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

Combinatorial images of sets of reals and semifilter trichotomy

Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to the Hurewicz 1927 problem is positive, it is shown here that semifilter trichotomy implies a negative answer to a slightly weaker form of this problem.Comment: Small update

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