Results about κ-normality

A regular topological space is called κ-normal if any two disjoint κ-closed subsets in it are separated. In this paper we present some results about κ-normality. In Section 1, the technique of adding isolated points to topological spaces has been used to construct three counterexamples. The first one shows the following statements: (1) uct of two linearly ordered topological spaces need not be κ-normal even if one of the factors is compact. uct of two normal countably compact topological spaces need not be κ-normal. uct of a normal topological space with a compact Hausdorff topological space need not be κ-normal. cond one presents a mad family R⊂[ω]ω such that the Mrówka space Ψ(R) is not κ-normal. The third one will show that a scattered locally compact countably compact topological space need not be κ-normal. In Section 2 we have proved the following κ-normal version of Stoneʹs theorem: If X is κ-normal and countably compact and Y is metrizable, then X×Y is κ-normal. For a κ-normal version of Dowkerʹs theorem, we have been able to prove one direction which is the following statement: If X is not κ-countably metacompact, then X×I is not κ-normal. We use I for the closed unit interval [0,1] with the usual topology. The converse is still unsettled. We will show that if X is a Dowker space, then the Alexandroff Duplicate space A(X) of X is a Dowker space with the property that A(X)×I is not κ-normal. Section 3 has been devoted to the notion of local κ-normality. It will be shown that not every locally κ-normal topological space is κ-normal, even if the space satisfies other topological properties such as locally compactness, metacompactness, or countable compactness.