Selective covering properties of product spaces, II: $\gamma$ spaces
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- by Arnold W. Miller, Boaz Tsaban and Lyubomyr Zdomskyy PDF
- Trans. Amer. Math. Soc. 368 (2016), 2865-2889 Request permission
Abstract:
We study productive properties of $\gamma$ spaces and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results:
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Solving a problem of F. Jordan, we show that for every unbounded tower set $X\subseteq \mathbb {R}$ of cardinality $\aleph _1$, the space $\operatorname {C}_\mathrm {p}(X)$ is productively Fréchet–Urysohn. In particular, the set $X$ is productively $\gamma$.
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Solving problems of Scheepers and Weiss and proving a conjecture of Babinkostova–Scheepers, we prove that, assuming the Continuum Hypothesis, there are $\gamma$ spaces whose product is not even Menger.
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Solving a problem of Scheepers–Tall, we show that the properties $\gamma$ and Gerlits–Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)).
We apply our results to solve a large number of additional problems and use Arhangel’skiĭ duality to obtain results concerning local properties of function spaces and countable topological groups.
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Additional Information
- Arnold W. Miller
- Affiliation: Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
- Email: miller@math.wisc.edu
- Boaz Tsaban
- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel – and – Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 7610001, Israel
- MR Author ID: 632515
- Email: tsaban@math.biu.ac.il
- Lyubomyr Zdomskyy
- Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Strasse 25, 1090 Vienna, Austria
- MR Author ID: 742789
- Email: lzdomsky@logic.univie.ac.at
- Received by editor(s): November 13, 2013
- Received by editor(s) in revised form: September 2, 2014
- Published electronically: October 2, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 2865-2889
- MSC (2010): Primary 26A03; Secondary 03E75, 03E17
- DOI: https://doi.org/10.1090/tran/6581
- MathSciNet review: 3449260