Local compactness and homeomorphisms of spaces of continuous functions

T. E. Khmyleva ; A. E. Kirienko

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2010), pp. 61-68

Résumé

In this paper we prove that 1) the spaces $C_p(S)$ and $C_p(T)$ of all continuous functions in the topology of pointwise convergence are not linearly homeomorphic if $S,T$ are not locally compact metrizable while the derivation set $T^{(1)}$ is compact and the derivation set $S^{(1)}$ is not; 2) the spaces $C_K(X)$ and $C_K(Y)$ of all continuous functions in the compact-open topology are not homeomorphic if $X$ and $Y$ are completely regular spaces while $X$ is locally compact and $\sigma$-compact and there is a point $y_0\in Y$ of countable character such that every neighborhood of it is not a pseudocompact.

Détail

Keywords: spaces of all continuous functions, linear homeomorphism, homeomorphism, metrizable space, locally compact space, topology of pointwise convergence, compact-open topology.