Variational principles and topological games

Let f be a bounded from below lower semicontinuous function defined in a completely regular topological space X. We show that there exists a continuous and bounded function g, defined in the same space, such that the perturbed function f+g attains its infimum in X. Moreover, the set of such good perturbations g (for which f+g attains its infimum) is dense in the space C⁎(X) of all bounded continuous functions in X with respect to the sup-norm. We give a sufficient condition under which this set of good perturbations contains a dense Gδ-subset of C⁎(X). The condition is in terms of existence of a winning strategy for one of the players in a certain topological game played in the space X. If the other player in the same game does not have a winning strategy, then the set of good perturbations is of the second Baire category in every open subset of C⁎(X). The game we consider is similar to a game introduced by E. Michael in the study of completeness properties of topological spaces and to a game used by Kenderov and Moors to characterize fragmentability of topological spaces.