More on ultrafilters and topological games
Keywords:open-point game, ultrafilter, G-space, Gp-space, countably compact
Two different open-point games are studied here, the G-game and the Gp-game, defined for each p âˆˆ ωâˆ—. We prove that for each p âˆˆ ωâˆ—, there exists a space in which none of the players of the Gp-game has a winning strategy.
Nevertheless a result of P. Nyikos, essentially shows that it is consistent, that there exists a countable space in which all these games are undetermined.
We construct a countably compact space in which player II of the Gp-game is the winner, for every p âˆˆ ωâˆ—. With the same technique of construction we built a countably compact space X, such that in X ×X player II of the G-game is the winner. Our last result is to construct ω1-many countably compact spaces, with player I of the G-game as a winner in any countable product of them, but player II is the winner in the product of all of them in the G-game.
A. V. Arkhangelskii, Classes of topological groups, Russian Math. Surveys 36 (1981), no. 3, 151-174.
A. R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. https://doi.org/10.4064/fm-66-2-185-193
D. Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970), 1-24. https://doi.org/10.1016/0003-4843(70)90005-7
A. Bouziad, The Ellis theorem and continuity in groups, Topology Appl. 50 (1993),73-80. https://doi.org/10.1016/0166-8641(93)90074-N
A. Blaszczyk and A. Szymanski, Cohen algebras and nowhere dense ultrafilters, Bulletin of the Polish Acad. of Sciences Math. 49 (2001), 15-25.
W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, Berlin, 1974. https://doi.org/10.1007/978-3-642-65780-1
A. Dow, A. V. Gubbi and A. Szyma'nski, Rigid Stone spaces within ZFC, Proc. Amer. Math. Soc. 102 (1988), 745-748. https://doi.org/10.1090/S0002-9939-1988-0929014-4
R. Engelking, General Topology, Sigma Series in Pure Mathematics Vol. 6, Heldermann Verlag Berlin, 1989.
Z. Frolık, Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87-91. https://doi.org/10.1090/S0002-9904-1967-11653-7
S. García-Ferreira, Three orderings on !âˆ—, Topology Appl. 50 (1993), 199-216. https://doi.org/10.1016/0166-8641(93)90021-5
S. García-Ferreira and R. A. Gonz'alez-Silva, Topological games defined by ultrafilters, Topology Appl. 137 (2004), 159-166. https://doi.org/10.1016/S0166-8641(03)00205-0
S. García-Ferreira and R. A. González-Silva, Topological games and product spaces, Coment. Math. Univ. Carolinae 43 (2002), no. 4, 675-685.
J. Gerlits and Zs. Nagy, Some properties of C(X), I, Topology Appl. 14 (1982), 151-161. https://doi.org/10.1016/0166-8641(82)90065-7
J. Ginsburg and V. Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418. https://doi.org/10.2140/pjm.1975.57.403
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
M. Hrusák, Fun with ultrafilters and special functions, manuscript.
M. Hrusák, Selectivity of almost disjoint families, Acta Univ. Caroline-Math. et Physica, 41 (2000), no. 2, 13-21.
Jan van Mill, An introduction to βω en: Handbook of Set-Theoretic Topology, editors K. Kunen y J. E. Vaughan, North-Holland, (1984), 505-567. https://doi.org/10.1016/B978-0-444-86580-9.50014-8
K. Kunen, Weak P-points in Nâˆ—, Colloq. Math. Soc. János Bolyai 23, Budapest (Hungary), 741-749.
W. F. Lindgren and A. Szymanski, A non-pseudocompact product of countably compact spaces via Seq, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3741-3746. https://doi.org/10.1090/S0002-9939-97-04013-6
P. Nyikos, Subsets of ωω and the Fréchet-Urysohn and αi -properties, Topology Appl. 48 (1992) 91-116. https://doi.org/10.1016/0166-8641(92)90021-Q
P. Simon, Applications of independent linked families, Colloq. Math. Soc. János Bolyai 41 (1983), 561-580.
P. L. Sharma, Some characterizations of the W-spaces and w-spaces, Gen. Topol. Appl. 9 (1978), 289-293. https://doi.org/10.1016/0016-660X(78)90032-6
J. E. Vaughan, Countably compact sequentially compact spaces, in: Handbook of Set-Theoretic Topology, editors J. van Mill and J. Vaughan, North-Holland, 571-600.
J. E. Vaughan, Two spaces homeomorphic to Seq(p), Coment. Math. Univ. Carolinae 42 (2001), no. 1, 209-218.
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