Products of normal spaces with Lašnev spaces

Normality and Collectionwise normality of product spaces, Ⅱ

“…is an open neighborhood of $y$ and Card $\Gamma_{y}$ is finite, we have shown that the collection (5) above is localy finite at $y$ .…”

“…Throughout this paper we assume all spaces to be Hausdorff, and all maps to be continuous. For two collectionwise normal spaces $X$ and $Y$ , the result which asserts normality of $X\times Y$ implies its collectionwise normality has been proved in cases $Y$ being metrizable, Las\v{n}ev, a paracompact M-space and $\sigma$ -locally compact paracompact by Okuyama [12], Hoshina [5], Rudin-Starbird [14] and Chiba [3], respectively. In a previous paper [16, Theorem 2.2], the author proved another case that the product of a paracompact $\Sigma$ -space $X$ and a collectionwise normal P-space $Y$ is collectionwise normal if and only if it is normal; this extends Nagami's theorem [9] with $Y$ being a paracompact $\sigma$ -space as well as affirmatively answers to the problem posed by Yang [17].…”

The Normality in Products with a Countably Compact Factor

“…ðX is not normal [3]. On the other hand, the equivalence of normality and countable paracompactness was established for many cases in the theory of product spaces, see [8], [9], [13], [15], [16], [19] and [20], in particular, it is well known by [16] that the product of a normal countably paracompact space with a metric space is normal iff it is countably paracompact. In section 2 of this paper, we prove that the product îð X of an uncountable regular cardinal î with a paracompact semi-stratifiable space is normal iff it is countably paracompact.…”

Countable paracompactness of Σ-products

“…The equivalence of normality and countable paracompactness in Cartesian products has been investigated by many authors [5,8,9,17,22] so that this topic constitutes a very interesting part in the theory of product spaces [7,14]. In this paper the equivalence of normality and countable paracompactness will be considered for Z-products.…”