Let X be a Tychonoff topological space, C (X, 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 argmin: C (X, R)? K (X)? X is defined as follows: argmin (f, K) = x? K:? (x) = min{? (y): y? K}. We present topologies on C (X, R)? K (X) under which argmin: C (X, R)? K (X)? X has a closed graph. We also extend a generic optimization theorem of G. Beer from Cech-complete spaces to locally Cech-complete ones.
Holá et al. (Wed,) studied this question.