Some remarks on Doitchinov completeness

We observe that the well-monotone (open covering) quasiuniformity of each topological space is left K-complete. On the other hand, we exhibit an example of a topological space the fine quasiuniformity of which is not D-complete. The semicontinuous quasiuniformity of a countably metacompact space X is shown to be D-complete if and only if X is closed-complete. Moreover it is proved that the well-monotone quasiuniformity of a ccc regular space X is D-complete if and only if X is almost realcompact. o note that a metrizable space admits a D-complete quasimetric if and only if it is an Fσδ-set in every metric space in which it is embedded. Each (Tychonoff) Čech complete quasimetrizable space is shown to admit a left K-complete quasimetric.