Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

Abstract

A space is called a μ-space if it can be embedded in a countable product of paracompact Fσ-metrizable spaces. The following are shown: (1) For a Tychonoff space X, if Cp(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces. (2) For a zero-dimensional space X, Cp(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces. In particular, let P be the space of irrational numbers. Then Cp(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.