Products of gamma-sets

Speaker(s)

Piotr Szewczak

Affiliation

Cardinal Wyszyński University in Warsaw

Date

March 6, 2019, 4:15 p.m.

Room

room 5050

Seminar

Topology and Set Theory Seminar

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Let X be a set of reals and Cp(X) be the set of all continuous real-valued functions on X with the pointwise convergence topology. By the result of Gerlits and Nagy the space Cp(X) has the Frechet-Urysohn property (a generalization of first-countability) if and only if the set X is a gamma-set (i.e., has a combinatorial covering property). The existence of uncountable gamma-sets of reals is independent with ZFC. Tsaban proved that sets with some specific combinatorial structure are gamma-sets. We generalize this class of sets and prove that their products have the property gamma. We also show that for every set X from our class and every gamma set Y, the product space X x Y have some strong property weaker than gamma. These investigations are motivated by the result of Miller, Tsaban and Zdomskyy that under CH, there are two gamma-sets whose product space is not even Menger (in particular it is not gamma). This is a joint work with Magdalena Włudecka.

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