Quasi-Metrizable Spaces Satisfying Certain Completeness Conditions

It is shown that a quasi-metrizable space possesses a -base if and only if it admits a left -complete quasi-metric and that a quasi-metrizable Moore space possesses a -base if and only if it admits a Smyth complete quasi-metric. Furthermore those quasi-metrizable Moore spaces are characterized that admit a right -complete (equivalently, bicomplete) quasi-metric. Finally, it is established that a topological space admits a Smyth complete quasi-uniformity if and only if it is quasi-sober.