Completion of quasi-topological groups

Abstract

A topology of a quasi-topological group is induced by several natural semi-uniformities, namely right, left, two-sided and Roelcke semi-uniformities. A quasi-topological group is called complete if every Cauchy (in some sense—we examine several generalizations of Cauchy properties) filter on the two-sided semi-uniformity converges. We use the theory of Hausdorff complete semi-uniform spaces, see [B. Batíková, Completion of semi-uniform spaces, Appl. Categor. Struct. 15 (2007) 483–491], and show that Hausdorff complete quasi-topological groups form an epireflective subcategory of Hausdorff quasi-topological groups. But the reflection arrows need not be embeddings. For several types of Cauchy-like properties we show examples of quasi-topological groups that cannot be embedded into a complete group.