Spaces with a regular Gδ-diagonal

A space X has a regular Gδ-diagonal if the diagonal in X×X can be represented as the intersection of the closures of a countable family of its neighbourhoods in the square.Below we generalize a theorem of McArthur [W.G. McArthur, Gδ-diagonals and metrization theorems, Pacific J. Math. 44 (1973) 613–617] to bounded subsets of spaces with a regular Gδ-diagonal showing that all such subsets are metrizable (Theorem 1). If a dense subspace Y of the product of some family of separable metrizable spaces has a regular Gδ-diagonal, then Y is submetrizable (Theorem 14).We also study the regular Gδ-diagonal property in the setting of paratopological groups. It is proved that every Hausdorff first countable Abelian paratopological group has a regular Gδ-diagonal (Theorem 17). However, it remains unknown whether “Abelian” in the above statement can be dropped. We also provide the first example of a countable (therefore, normal) Abelian paratopological group G with a countable π-base such that the space G is not Fréchet–Urysohn and hence, is not first countable. This is in contrast with the fact that every Hausdorff topological group with a countable π-base is metrizable. Several related results on submetrizability are obtained, and new open questions are formulated.