The rank of the diagonal and submetrizability

Abstract

summary:Several topological properties lying between the submetrizability and the $G_\delta $-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_\delta $-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_\delta $-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable space. The rank $5$-diagonal plays a crucial role in this criterion. Every closed bounded subset of a Tychonoff space with a $G_\delta $-diagonal is shown to be Čech-complete. Under a slightly stronger condition, any such subset is shown to be a Moore space. We also establish that every closed bounded subset of a Tychonoff space with a regular $G_\delta $-diagonal is metrizable by a complete metric and, therefore, has the Baire property. Some further results are obtained, and new open problems are posed.