A note on unibasic spaces and transitive quasi-proximities

Adalberto García-Máynez, Adolfo Pimienta Acosta
DOI: https://doi.org/10.4995/agt.2017.4092

Abstract

In this paper we prove there is a bijection between the set of all annular bases of a topological spaces $(X,\tau)$ and the set of all transitive quasi-proximities on $X$ inducing $\tau$. We establish some properties of those topological spaces $(X,\tau)$ which imply that $\tau$ is the only annular basis.


Keywords

annular basis; entourage; semi-block; quasi-proximity, transitive quasi-proximity-uniformity; unibasic spaces

Subject classification

Primary 54E05; 54E15; 54D35; Secondary 05C50

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References

P. Fletcher and W. Lindgren, Quasi-uniformity spaces, vol 77, Marcel Dekker, Inc., New York, First edition, 1982.

J. Ferrer, On topological spaces with a unique quasi-proximity, Quaest. Math 17 (1994), 479-486. https://doi.org/10.1080/16073606.1994.9631779

H.-P. Künzi, Nontransitive quasi-uniformities in the Pervin quasi-proximity class, Proc. Amer. Math. Soc. 130 (2002), 3725-3730. https://doi.org/10.1090/S0002-9939-02-06477-8

H.-P. Künzi, Quasi-uniform spaces-eleven years later, Topology Proceedings 18 (1993), 143-171.

H.-P. Künzi, Topological spaces with a unique compatible quasi-proximity, Arch. Math. 43 (1984), 559-561. https://doi.org/10.1007/BF01190960

H.-P. Künzi and M. J. Pérez-Peñalver, The number of compatible totally bounded quasi-uniformities, Act. Math. Hung. 88 (2000), 15-23. https://doi.org/10.1023/A:1006788124139

H.-P. Künzi and S. Watson, A nontrivial $T_1$-spaces admitting a unique quasi-proximity, Glasg. Math. J. 38 (1996), 207-213. https://doi.org/10.1017/S0017089500031451

W. J. Pervin. Quasi-uniformization of topological spaces, Math. Annalen 147 (1962), 116-117. https://doi.org/10.1007/BF01440953

W. J. Pervin, Quasi-proximities for topological spaces, Math. Annalen 150 (1963), 325-326. https://doi.org/10.1007/BF01470761

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1. Completion of pre-quasi-uniform spaces
Adalberto García-Máynez, Adolfo Pimienta Acosta
Topology and its Applications  vol: 221  first page: 491  year: 2017  
doi: 10.1016/j.topol.2017.02.025



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt