Infinite games and quasi-uniform box products

Hope Sabao; Olivier Olela Otafudu

doi:10.4995/agt.2019.9679

Authors

Hope Sabao University of Zambia
Olivier Olela Otafudu University of the Witwatersrand

DOI:

https://doi.org/10.4995/agt.2019.9679

Keywords:

infinite games, W-spaces, Σ-products, quasi-uniform spaces, quasi-uniform box products

Abstract

We introduce new infinite games, played in a quasi-uniform space, that generalise the proximal game to the framework of quasi-uniform spaces. We then introduce bi-proximal spaces, a concept that generalises proximal spaces to the quasi-uniform setting. We show that every bi-proximal space is a W-space and as consequence of this, the bi-proximal property is preserved under Σ-products and closed subsets. It is known that the Sorgenfrey line is almost proximal but not proximal. However, in this paper we show that the Sorgenfrey line is bi-proximal, which shows that our concept of bi-proximal spaces is more general than that of proximal spaces. We then present separation properties of certain bi-proximal spaces and apply them to quasi-uniform box products.

Downloads

Download data is not yet available.

Author Biographies

Hope Sabao, University of Zambia

Department of Mathematics and Statistics

Olivier Olela Otafudu, University of the Witwatersrand

School of Mathematics

References

J. R. Bell, The uniform box product, Proc. Amer. Soc. 142 (2014), 2161-2171. https://doi.org/10.1090/S0002-9939-2014-11910-1

J. R. Bell, An infinite game with topological consequences, Topol. Appl. 175 (2014), 1-14. https://doi.org/10.1016/j.topol.2014.06.014

T. Daniel and G. Gruenhage, Some nonnormal $sum$-products, Topol. Appl. 43, no. 1 (1992), 19-25. https://doi.org/10.1016/0166-8641(92)90150-X

P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics., vol. 77, Marcel Dekker Inc., New York, 1982.

G. Gruenhage, Infinite games and generalizations of first-countable spaces, Gen. Topol. Appl. 6 (1976), 339-352. https://doi.org/10.1016/0016-660X(76)90024-6

F. Ishikawa, On countably paracompact spaces, Proc. Japan Acad. 31, no. 10 (1955), 686-687. https://doi.org/10.3792/pja/1195525547

K. Kunen, Paracompactness of box products of compact spaces, Trans. Amer. Math. Soc. 240 (1978), 307-316. https://doi.org/10.1090/S0002-9947-1978-0514975-6

H.-P. A. Künzi, An introduction to quasi-uniform spaces, in: Beyond Topology (F. Mynard and E. Pearl, eds.), Contemporary Mathematics, vol. 486, AMS, 2009, pp. 239-304. https://doi.org/10.1090/conm/486/09511

H.-P. A. Künzi and S. Watson, A quasi-metric space without a complete quasi-uniformity, Topol. Appl. 70, no. 2-3 (1996), 175-178. https://doi.org/10.1016/0166-8641(96)88666-4

K. Morita, Paracompactness and product spaces, Fundam. Math. 50, no. 3 (1962), 223-236. https://doi.org/10.4064/fm-50-3-223-236

O. Olela Otafudu and H. Sabao, On quasi-uniform box products, Appl. Gen. Topol. 18, no. 1 (2017), 61-74. https://doi.org/10.4995/agt.2017.5818

W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316-317. https://doi.org/10.1007/BF01440953

J. Roitman, Paracompactness and normality in box products: old and new, Set theory and its Applications, Contemp. Math. 533 (2011), 157-181. https://doi.org/10.1090/conm/533/10507

R. Stoltenberg, Some properties of quasi-uniform spaces, Proc. London Math. Soc. 17 (1967), 226-240. https://doi.org/10.1112/plms/s3-17.2.226

S. Willard, General Topology, Dover Publications, INC. Mineols, New York, 2004.