Results about κ-normality

Abstract

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) A product of two linearly ordered topological spaces need not be κ- normal even if one of the factors is compact. (2) A product of two normal countably compact topological spaces need not be κ- normal. (3) A product of a normal topological space with a compact Hausdorff topological space need not be κ- normal. The second 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 Stone's 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 Dowker's 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.