Regular completions of Cauchy spaces

Completion of a Cauchy space without the T2‐restriction on the space

“…The completion of Cauchy spaces is already well known and familiar to most of us. In fact, since Keller [5] introduced the axiomatic definition of Cauchy spaces a very rich and extensive completion theory has been developed for Cauchy spaces during the last three decades [2,3,7,10,12]. It seems that Cauchy space rather than uniform convergence space is a natural generalization of completion of uniform space.…”

A note on regular Cauchy spaces

“…Since <Sf' is regular, ^ ^ <g*' ^ if*. But a minimal regular convergence space is shown in [1] to be regular-closed, and it follows from Theorem 2.2 that ϋ^ = <£" = q -^. Thus (X, ^) is a minimal regular Cauchy space.…”

“…Characterizations of regularclosed and Urysohn-closed convergence spaces are given in [1]; these results are restated, in modified form, in the following lemma.…”

“…3* Minimal regular Cauchy spaces* A regular convergence space (X, q) is minimal regular if, whenever p is a regular convergence structure on X and p ^ q, then p = q. A characterization of minimal regular convergence spaces, which we shall not make use of, is given in [1], A minimal regular Cauchy space is defined in the analogous (and obvious) way. THEOREM …”

On finest unitary extensions of topological monoids

“…Each Cauchy space has many completions which have been studied in papers of many authors (see [2][3][4][5][6][7][8][9][10][11]). The most important of them is perhaps the Wyler completion possessing the universal property over other completions.…”