Abstract
AbstractLet X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent:
X is α-favourable (in the sense of the Banach-Mazur game);
The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X).
In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties:
X contains some dense completely metrizable subset;
the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X).
If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).
Funder
Bulgarian National Science Fund
European Fund for Regional Development
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Geometry and Topology,Numerical Analysis,Statistics and Probability,Analysis
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