# Article

Full entry | PDF   (0.3 MB)
Keywords:
topological game; strategy; separability; $\theta$-separability; $\Omega$-separability; point-open game
Summary:
The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space $X$ as follows: at the $n$-th move the first player picks a point $x_n\in X$ and the second responds with choosing an open $U_n\ni x_n$. The game stops after $\omega$ moves and the first player wins if $\cup\{U_n:n\in\omega\}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta$ and $\Omega$. In $\theta$ the moves are made exactly as in the point-open game, but the first player wins iff $\cup\{U_n:n\in\omega\}$ is dense in $X$. In the game $\Omega$ the first player also takes a point $x_n\in X$ at his (or her) $n$-th move while the second picks an open $U_n\subset X$ with $x_n\in\overline{U}_n$. The conclusion is the same as in $\theta$, i.e\. the first player wins iff $\cup\{U_n:n\in\omega\}$ is dense in $X$. It is clear that if the first player has a winning strategy on a space $X$ for the game $\theta$ or $\Omega$, then $X$ is in some way similar to a separable space. We study here such spaces $X$ calling them $\theta$-separable and $\Omega$-separable respectively. Examples are given of compact spaces on which neither $\theta$ nor $\Omega$ are determined. It is established that first countable $\theta$-separable (or $\Omega$-separable) spaces are separable. We also prove that \newline 1) all dyadic spaces are $\theta$-separable; \newline 2) all Dugundji spaces as well as all products of separable spaces are $\Omega$-separable; \newline 3) $\Omega$-separability implies the Souslin property while $\theta$-separability does not.
References:
[1] Amirdjanov G.P., Šapirovsky B.E.: On dense subsets of topological spaces (in Russian). Doklady Akad. Nauk SSSR 214 No. 4 (1974), 249-252. MR 0343228
[2] Arhangel'skii A.V.: On bicompacta which satisfy hereditary Souslin condition (in Russian). Doklady Akad. Nauk SSSR 199 No. 6 (1971), 1227-1230. MR 0288718
[3] Arhangel'skii A.V.: The structure and classification of topological spaces and cardinal invariants (in Russian). Uspehi Mat. Nauk (1978), 33 No. 6 29-84. MR 0526012
[4] Baldwin S.: Possible point-open types of subsets of the reals. Topology Appl. (1991), 38 219-223. MR 1098902 | Zbl 0719.54003
[5] Daniels P., Gruenhage G.: The point-open types of subsets of the reals. Topology Appl. (1990), 37 53-64. MR 1075373
[6] Engelking R.: General Topology. PWN, Warszawa, 1977. MR 0500780 | Zbl 0684.54001
[7] Galvin F.: Indeterminacy of point-open games. Bull. Acad. Polon. Sci., Sér. Math. (1978), 26 No. 5 445-449. MR 0493925 | Zbl 0392.90101
[8] Gruenhage G.: Infinite games and generalizations of first countable spaces. Gen. Topol. Appl. (1976), 6 No. 3 339-352. MR 0413049 | Zbl 0327.54019
[9] Gul'ko S.P.: On structure of spaces of continuous functions and on their hereditary paracompactness (in Russian). Uspehi Mat. Nauk (1979), 34 No. 6 33-40. MR 0562814
[10] Juhasz I.: On point-picking games. Topology Proc. (1985), 10 No. 1 103-110. MR 0851205 | Zbl 0604.54006
[11] Kunen K.: Set theory. Introduction to independence proofs. North Holland P.C., Amsterdam, 1980. MR 0597342
[12] Lutzer D.J., McCoy R.A.: Category in function spaces. Pacific J. Math. (1980), 90 No. 1 145-168. MR 0599327 | Zbl 0481.54017
[13] Malyhin V.I., Rančin D.V., Ul'ianov V.M., Šapirovsky B.E.: On topological games (in Russian). Vestnik MGU, Matem., Mech., 1977, No. 6, pp. 41-48.
[14] Preiss D., Simon P.: A weakly pseudocompact subspace of Banach space is weakly compact. Comment. Math. Univ. Carolinae (1974), 15 603-609. MR 0374875 | Zbl 0306.54033
[15] Šapirovsky B.E.: On tightness, $\pi$-weight and related notions (in Russian). Scientific notes of Riga University, 1976, No. 3, pp. 88-89.
[16] Shakhmatov D.B.: Compact spaces and their generalizations. Recent Progress in General Topology, 1992, Elsevier S.P. B.V., pp. 572-640. MR 1229139 | Zbl 0801.54001
[17] Telgársky R.: Spaces defined by topological games. Fund. Math. (1975), 88 193-223. MR 0380708
[18] Telgársky R.: Spaces defined by topological games, II. Fund. Math. (1983), 116 No. 3 189-207. MR 0716219
[19] Tkachuk V.V.: Topological applications of game theory (in Russian). Moscow State University P.H., Moscow, 1992.
[20] Uspensky V.V.: Topological groups and Dugundji compact spaces. Math. USSR Sbornik (1989), 67 No. 2 555-580. MR 1019483
[21] White H.E.: Topological spaces that are $\alpha$-favorable for a player with perfect information. Proc. Amer. Math. Soc. (1975), 50 No. 3 477-482. MR 0367941

Partner of