Locally Compact Topological Groups and Cofinal Completeness

Abstract

It is proved that a Tychonoff topological group is locally compact if and only if it is of pointwise countable type and its left uniformity is cofinally complete. From this result a characterization is derived of those T 0 paratopological groups ( X , τ) of pointwise countable type for which ( X , τ ∨ τ −1 ) is locally compact and also a characterization is deduced of locally pseudocompact topological groups in terms of cofinal completeness. Also characterized are the Tychonoff topological groups of pointwise countable type for which their left uniformity has property U . Finally, cofinal completeness of the Hausdorff–Bourbaki uniformity of a topological group is studied.