The quasi-uniform box product and infinite games

PhD (Mathematics), North-West University, Mafikeng Campus, 2017In t his PhD thesis, we present t he quasi-uniform box product, a concept that generalise the uniform box product to the framework of quasi-uniform spaces . We show that the quasi-uniform box product is a topology, on the product of countably many copies of a quasi-uniform space, that is finer than t he Tychonov product topology but coarser than the uniform box product. This topology is generated by a quasi-uniformity called the constant quasi-uniformity, whose symmetrised uniformity 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 and study the connections between the quasi-uniform box product of the prefilter space associated with a quasiuniform space and t he prefilter space of t he quasi-uniform box product of the same quasi-uniform space. Furthermore, we generalise an infinite game of two players, played in a uniform space, to the quasi-uniform setting and use our new infinite game to show that the quasi-uniform box product of countably many copies of a Fort-space is collectionwise normal, countably paracompact and collectionwise Hausdorff.Doctora