Singular compactifications of product spaces

Abstract

Assume that both $X$ and $Y$ are non-compact locally compact spaces. Let $\delta(X\times Y)$ be a compactification of $X\times Y$ such that $\delta(X\times Y)\geq\omega X\times\omega Y$, where $\omega X$ and $\omega Y$ are the one-point compactifications of $X$ and $Y$, respectively. Then J. L. Blasco [2] proved the theorem that $\delta(X\times Y)$ is not a weakly singular compactification of $X\times Y$ if $X$ is pseudcompact. In this paper we give an alternative, simpler proof for the above theorem. Furthermore, in the case $X$ is either a non-separable metrizable space or a separable metrizable space with a non-compact quasi-component space $Q(X)$ and $d(Y)\leq d(X)$, where $d(X)$ is the density of $X$, for any compact space $S$ we establish a theorem that $X\times Y$ has a singular compactification with $S$ as a remainder if and only if $X$ has a singular compactification with $S$ as a remainder.