TY - JOUR

T1 - Selective covering properties of product spaces, II

T2 - γ spaces

AU - Miller, Arnold W.

AU - Tsaban, Boaz

AU - Zdomskyy, Lyubomyr

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2016/4

Y1 - 2016/4

N2 - We study productive properties of γ 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 ⊆ R of cardinality ℵ1, the space Cp(X) is productively Fréchet– Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss and proving a conjecture of Babinkostova–Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ 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.

AB - We study productive properties of γ 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 ⊆ R of cardinality ℵ1, the space Cp(X) is productively Fréchet– Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss and proving a conjecture of Babinkostova–Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ 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.

UR - http://www.scopus.com/inward/record.url?scp=84955137157&partnerID=8YFLogxK

U2 - 10.1090/tran/6581

DO - 10.1090/tran/6581

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AN - SCOPUS:84955137157

SN - 0002-9947

VL - 368

SP - 2865

EP - 2889

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 4

ER -