A construction of the B-completion of a $T_0$-quasi-metric space

Abstract

We present a construction of the B-completion of a $T_0$-quasi-metric space $(X,d)$ that is based on the injective hull of $(X,d)$. The B-completion of a $T_0$-quasi-metric space $(X,d)$ is a $T_0$-quasi-metric extension of $(X,d)$ studied by H.-P.A. Künzi and C. Makitu Kivuvu.

In general, the B-completion of a $T_0$-quasi-metric space is larger than the better known bicompletion of $(X,d)$. In the special case of a balanced $T_0$-quasi-metric space the B-completion coincides with a completion that was first investigated by D. Doitchinov.

Among other things, in this note we show that the B-completion of the natural $T_0$-quasi-metric of a partially ordered set (possessing a smallest and a largest element) can be identified with the Dedekind–MacNeille completion of that partially ordered set.