| 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:
|
http://hdl.handle.net/10338.dmlcz/146994 |
| . |
| 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, http://at.yorku.ca/v/a/a/a/27.htm. |
| Reference:
|
[11] Pavlov O.: Problems on (ir)resolvability.in Open Problems in Topology II, Elsevier, 2007, pp. 51–59. |
| Reference:
|
[12] Gruenhage G.: Generalized metrizable spaces.in 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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