Nowhere constant families of maps and resolvability

Juhász, István and van Mill, Jan (2024) Nowhere constant families of maps and resolvability. CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, Publis. ISSN 0008-4395

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Abstract

If X is a topological space and Y is any set, then we call a family F of maps from X to Y nowhere constant if for every non-empty open set U in X there is f ∈ F with ∣ f [U]∣ > 1, i.e., f is not constant on U. We prove the following result that improves several earlier results in the literature. If X is a topological space for which C(X), the family of all continuous maps of X to R, is nowhere constant and X has a π-base consisting of connected sets then X is c-resolvable.

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