Quasi-convergence spaces and biconvergence spaces (Q6547330)

In this paper the author considers two types of general convergence named by quasiconvergence and biconvergence, respectively. These notions of convergence become of some importance if one is not only looking at quasimetric or quasi-uniform spaces but also at more general spaces like quasi-uniform limit spaces or quasi-Cauchy spaces, where in general convergence is not topological. If supposing a so-called ``limit space axiom'', the resulting categories of biconvergence spaces and of quasi-convergence spaces are isomorphic. The author points out that the advantage of quasi-convergence lies in the theory of completion of quasi-uniform spaces, and proofs can be borrowed directly from the theory of convergence spaces, replacing filters by pairs of filters. Analogously, similar results are obtained when one is considering quasi-Cauchy spaces, but in principle both approaches can be used. In these respective categories differences occur only by looking at certain Hausdorff separation axioms for quasi-convergence and biconvergence, and the author compares them in the special case of quasi-uniform spaces. \N\NNext the author defines some basic principles of the above-mentioned approaches, and he shows that the categories Q-Conv of quasiconvergence spaces and corresponding maps respectively Q-Lim of quasi-limit spaces and corresponding maps are cartesian closed topological categories. Similar to the above, he defines biconvergence and bilimit spaces. As one main result, he states that the category Q-Lim is isomorphic to the category BiLim. In the next two sections, the author studies pretopological and topological quasi-convergence spaces by considering the relationships between them and bitopological spaces, respectively. And at the end of paper, the author illustrates essential differences between ``natural'' definitions in biconvergence and quasi-convergence spaces by considering the Haudorff separation axiom.