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On \(\alpha \)-normal and \(\beta \)-normal spaces. (English) Zbl 1053.54030

A topological space \(X\) is said to be \(\alpha \)-normal (or \(\beta \)-normal) if for all disjoint closed subsets \(A,B\) there exist disjoint open subsets \(U,V\) such that \(A\cap U\) is dense in \(A\) and \(B\cap V\) is dense in \(B\) (and \(\bar U\cap \bar V=\emptyset \), resp.). Some results: 1.9. Every extremally disconnected \(\alpha \)-normal space is normal; 2.6. For any Tychonoff space \(X\), the space \(C_p(X)\) is normal if it is \(\beta \)-normal; 3.4. [CH] There exists \(X\) such that \(C_p(X)\) is hereditarily \(\alpha \)-normal and not normal (there is a note that E. Murtinová has recently constructed a \(\beta \)-normal non normal space); 3.12 There exists a pseudocompact \(\alpha \)-normal and not \(\beta \)-normal space; 4.4. \(\beta \)-normal spaces have the (Nyikos) property \(\omega D\).

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Keywords:

normality
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