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

A space is called a mu-space if it can be embedded in a countable product of paracompact Fsigma-metrizable spaces. The following are shown: (1) For a Tychonoff space X, if C-p(X, R) is a g-space, then X is a countable union of compact metrizable subspaces. (2) For a zero-dimensional space X, C-p (X, 2) is a A-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 C-p (P, 2) is a cosmic space (i.e., a space with a countable network) which is not a A-space. (C) 2004 Elsevier B.V. All rights reserved.