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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
Zbl 1121.54051
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