On precompact (quasi-) uniform structures
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- by P. Th. Lambrinos PDF
- Proc. Amer. Math. Soc. 62 (1977), 365-366 Request permission
Abstract:
The following are shown: (1) The subbasis theorem on precompactness does not hold even in uniform spaces. (2) The supremum of (even finitely many) precompact quasi-uniform structures is not necessarily precompact. (3) A compact quasi-uniform space is not necessarily totally bounded. These results contradict corresponding assertions in the literature.References
- Steven A. Gaal, Point set topology, Pure and Applied Mathematics, Vol. XVI, Academic Press, New York-London, 1964. MR 0171253
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- M. G. Murdeshwar and S. A. Naimpally, Quasi-uniform topological spaces, P. Noordhoff Ltd., Groningen, 1966. MR 0211386
- M. G. Murdeshwar and K. K. Theckedath, Boundedness in a quasi-uniform space, Canad. Math. Bull. 13 (1970), 367–370. MR 270333, DOI 10.4153/CMB-1970-069-5
- J. L. Sieber and W. J. Pervin, Completeness in quasi-uniform spaces, Math. Ann. 158 (1965), 79–81. MR 172229, DOI 10.1007/BF01370731
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 62 (1977), 365-366
- MSC: Primary 54E15
- DOI: https://doi.org/10.1090/S0002-9939-1977-0428292-0
- MathSciNet review: 0428292