Hurewicz sets of reals without perfect subsets

We show that even for subsets X of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Gamma,Gamma), asserting that for each countable family of open gamma-covers of X, there is a choice function whose image is a gamma-cover of X. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszynski and Tsaban, and implies that for C_p(X), the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangelskii's property alpha_2

First-countable Lindel\"of scattered spaces

We study the class of first-countable Lindel\"of scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a scattered subspace of $\mathbb Q$, the class of $T_2$ FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to $Q$-sets, Lusin sets, and their relatives, and to the cardinals $\mathfrak{b}$ and $\mathfrak{d}$. Many natural questions about FLS spaces turn out to be independent of $\mathsf{ZFC}$. We prove that there exist uncountable FLS spaces with scattered height $\omega$. On the other hand, an uncountable FLS space with finite scattered height exists if and only if $\mathfrak{b} = \aleph_1$. We prove some independence results concerning the possible cardinalities of FLS spaces, and concerning what ordinals can be the scattered height of an FLS space. Several open problems are included

Independence in topological and C*-dynamics

We develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods. One of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties. We also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to C*-dynamics.Comment: 54 pages; Definition 2.2 changed; to appear in Mathematische Annale