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A Construction of the Pair Completion of a Quasi-Uniform Space

Published online by Cambridge University Press:  20 November 2018

William F. Lindgren  and
Peter Fletcher
William F. Lindgren
Affiliation:
Slippery Rock State College, Slippery Rock PA.16057
Peter Fletcher
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
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Although L. Nachbin introduced the concept of a quasiuniform space in connection with the study of uniform ordered spaces [7], it was A. Császár who first developed a theory of completion for these spaces. In the review of Császár′s work [3], J. Isbell wrote: “A topogenic space is already complete according to the present definition; an unfortunate side effect is that not every convergent filter is Cauchy.” This aspect of Császár′s theory of completeness seems to have stalled the investigation of its applications, so that even in the study of uniform ordered spaces Császár′s important work has often been overlooked. While it may have seemed that Isbell had pointed out the Achilles′ heel of Császár′s theory, in this paper we outline a construction of Császár′s completion for quasi-uniform spaces, which differs from Császár′s original construction and the construction given by S. Salbany [8], in which Isbell′s objection does not obtain.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 1 , 01 March 1978 , pp. 53 - 59
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Bushaw, D., Elements of General Topology, John Wiley and Sons, Inc., New York (1963).Google Scholar
2. Csâszâr, A., Fundaments de la Topologie Générale, Paris (1960).Google Scholar
3. Isbell, J., Math Reviews 22, (1961) ࿋.Google Scholar
4. Lindgren, W. F. and Fletcher, P., Equinormal quasi-uniformities and quasi-metrics, Glasnik Mat., to appear.Google Scholar
5. Lindgren, W. F. and Fletcher, P., A theory of uniformities for generalized ordered spaces, submitted for publication.Google Scholar
6. Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform Topological Spaces, Amsterdam, Noordhoff, (1966).Google Scholar
7. Nachbin, L., Topology and Order, D. Van Nostrand Co., Inc., Princeton, N.J., (1965).Google Scholar
8. Salbany, S., Bitopological spaces, compactifications and completions, Mathematical monographs of the University of Cape Town, No. 1. Cape Town (1974).Google Scholar
9. Sieber, J. L. and Pervin, W. J., Completeness in quasi-uniform spaces, Math. Ann. 158 (1965) 79-81.Google Scholar