∗-half completeness in quasi-uniform spaces

Abstract

[EN] Romaguera and Sánchez-Granero (2003) 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.