Spaces with a regular Gδ-diagonal

A space X has a regular G(delta)-diagonal if the diagonal in X 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, G(delta)-diagonals and metrization theorems, Pacific J. Math. 44 (1973) 613-617] to bounded subsets of spaces with a regular G(delta)-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 G(delta)-diagonal, then Y is submetrizable (Theorem 14). We also study the regular G(delta)-diagonal property in the setting of paratopological groups. It is proved that every Hausdorff first countable Abelian paratopological group has a regular G(delta)-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 pi-base such that the space G is not Frechet-Urysohn and hence, is not first countable. This is in contrast with the fact that every Hausdorff topological group with a countable pi-base is metrizable. Several related results on submetrizability are obtained, and new open questions are formulated. (C) 2005 Published by Elsevier B.V.