A construction of the B -completion of a T0T0-quasi-metric space

We present a construction of the B  -completion of a T0T0-quasi-metric space (X,d)(X,d) that is based on the injective hull of (X,d)(X,d). The B  -completion of a T0T0-quasi-metric space (X,d)(X,d) is a T0T0-quasi-metric extension of (X,d)(X,d) studied by H.-P.A. Künzi and C. Makitu Kivuvu.In general, the B  -completion of a T0T0-quasi-metric space is larger than the better known bicompletion of (X,d)(X,d). In the special case of a balanced T0T0-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 T0T0-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.