove that argmin$:\left(C\right(X),{\tau }_U)\times \left(K\right(X),{\tau }_V)\to X$ is minimal usco and extend Kenderov's generic optimization theorem to Tychonoff almost Čech-complete spaces.