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Selective covering properties of product spaces, II: $ \gamma$ spaces


Authors: Arnold W. Miller, Boaz Tsaban and Lyubomyr Zdomskyy
Journal: Trans. Amer. Math. Soc. 368 (2016), 2865-2889
MSC (2010): Primary 26A03; Secondary 03E75, 03E17
Published electronically: October 2, 2015
MathSciNet review: 3449260
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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:

  1. 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 $.
  2. 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.
  3. 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
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
Email: lzdomsky@logic.univie.ac.at

DOI: https://doi.org/10.1090/tran/6581
Keywords: Gerlits--Nagy property $\gamma$, Gerlits--Nagy property (*), Menger property, Hurewicz property, Rothberger property, Sierpi\'nski set, selection principles, Cohen forcing, special sets of real numbers, $\operatorname{C}_\mathrm{p}$ theory, selectively separable, countable fan tightness.
Received by editor(s): November 13, 2013
Received by editor(s) in revised form: September 2, 2014
Published electronically: October 2, 2015
Article copyright: © Copyright 2015 American Mathematical Society