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On Quasi-Metric Completion
by
Elena Alemany
In this paper we deal with the quasi-metric completion of quasi-metric spaces.
It is well known that every metric space has an (up to isometry) unique metric completion. In the non-symmetric case, it is possible to find examples of Hausdorff quasi-metric spaces which do not admit a quasi-metric completion (not even for a very general notion of it, such as half-completion. In fact, there exist many different notions of quasi-metric completeness in the literature). This takes us to the problem of characterizing the quasi-metric spaces that admit a quasi-metric completion.
In this direction, H.P.A. Künzi proved that a quasi-metric space (X, d) has a quasi-metric sequential completion iff whenever <xn > is a Cauchy sequence (in the sense of Fletcher and Lindgren) in (X, d) and x in X is a T(d-1)-cluster point of the sequence <xn>, then x is a T(d)-cluster point of the sequence <xn>.
Later on, S. Romaguera proved that a quasi-metric space (X, d) has a left K-sequential completion iff whenever <xn > is a left K-Cauchy sequence in (X, d) which is T(d-1)-convergent to a point x in X, then <xn > is T(d)-convergent to x.
In this talk we focus our attention in studying the problem of quasi-metric completion cited above for the notions of half-completeness, bicompleteness and right K-completeness. In particular we characterize those quasi-metric spaces that admit a quasi-metric half-completion and a quasi-metric right K-completion respectively. We also present a characterization of the quasi-metric spaces whose bicompletion is quasi-metric. Some consequences of these results are also given.
*These results are a part of joint works with S. Romaguera, namely ''On half-completion and bicompletion of quasi-metric spaces'' (to appear in Comm. Math. Univ. Carolinae) and ''On right K-sequentially complete quasi-metric spaces'' (to appear in Acta Math. Hungar.).
Date received: June 24, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caah-04.