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Weak normality properties and factorizations of normality. (English) Zbl 1104.54009

J. Mack [Trans. Am. Math. Soc. 148, 265–272 (1970; Zbl 0209.26904)] defined a weak form of normality called \(\delta\)-normality (any two disjoint regular \(G_{\delta}\)-sets have disjoint neighborhoods). The authors study the question whether there is a weak form of normality which together with \(\delta\)-normality gives normality. They find such a notion, resp., such a class of spaces; it is the class of \(\Sigma\)-normal spaces (for each closed set \(F\) and each its neighborhood \(U\) there is a regular \(F_{\sigma}\)-set \(V\) with \(F \subset V\subset U\)). They also define the class of functionally \(\delta\)-normal spaces (for any two disjoint closed subsets, one of which is a regular \(G_{\delta}\)-set, there is a Urysohn function) and prove that a \(\Sigma\)-normal functionally \(\delta\)-normal space is normal. A kind of Urysohn lemma for functionally \(\delta\)-normal spaces is also shown.

MSC:

54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness

Citations:

Zbl 0209.26904
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