Relative Collectionwise Normality

Abstract

Abstract.In this paper we study properties of relative collectionwise normality type based on relative properties of normality type introduced by Arhangel'skii and Genedi. Theorem Suppose Y is strongly regular in the space X. If Y is paracompact in X then Y is collectionwise normal in X. Example A T2 space X having a subspace which is 1− paracompact in X but not collectionwise normal in X. Theorem Suppose that Y is s-regular in the space X. If Y is metacompact in X and strongly collectionwise normal in X then Y is paracompact in X.2000 AMS Classification: Primary 54D20, Secondary 54A35

Relative star normal type

“…Note that Y is Aull paracompact in X then Y is 2-paracompact in X. The following is a variation of Lemma 17 of [11].…”

“…The subspace Y ∪ Z ∪ Q is 1-paracompact in X but not collectionwise normal in X [11] and thus Y ∪ Z ∪ Q is not 2-fully normal in X. On the other hand it is readily seen that Y ∪ Z ∪ Q is 2-strongly star normal in X.…”

“…Example 23 [11]. A T 2 Lindelöf space X having a subspace which is 2-strongly star normal in X but not 2-fully normal in X.…”

“…We say that Y is (strongly) collectionwise normal in a space X provided for every discrete collection (collection discrete with respect to Y ) of closed subsets of X, [11]. Note that if Y is collectionwise normal in X then Y is regular in X.…”