On quasi-uniform box products

[EN] We revisit the computation of entourage sections of the constant uniformity of the product of countably many copies the Alexandroff one-point compactification called the Fort space. Furthermore, we define the concept of a quasi-uniformity on a product of countably many copies of a quasi-uniform space, where the symmetrised uniformity of our quasiuniformity coincides with the constant uniformity. We use the concept of Cauchy filter pairs on a quasi-uniform space to discuss the completeness of its quasi-uniform box product.Olela Otafudu, O.; Sabao, H. (2017). On quasi-uniform box products. Applied General Topology. 18(1):61-74. doi:10.4995/agt.2017.5818.SWORD6174181Bell, J. R. (2014). The uniform box product. Proceedings of the American Mathematical Society, 142(6), 2161-2171. doi:10.1090/s0002-9939-2014-11910-1Bell, J. R. (2014). An infinite game with topological consequences. Topology and its Applications, 175, 1-14. doi:10.1016/j.topol.2014.06.014Fletcher, P., & Hunsaker, W. (1992). Completeness using pairs of filters. Topology and its Applications, 44(1-3), 149-155. doi:10.1016/0166-8641(92)90087-gGruenhage, G. (1976). Infinite games and generalizations of first-countable spaces. General Topology and its Applications, 6(3), 339-352. doi:10.1016/0016-660x(76)90024-6Künzi, H.-P. A., & Kivuvu, C. M. (2008). A double completion for an arbitrary T0-quasi-metric space. The Journal of Logic and Algebraic Programming, 76(2), 251-269. doi:10.1016/j.jlap.2008.02.00

Completion of continuity spaces with uniformly vanishing asymmetry

The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in question. For qausi-metric spaces and quasi-uniform spaces various non-equivalent completions exist, often defined on a certain subcategory of spaces that satisfy a key property required for the particular completion to exist. The classical filter completion of a uniform space can be adapted to yield a filter completion of a metric space. We show that this completion by filters generalizes to continuity spaces that satisfy a form of symmetry which we call uniformly vanishing asymmetry

The Katětov construction modified for a T0-quasi-metric space

AbstractWe discuss the existence and uniqueness of a T0-quasi-metric space qU defined by the following three conditions: (i) qU is bicomplete and supseparable, (ii) every isometry between two finite subspaces of qU extends to an isometry of qU onto itself, and (iii) qU contains an isometric copy of every supseparable T0-quasi-metric space