âˆ—-half completeness in quasi-uniform spaces
Keywords:Quasi-uniform, âˆ—-half completion, âˆ—-compactification
Romaguera and Sánchez-Granero (2003) have introduced the notion of T1âˆ—-half completion and used it to see when a quasi-uniform space has a âˆ—-compactification. In this paper, for any quasi-uniform space, we construct a âˆ—-half completion, called standard âˆ—-half completion. The constructed âˆ—-half completion coincides with the usual uniform completion in the uniform spaces and is the unique (up to quasi-isomorphism) T1 âˆ—-half completion of a symmetrizable quasi-uniform space. Moreover, it constitutes a âˆ—-compactification for âˆ—-Cauchy bounded quasi-uniform spaces. Finally, we give an example which shows that the standard âˆ—-half completion differs from the bicompletion construction.
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