Ultra-pseudo metric spaces and their characterization with completely regular topological spaces (Q6559421)

Completely regular topological spaces were introduced by Tychonoff in 1929. They are characterized based on the collection of all continuous real-valued functions defined on a space. \textit{A. Weil} [Sur les espaces à structure uniforme et sur la topologie générale. Hermann, Paris (1937; JFM 63.0569.04)] was the first to document the first accurate delineation of a uniform space. Uniform spaces are an extension of the concepts of metric spaces, pseudo-metric spaces, and topological spaces, by providing a more generalized framework.\N\NWe call a topological space \((X,\tau)\) ``uniformizable'' when a uniformity on \(X\) induces a topology that is homeomorphic to \(\tau\). There exists an equivalence between uniformizable spaces, subspaces of the product of pseudometric spaces, and completely regular topological spaces.\N\NIn this article, the authors examine partial uniform spaces and ultra-pseudo metric spaces, providing relevant examples. They establish the characterization of completely regular topological spaces as the subspaces of the product of ultra-pseudo metric spaces.