publication . Preprint . 2020

Maximal almost disjoint families and pseudocompactness of hyperspaces

Guzmán, Osvaldo; Hrušák, Michael; Rodrigues, Vinicius de Oliveira; Todorčević, Stevo; Tomita, Artur Hideyuki;
Open Access English
  • Published: 18 Apr 2020
We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(\omega)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and prove that consistently such families do not exist \emph{genericaly}, by constructing a consistent example of a maximal almost disjoint family $\mathcal A$ of size less than $\mathfrak c$ whose hyperspace is not pseudocompact.
arXiv: Mathematics::General Topology
free text keywords: Mathematics - General Topology, Primary 54D20, 03E35, Secondary 54D35, 03E17
Funded by
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
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26 references, page 1 of 2

[1] Bohuslav Balcar, Jan Pelant, and Petr Simon. The space of ultrafilters on N covered by nowhere dense sets. Fundamenta Mathematicae, 110:11-24, 1980. [OpenAIRE]

[2] Andreas Blass. Combinatorial cardinal characteristics of the continuum. In M. Foreman and A. Kanamori, editors, Handbook of Set Theory, pages 395-489. Springer Netherlands, 2009. [OpenAIRE]

[3] Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel J. Math., 65(3):259-271, 1989. [OpenAIRE]

[4] Jo¨rg Brendle. Van Douwen's diagram for dense sets of rationals. Ann. Pure Appl. Logic, 143(1-3):54-69, 2006.

[5] Jo¨rg Brendle and Vera Fischer. Mad families, splitting families and large continuum. J. Symbolic Logic, 76(1):198-208, 2011.

[6] Jo¨rg Brendle and Saharon Shelah. Ultrafilters on ω -their ideals and their cardinal characteristics. Transactions of the American Mathematical Society, 351(7):2643-2674, 1999. [OpenAIRE]

[7] Jiling Cao, Tsugunori Nogura, and A.H. Tomita. Countable compactness of hyperspaces and Ginsburg's questions. Topology and its Applications, 144(1):133 - 145, 2004.

[8] Alan Dow and Saharon Shelah. On the cofinality of the splitting number. Indag. Math. (N.S.), 29(1):382-395, 2018.

[9] Alan Dow and Saharon Shelah. Pseudo P-points and splitting number. Arch. Math. Logic, 58(7-8):1005-1027, 2019.

[10] Ilijas Farah. OCA and towers in P(N)/fin. Comment. Math. Univ. Carolin., 37(4):861-866, 1996.

[11] Vera Fischer, Sy D. Friedman, Diego A. Mej´ıa, and Diana C. Montoya. Coherent systems of finite support iterations. J. Symb. Log., 83(1):208-236, 2018.

[12] Vera Fischer and Diego Alejandro Mejia. Splitting, bounding, and almost disjointness can be quite different. Canad. J. Math., 69(3):502-531, 2017.

[13] John Ginsburg. Some results on the countable compactness and pseudocompactness of hyperspaces. Canadian Journal of Mathematics, 27(6):1392-1399, 1975. [OpenAIRE]

[14] Osvaldo Guzm´an-Gonz´alez, Michael Hruˇs´ak, Carlos Azarel Mart´ınez-Ranero, and Ulises Ariet Ramos-Garc´ıa. Generic existence of MAD families. J. Symb. Log., 82(1):303-316, 2017.

[15] Stephen H. Hechler. Short complete nested sequences in βN\N and small maximal almostdisjoint families. General Topology and Appl., 2:139-149, 1972. [OpenAIRE]

26 references, page 1 of 2
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