Normality and Collectionwise normality of product spaces, Ⅱ

Abstract

We prove that the product space $X\times Y$ is collectionwise normal if and only if $X\times Y$ is normal in the following cases; (1) $X$ is a collectionwise normal $\Sigma$-space and $Y$ is a collectionwise normal first countable $P$-space, (2) $X$ is the closed image of a normal M-space and $Y$ is a collectionwise normal first countable $P$-space, (3) $X$ is the closed image of a paracompact M-space and $Y$ is a collectionwise normal $P$-space. In particular, (2) and (3) essentially generalize K. Chiba’s theorems [3].