A new generalization of normality called almost β-normality is introduced and studied which is a simultaneous generalization of almost normality and β-normality. A topological space is called almost β-normal if for every pair of disjoint closed sets A and B one of which is regularly closed, there exist disjoint open sets U
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
Abstract. We give an example of a κ-normal space which is not densely normal.In [2] Arhangelskii introduced the definition of a densely normal topological space and noted that every densely normal space is κ-normal. The definition of a κ-normal topological space was introduced by Stchepin in [1]. Problem 25 of [2] asked whether every κ-normal space is densely normal. Here we show that the answer is negative.1 Definition 1. A space X is κ-normal if whenever E, F are disjoint canonical closed subsets of X there exist disjoint open subsets of X, U and V, such that E ⊆ U andRecall that a canonical closed set is a set which is equal to the closure of its interior.One version of relative normality is the idea of X being normal on a subspace Y , which was introduced by Arhangelskii in [2]. If A and Y are subsets of a space X, A is concentrated on Y if A ⊆ A ∩ Y . A space X is normal on a subspace Y if whenever E and F are disjoint closed subsets of X concentrated on Y , then there are disjoint open U, V ⊆ X such that E ⊆ U and F ⊆ V. Definition 2.A space X is densely normal if there exists a dense subspace Y of X such that X is normal on Y . Theorem 1. There is a Tychonoff space which is κ-normal but not densely normal.Let C R denote the Cantor set. For each bounded I ⊆ R, let l(I) denote the infimum of I and let r(I) denote the supremum of I. Let D ⊆ R\(Q ∪ C R ) be a countable dense subset of R. Let X = R\[D ∪ (C R ∩ Q)] and let τ R be the subspace topology on X inherited from R. We will define a topology τ on X such that τ R ⊆ τ . In order to distinguish (X, τ R ) from (X, τ ) we will write X R when considering X as a subspace of R. For each x ∈ X we will describe the basic open neighborhoods of x and define a local base B x at x. If x ∈ Q, an open neighborhood of x is any element of τ R that contains x. Thus, if x ∈ Q, let B x = {U ⊆ X: U is open in X R , x ∈ U}. We must go to some length to describe B x for x / ∈ Q.
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