Some remarks on Doitchinov completeness (Q675103)

This paper studies left \(K\)-completeness, \(D\)-completeness, and convergence completeness of the fine, semi-continuous, and well-monotone quasi-uniformities and considers conditions under which a (quasi-)metrizable space admits a \(D\)-complete or left \(K\)-complete quasi-metric. The well-monotone quasi-uniformity is shown to be always left \(K\)-complete and to be convergence complete if the underlying space is regular and almost realcompact. Moreover, a topological space satisfying the countable-chain condition whose well-monotone quasi-uniformity is convergence complete is almost realcompact. A quasi-metrizable space admits a left \(K\)-complete quasi-metric if it is either scattered or Čech complete, and a metrizable space admits a \(D\)-complete quasi-metric if, and only if, it is an \(F_{\sigma \delta}\)-set in each metric space in which it is embedded. The authors pose several challenging problems, among which are the problems of finding a regular topological space whose fine quasi-uniformity is not convergence complete and of finding a Moore space whose fine quasi-uniformity is not convergence complete.