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Title: Resolvability in c.c.c. generic extensions (English)
Author: Soukup, Lajos
Author: Stanley, Adrienne
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 58
Issue: 4
Year: 2017
Pages: 519-529
Summary lang: English
Category: math
Summary: Every crowded space $X$ is ${\omega}$-resolvable in the c.c.c. generic extension $V^{\operatorname{Fn}(|X|,2)}$ of the ground model. We investigate what we can say about ${\lambda}$-resolvability in c.c.c. generic extensions for $\lambda > \omega$. A topological space is monotonically $\omega _1$-resolvable if there is a function $f:X\to \omega _1$ such that \begin{displaymath} \{x\in X: f(x)\geq {\alpha}\}\subset^{dense}X \end{displaymath} for each ${\alpha}< \omega _1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\omega}_1$-resolvable in some c.c.c. generic extension; (2) $X$ is monotonically $\omega _1$-resolvable; (3) $X$ is ${\omega}_1$-resolvable in the Cohen-generic extension $V^{\operatorname{Fn}(\omega _1,2)}$. We investigate which spaces are monotonically $\omega _1$-resolvable. We show that if a topological space $X$ is c.c.c., and ${\omega}_1\le \Delta(X)\le |X|<{\omega}_{\omega}$, where $\Delta(X) = \min\{ |G| : G \ne \emptyset \mbox{ open}\}$, then $X$ is monotonically $\omega _1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=\Delta(Y)=\aleph_\omega$ which is not monotonically $\omega _1$-resolvable. The characterization of $\omega _1$-resolvability in c.c.c. generic extension raises the following question: is it true that crowded spaces from the ground model are ${\omega}$-resolvable in $V^{\operatorname{Fn}({\omega},2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is ${\omega}$-resolvable in $V^{\operatorname{Fn}({\omega},2)}$, (ii) if there are no weakly inaccessible cardinals, then every crowded space $X$ is ${\omega}$-resolvable in $V^{\operatorname{Fn}({\omega}_1,2)}$. Moreover, it is also consistent, modulo a measurable cardinal, that there is a crowded space $X$ with $|X|=\Delta(X)=\omega _1$ such that $X$ remains irresolvable after adding a single Cohen real. (English)
Keyword: resolvable
Keyword: monotonically $\omega _1$-resolvable
Keyword: measurable cardinal
MSC: 03E35
MSC: 54A25
MSC: 54A35
idZBL: Zbl 06837083
idMR: MR3737122
DOI: 10.14712/1213-7243.2015.226
Date available: 2017-12-12T06:53:42Z
Last updated: 2020-01-05
Stable URL:
Reference: [1] Angoa J., Ibarra M., Tamariz-Mascarúa A.: On $\omega$-resolvable and almost-${\omega}$-resolvable spaces.Comment. Math. Univ. Carolin. 49 (2008), no. 3, 485–508. Zbl 1212.54069, MR 2490442
Reference: [2] Bolstein R.: Sets of points of discontinuity.Proc. Amer. Math. Soc. 38 (1973), no. 1, 193–197. Zbl 0232.54014, MR 0312457, 10.1090/S0002-9939-1973-0312457-9
Reference: [3] Dorantes-Aldama A.: Baire irresolvable spaces with countable Souslin number.Topology Appl. 188 (2015), 16–26. Zbl 1317.54011, MR 3339107, 10.1016/j.topol.2015.03.005
Reference: [4] Hewitt E.: A problem of set theoretic topology.Duke Math. J. 10 (1943), 309–333. Zbl 0060.39407, MR 0008692, 10.1215/S0012-7094-43-01029-4
Reference: [5] Juhász I., Magidor M.: On the maximal resolvability of monotonically normal spaces.Israel J. Math. 192 (2012), 637–666. Zbl 1264.54004, MR 3009737, 10.1007/s11856-012-0042-z
Reference: [6] Juhász I., Soukup L., Szentmiklóssy Z.: Resolvability and monotone normality.Israel J. Math. 166 (2008), 1–16. Zbl 1155.54006, 10.1007/s11856-008-1017-y
Reference: [7] Kunen K.: Maximal $\sigma$-independent families.Fund. Math. 117 (1983), no. 1, 75–80. Zbl 0532.03024, MR 0712215, 10.4064/fm-117-1-75-80
Reference: [8] Kunen K., Prikry K.: On descendingly incomplete ultrafilters.J. Symbolic Logic 36 (1971), no. 4, 650–652. Zbl 0259.02053, MR 0302441, 10.2307/2272467
Reference: [9] Kunen K., Szymanski A., Tall F.: Naire irresolvable spaces and ideal theory.Ann. Math. Sil. no. 14 (1986), 98–107. MR 0861505
Reference: [10] Kunen K., Tall F.: On the consistency of the non-existence of Baire irresolvable spaces.manuscript privately circulated, Topology Atlas, 1998,
Reference: [11] Pavlov O.: Problems on (ir) Open Problems in Topology II, Elsevier, 2007, pp. 51–59.
Reference: [12] Gruenhage G.: Generalized metrizable Recent Progress in General Topology, III, Springer Science & Business Media, 2013, pp. 471–505. Zbl 1314.54001, MR 3205490
Reference: [13] Tamariz-Mascarúa A., Villegas-Rodríguez H.: Spaces of continuous functions, box products and almost-$\omega$-resolvable spaces.Comment. Math. Univ. Carolin. 43 (2002), no. 4, 687–705. Zbl 1090.54011, MR 2045790
Reference: [14] Ulam S.: Zur Masstheorie in der allgemeinen Mengenlehre.Fund. Math. 16 (1930), 140–150. 10.4064/fm-16-1-140-150
Reference: [15] Woodin W.H.: Descendingly complete ultrafilter on $\aleph_{\omega}$.personal communication.


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