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• W2073057400 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." @default.
• W2073057400 created "2016-06-24" @default.
• W2073057400 creator A5047802215 @default.
• W2073057400 date "2002-10-01" @default.
• W2073057400 modified "2023-09-29" @default.
• W2073057400 title "Results about κ-normality" @default.
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• W2073057400 doi "https://doi.org/10.1016/s0166-8641(01)00258-9" @default.
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