A semifilter approach to selection principles II: tau*-covers

In this paper we settle all questions whether (it is consistent that) the properties P and Q [do not] coincide, where P and Q run over selection principles of the type U_fin(O,A).Comment: 9 pages; Latex2e; 1 table; Submitted to CMU

Productively Lindel\"of spaces of countable tightness

Michael asked whether every productively Lindel\"of space is powerfully Lindel\"of. Building of work of Alster and De la Vega, assuming the Continuum Hypothesis, we show that every productively Lindel\"of space of countable tightness is powerfully Lindel\"of. This strengthens a result of Tall and Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a productively Lindel\"of space $X$ is powerfully Lindel\"of if every open cover of $X^\omega$ admits a point-continuum refinement consisting of basic open sets. This strengthens a result of Burton and Tall. Finally, we show that separation axioms are not relevant to Michael's question: if there exists a counterexample (possibly not even $\mathsf{T}_0$), then there exists a regular (actually, zero-dimensional) counterexample.Comment: 7 page

Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel'skii alpha_1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C_p(X) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our main result is that C_p(X) is an alpha_1 space if, and only if, each Borel image of X in the Baire space is bounded. Using this characterization, we solve a variety of problems posed in the literature concerning spaces of continuous functions.Comment: To appear in Jouranl of the European Mathematical Societ

Locally compact, ω1\omega_1-compact spaces

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Comment: 21 pages, submitted, comments are welcom

Products of HH-separable spaces in the Laver model

We prove that in the Laver model for the consistency of the Borel's conjecture, the product of any two $H$-separable spaces is $M$-separable