96 research outputs found
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, -compact spaces
An -compact space is a space in which every closed discrete
subspace is countable. We give various general conditions under which a locally
compact, -compact space is -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
- …