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

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. eral, 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. 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.