Nowhere constant families of maps and resolvability

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.