(metrically) Quarter-stratifiable Spaces and Their Applications in the Theory of Separately Continuous Functions

We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions. The starting point for writing this paper was the desire to improve the results of V.K. Maslyuchenko et al. [MMMS], [MS], [KM], [KMM] who generalized a classical theorem of W.Rudin [Ru] which states that every separately continuous function f : X × Y → R on the product of a metrizable space X and a topological space Y belongs to the first Baire class. It was proven in [MMMS] that the metrizability of X in the Rudin theorem can be weakened to the σ-metrizability and paracompactness of X . A subtle analysis of Rudin’s original proof reveals that this theorem is still valid for a much wider class of spaces X . These spaces are of independent interest, so we decided to give them a special name — metrically quarter-stratifiable spaces. (Metrically) quarter-stratifiable spaces are introduced and studied in details in the first three sections of this paper, where we investigate relationships between the class of (metrically) quarter-stratifiable spaces and other known classes of generalized metric spaces. It turns out that each semi-stratifiable space is quarter-stratifiable (this is a reason for the choice of the term “quarter-stratifiable”), while each quarter-stratifiable Hausdorff space has Gδ-diagonal. Because of this, the class of quarter-stratifiable spaces is “orthogonal” to the class of compacta — their intersection contains only metrizable compacta. The class of quarter-stratifiable spaces is quite wide and has many nice inheritance properties. Moreover, every (submetrizable) space with Gδ-diagonal is homeomorphic to a closed subset of a (metrically) quarter-stratifiable T1-space. The following diagram describes the interplay between the class of (metrically) quarter-stratifiable spaces and other classes of generalized metric spaces in the framework of Hausdorff spaces. Typeset by AMS-TEX 1