# Article

 Title: On meager function spaces, network character and meager convergence in topological spaces  (English) Author: Banakh, Taras Author: Mykhaylyuk, Volodymyr Author: Zdomskyy, Lyubomyr Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 52 Issue: 2 Year: 2011 Pages: 273-281 Summary lang: English . Category: math . Summary: For a non-isolated point $x$ of a topological space $X$ let $\mathrm{nw}_\chi (x)$ be the smallest cardinality of a family $\mathcal N$ of infinite subsets of $X$ such that each neighborhood $O(x)\subset X$ of $x$ contains a set $N\in \mathcal N$. We prove that (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $\mathrm{nw}_\chi (x)=\aleph_0$; (b) for each point $x\in X$ with $\mathrm{nw}_\chi (x)=\aleph_0$ there is an injective sequence $(x_n)_{n\in \omega }$ in $X$ that $\mathcal F$-converges to $x$ for some meager filter $\mathcal F$ on $\omega$; (c) if a functionally Hausdorff space $X$ contains an $\mathcal F$-convergent injective sequence for some meager filter $\mathcal F$, then for every path-connected space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. Also we investigate properties of filters $\mathcal F$ admitting an injective $\mathcal F$-convergent sequence in $\beta \omega$. Keyword: network character Keyword: meager convergent sequence Keyword: meager filter Keyword: meager space Keyword: function space MSC: 54A20 MSC: 54C35 MSC: 54E52 . Date available: 2011-05-17T08:40:20Z Last updated: 2012-08-13 Stable URL: http://hdl.handle.net/10338.dmlcz/141493 . Reference: [1] Aviles Lopez A., Cascales S., Kadets V., Leonov A.: The Schur $l_1$ theorem for filters.Zh. Mat. Fiz. Anal. Geom. 3 (2007), no. 4, 383–398. MR 2376601 Reference: [2] Bartoszynski T., Goldstern M., Judah H., Shelah S.: All meager filters may be null.Proc. Amer. Math. Soc. 117 (1993), no. 2, 515–521. Zbl 0776.03023, MR 1111433 Reference: [3] van Douwen E.: The integers and topology.in: Handbook of Set-Theoretic Topology (K. Kunen, J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 111–167. Zbl 0561.54004, MR 0776622 Reference: [4] Ganichev M., Kadets V.: Filter convergence in Banach spaces and generalized bases.in General Topology in Banach Spaces (T. Banakh, ed.), Nova Sci. Publ., Huntington, NY, 2001, pp. 61–69. Zbl 1035.46009, MR 1901534 Reference: [5] García-Ferreira S., Malykhin V., A. Tamariz-Mascarúa A.: Solutions and problems on convergence structures to ultrafilters.Questions Answers Gen. Topology 13 (1995), no. 2, 103–122. MR 1350228 Reference: [6] Engelking R.: General Topology.Heldermann Verlag, Berlin, 1989. Zbl 0684.54001, MR 1039321 Reference: [7] Hart K.P.: Efimov's problem.in: Open Problems in Topology II (E. Pearl, ed.), Elsevier, Amsterdam, 2007, 171–177. MR 2367385 Reference: [8] Lašnev N.: On continuous decompositions and closed mappings of metric spaces.Dokl. Akad. Nauk SSSR 165 (1965), 756–758 (in Russian). MR 0192478 Reference: [9] Lutzer D.J., McCoy R.A.: Category in function spaces. I..Pacific J. Math. 90 (1980), no. 1, 145–168. Zbl 0481.54017, MR 0599327 Reference: [10] Ketonen J.: On the existence of $P$-points in the Stone-Cech compactification of integers.Fund. Math. 92 (1976), no. 2, 91–94. Zbl 0339.54035, MR 0433387 Reference: [11] Malykhin V.I., Tironi G.: Weakly Fréchet-Urysohn and Pytkeev spaces.Topology Appl. 104 (2000), 181–190. Zbl 0952.54003, MR 1780904 Reference: [12] Mazur K.: $F_\sigma$-ideals and $\omega_1\omega_1^*$-gaps in the Boolean algebras $P(\omega)/I$.Fund. Math. 138 (1991), no. 2, 103–111. MR 1124539 Reference: [13] Mykhaylyuk V.: On questions connected with Talagrand's problem.Mat. Stud. 29 (2008), 81–88. MR 2424602 Reference: [14] Pytkeev E.G.: The Baire property of spaces of continuous functions.Mat. Zametki 38 (1985), no. 5, 726–740. MR 0819632 Reference: [15] Pytkeev E.G.: Spaces of continuous and Baire functions in weak topologies.Doktor Sci. Dissertation, Ekaterinburg, 1993 (in Russian). Reference: [16] Rudin M.E.: Types of ultrafilters.1966 Topology Seminar (Wisconsin, 1965), pp. 147–151, Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J. Zbl 0431.03033, MR 0216451 Reference: [17] Solecki S.: Analytic ideals.Bull. Symbolic Logic 2 (1996), no. 3, 339–348. Zbl 0932.03060, MR 1416872 Reference: [18] Talagrand M.: Compacts de fonctions mesurables et filtres non mesurables.Studia Math. 67 (1980), no. 1, 13–43. Zbl 0435.46023, MR 0579439 Reference: [19] Tkachuk V.: Characterization of the Baire property of $C_p(X)$ by the properties of the space $X$.Cardinal Invariants and Mappings of Topological Spaces, Izhevsk, 1984, pp. 76–77 (in Russian). Reference: [20] Vaughan J.: Small uncountable cardinals and topology.in: Open Problems in Topology (J. van Mill, G. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 195–218. MR 1078647 .

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