On Completely Separable MAD Families (Set Theory : Reals and Topology)

We survey some constructions of completely separable MAD families

A non-discrete space X with Cp(X) Menger at infinity

[EN] In a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of Cp(X) in some compactification is Menger but not σ-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM. The second-named author was also supported by the 2017 PRODEP grant UAM-PTC-636 awarded by the Mexican Secretariat of Public Education (SEP).Bella, A.; Hernández-Gutiérrez, R. (2019). A non-discrete space X with Cp(X) Menger at infinity. Applied General Topology. 20(1):223-230. https://doi.org/10.4995/agt.2019.10714SWORD223230201A. V. Arkhangel'skii, Topological Function Spaces, Mathematics and its Applications (Soviet Series), 78. Kluwer Academic Publishers Group, Dordrecht, 1992. x+205 pp. ISBN: 0-7923-1531-6L. F. Aurichi and A. Bella, When is a space Menger at infinity?, Appl. Gen. Topol. 16, no. 1 (2015), 75-80. https://doi.org/10.4995/agt.2015.3244T. Bartoszynski and H. Judah, Set theory. On the Structure of the Real Line, A K Peters, Ltd., Wellesley, MA, 1995. xii+546 pp. ISBN: 1-56881-044-X https://doi.org/10.1201/9781439863466A. Bella, S. Tokgöz and L. Zdomskyy, Menger remainders of topological groups, Arch. Math. Logic 55, no. 5-6 (2016), 767-784. https://doi.org/10.1007/s00153-016-0493-8D. Chodounsky, D. Repovs and L. Zdomskyy, Mathias forcing and combinatorial covering properties of filters, J. Symb. Log. 80, no. 4 (2015), 1398-1410. https://doi.org/10.1017/jsl.2014.73O. Guzmàn, M. Hrusák and A. Martínez-Celis, Canjar filters, Notre Dame J. Form. Log. 58, no. 1 (2017), 79-95. https://doi.org/10.1215/00294527-3496040M. Henriksen and J. R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958), 83-105. https://doi.org/10.1215/S0012-7094-58-02509-2R. Hernández-Gutiérrez and P. J. Szeptycki, Some observations on filters with properties defined by open covers, Comment. Math. Univ. Carolin. 56, no. 3 (2015), 355-364. https://doi.org/10.14712/1213-7243.2015.125W. Just, A. W. Miller, M. Scheepers and P. J. Szeptycki, The combinatorics of open covers. II, Topology Appl. 73, no. 3 (1996), 241-266. https://doi.org/10.1016/S0166-8641(96)00075-2W. Marciszewski, P-filters and hereditary Baire function spaces, Topology Appl. 89, no. 3 (1998), 241-247. https://doi.org/10.1016/S0166-8641(97)00216-2W. Marciszewski, On analytic and coanalytic function spaces Cp(X) , Topology Appl. 50 (1993), 341-248. https://doi.org/10.1016/0166-8641(93)90023-7B. Tsaban, Menger's and Hurewicz's problems: solutions from "the book" and refinements, in: Set theory and its applications, 211-226, Contemp. Math., 533, Amer. Math. Soc., Providence, RI, 2011

Games and hereditary Baireness in hyperspaces and spaces of probability measures

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter $\mathcal{F}$ and prove that it is equivalent to $K(\mathcal{F})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_r(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ which is not completely metrizable with $P_r(X)$ hereditarily Baire. As far as we know this is the first example of this kind