Infinite games and quasi-uniform box products
Keywords:infinite games, W-spaces, Σ-products, quasi-uniform spaces, quasi-uniform box products
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.
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.
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