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Spaces with a regular \(G_{\delta}\)-diagonal. (English) Zbl 1117.54004
A space \(X\) has a regular \(G_\delta\)-diagonal if the diagonal in \(X\times X\) can be represented as the intersection of the closures of a countable family of its neighborhoods in the square. The authors study what restrictions on a topological space can be formulated whenever \(X\) has a regular \(G_\delta\)-diagonal. They prove that if a dense subspace \(Y\) of the product of some family of separable metrizable spaces has a regular \(G_\delta\)-diagonal, then \(Y\) is submetrizable. This result should be compared with the result of [R. Z. Buzyakova, Commentat. Math. Univ. Carol. 46, 469–473 (2005; Zbl 1121.54051)]. The authors also study the regular \(G_\delta\)-diagonal property in the more narrow setting of paratopological groups. A family \(\gamma\) of non-empty open subsets of a space \(X\) is called a local \(\pi\)-base at a point \(p\in X\) if every open neighborhood \(O_p\) of \(p\) contains some \(U\in \gamma\). The authors show that every Hausdorff first countable Abelian paratopological group has a regular \(G_\delta\)-diagonal. Several new open questions are formulated, as for example, whether the word Abelian can be dropped.

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
22A05 Structure of general topological groups
54A35 Consistency and independence results in general topology
Zbl 1121.54051
Full Text: DOI
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