Selective covering properties of product spaces, II: 𝛾 spaces

Miller, Arnold; Tsaban, Boaz; Zdomskyy, Lyubomyr

doi:10.1090/tran/6581

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:

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 $\gamma$.
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.
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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