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
The main theorem characterizes the quasi‐perfect and perfect preimages of T1‐spaces having a base of countable order. The preimages of T1‐spaces having a base of countable order are connected with the preimages of T1‐spaces having a primitive base under the same type of mappings.
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