A.V. Arhangel'skii, R.Z. Buzyakova
The rank of the diagonal and submetrizability

Comment.Math.Univ.Carolin. 47,4 (2006) 585-597.

Abstract: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 \v Cech-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.

Keywords: $G_\delta $-diagonal, rank $k$-diagonal, submetrizability, condensation, regular $G_\delta $-diagonal, zero-set diagonal, \v Cech-completeness, pseudocompact space, Moore space, Mrowka space, bounded subset, extent, Souslin number
AMS Subject Classification: 54D20, 54F99