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arXiv · 1310.8622

Selective covering properties of product spaces, II: gamma spaces

Arnold W. Miller·Boaz Tsaban·Lyubomyr Zdomskyy
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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 of reals X of cardinality aleph_1, the space Cp(X) is productively FU. 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 CH, 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'skii duality to obtain results concerning local properties of function spaces and countable topological groups.

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Arnold W. Miller, Boaz Tsaban, Lyubomyr Zdomskyy. 2013-10-31. Selective covering properties of product spaces, II: gamma spaces. https://doi.org/10.1090/tran/6581

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