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Some Results on Quasi-Uniform Spaces

Published online by Cambridge University Press:  20 November 2018

Karen S. Carter  and
T. L. Hicks
Karen S. Carter
Affiliation:
Department of Mathematics, University of Missouri-Rolla, Rolla, Missouri 65401, U.S.A.
T. L. Hicks
Affiliation:
Department of Mathematics, University of Missouri-Rolla, Rolla, Missouri 65401, U.S.A.
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Constructions are made of a T 1 space which does not have a T 1 completion and of a quasi-uniform space which is complete, but not strongly complete. An example relating to a completion due to Popa is given. An alternate definition for Cauchy filter, called C-filter, is examined and a construction of a C-completion is given. We discuss quasi-pseudometrics over a Tikhonov semifield R Δ. Every topological space is quasi-pseudometrizable over a suitable R Δ. It is shown that if a quasi-pseudometric space over R Δ is complete, the corresponding quasi-uniform structure is C-complete. A general method for constructing compatible quasiuniform structures is given.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 1 , 01 March 1976 , pp. 39 - 51
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Antonovskii, M. Ya., Boltyanskii, V. G., and Sarymsakov, T. A., A survey of the theory of topological semi-fields, Russian Math. Surveys 21, no. 4, 163–192 (1966).Google Scholar
2. Boltjanskii, V. G., Separability axioms and metric, Soviet Math. Dokl., 12 (1971) 639–643.Google Scholar
3. Carlson, J. W. and Hicks, T. L., On completeness in a quasi-uniform space, J. Math. Anal. Appl. 34 (1971) 618–627.Google Scholar
4. Fletcher, P., Finite topological spaces and quasi-uniform structures, Canad. Math. Bull. 12, 771–775 (1969).Google Scholar
5. Fletcher, P., On completeness of quasi-uniform spaces, Archiv. Math. 22 (1971) 200–204.Google Scholar
6. Fletcher, P. and Lindgren, W. F., Quasi-uniformities with a transitive base, Pacific J. Math. 43 (1972)619–631.Google Scholar
7. Gaal, S. A., Point set topology, New York and London: Academic Press 1964.Google Scholar
8. Lindgren, W. F., Topological spaces with a unique quasi-uniform structure, Arch. Math., 22 (1971) 417–419.Google Scholar
9. Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform topological spaces, Groningen 1966.Google Scholar
10. Pervin, W. J., Quasi-uniformization of topological spaces, Math. Ann., 147 (1962) 316–317.Google Scholar
11. Popa, Eugen, Completion of quasi-uniform spaces., Math. Ann. 186 (1970) 297–298.Google Scholar
12. Sieber, J. L. and Pervin, W. J., Completeness in quasi-uniform spaces, Math. Ann., 158 (1965) 79–81.Google Scholar