On paracompactness in function spaces with the compact-open topology.

Abstract:

A k-network $\mathcal {P}$ for a space X is a family of subsets of X such that if $C \subset U$, with C compact and U open, then there is a finite union R of members of $\mathcal {P}$ such that $C \subset R \subset U$. An ${\aleph _0}$-space is a ${T_3}$-space having a countable k-network and an $\aleph$-space is a ${T_3}$-space having a $\sigma$-locally finite k-network. In this paper, it is shown that if X is an ${\aleph _0}$-space and Y is a paracompact $\aleph$-space, then $\mathcal {C}(X,Y)$, with the compact-open topology is a paracompact $\aleph$-space. The result implies that if X is separable metric and Y is metric, then $\mathcal {C}(X,Y)$ is paracompact.