Conditional Completeness of C(X,ℝτ) for Weak P-spaces ℝτ

Abstract

Let $\tau$ be a topology on the real numbers which is finer than the usual topology such that $\R_\tau$ is a weak $P$-space. In this paper, conditional completeness and conditional $\sigma$-completeness of $C(X)$ and $C(X,\R_\tau)$ are compared. In particular, it is shown for zero-dimensional spaces that $C(X,\R_\tau)$ is conditionally $\sigma$-complete if and only if $X$ is a $P$-space and that $C(X,\R_\tau)$ is conditionally complete if and only if $X$ is an extremally disconnected $P$-space.