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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.

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