*-Compactifications of quasi-uniform spaces

Abstract

Abstract.Romaguera and have introduced the notion of T1 * -half completion and used it to see when a quasi-uniform space has a * -compactification. In this paper, for any quasi-uniform space, we construct a * -half completion, called standard * -half completion. The constructed * -half completion coincides with the usual uniform completion in the uniform spaces and is the unique (up to quasi-isomorphism) T1* -half completion of a symmetrizable quasi-uniform space. Moreover, it constitutes a * -compactification for * -Cauchy bounded quasi-uniform spaces. Finally, we give an example which shows that the standard * -half completion differs from the bicompletion construction.2000 AMS Classification: 54E15, 54D35.

T0 *-compactification in the hyperspace

“…Later on, a study and description of the structure of T 0 *-compactifications of a quasi-uniform space was carried out in [17], while T 1 *-compactifications on the hyperspace were studied in [16], where some characterizations of T 1 *-compactifiability of the hyperspace were given.…”

∗-half completeness in quasi-uniform spaces

“…Moreover, a point symmetric totally bounded T 1 quasi-uniform space may have many totally bounded compactifications (see [5, page 34]) . Contrary to this notion, Romaguera and Sánchez-Granero have introduced the notion of * -compactification of a T 1 quasiuniform space (see [8], [10] and [11]) and prove that: (a) Each T 1 quasi-uniform space having a T 1 * -compactification has an (up to quasi-isomorphism) unique T 1 * -compactification ([11, Corollary of Theorem 1]); and (b) All the Wallmantype compactifications of a T 1 topological space can be characterized in terms of the * -compactification of its point symmetric totally transitive compatible quasi-uniformities ([9, Theorem 1]). The proof of (a) is achieved with the help of the notion of T 1 * -half completion of a quasi-uniform space, which is introduced in [11].…”