Murray Bell

Countable spread of exp Y and λY

Abstract We show that it is consistent with ZFC that there exists a compact 0-dimensional Hausdorff space X for which exp X has countable spread, but X is not metrizable. This establishes the independence of Malyhins problem. The space X also has no uncountable weakly separated subspaces, its superextension is first countable, and its square is a strong S -space. For 0-dimensional Y we prove that λ Y has countable spread iff Y is compact and metrizable. We show that it is consistent with ZFC that if Y is 0-dimensional and λ Y is first countable, then Y is compact and metrizable.

First countable pseudocompactifications

Abstract A general construction for creating first countable pseudo-compact extensions of certain spaces is given. We show that there exists a first countable σ-compact space with no first countable pseudocompactification. We construct a new Suslinean example, namely, a first countable, pseudocompact, ccc, non-separable space. Assuming Martins Axiom, we can further achieve a point-countable base. A second application is to unify and clarify examples of Scott and Watson of pseudocompact, metaLindelof and point-countable base non-compact spaces.

A first countable compact space that is not an N∗ image

Let N∗ be the nonisolated points of the Stone-Cech compactification of the countable discrete space N. We show that, in the absence of CH, there can be a first countable compact space which is not an N∗ image. This answers a question of van Douwen and Przymusinski. Our example is even hereditarily metacompact.

Polyadic spaces of countable tightness

Abstract We answer a question of J. Gerlits by constructing a polyadic space of countable tightness which is not a continuous image of Aκω (Aκ is the one point compactification of the discrete space κ). The space is a Uniform Eberlein compact space of weight ω1. It will follow that being an Aκω image is not preserved by countable inverse limits.

The hyperspace of a compact space, I

Abstract We investigate the properties monolithic and d-separable for the hyperspace H(X) of all nonempty closed subsets of a compact Hausdorff space X . A. Arhangelskii has asked whether H(X) monolithic is equivalent to X metrizable. We answer this with: Let X be a compact orderable space. Then H(X) is monolithic iff X is monolithic and hereditarily Lindelof. So, a Suslin continuum has a monolithic hyperspace. In contrast, MA(ω 1 ) implies that for any compact Hausdorff space X , H(X) is monolithic iff X is metrizable. We prove that H(X) is always d-separable. A special case of this yields that every locally compact Hausdorff space X has a discrete (in H(X) ) π-net.

Topological groups homeomorphic to products of discrete spaces

Abstract We give necessary and sufficient conditions for a topological group to be homeomorphic to a product of the form Nκ × Dτ, where κ and τ are infinite cardinals and N and D denote countable (infinite) and two-point discrete spaces respectively. These conditions are purely topological: (a) zero-dimensionality in the sense of dim; and (b) being an absolute extensor in the dimension zero (briefly, an AE(0) space).

Chains and discrete sets in zero-dimensional compact spaces

Let X be a compact zero-dimensional space and let B{X) denote the Boolean algebra of all clopen subsets of X. Let k be an infinite cardinal. It is shown that if B( X) contains a chain of cardinality k then X x X contains a discrete subset of cardinality k. This complements a recent result of J. Baumgartner and P. Komjath relating antichains in B(X) to the w-weight of X.

On complete partially ordered sets and compatible topologies

We describe two complete partially ordered sets which are the intersection of complete linear orderings but which have no compatible Hausdorff topology. One is two-dimensional, while the second is countable, and leads to an example of a countable, compact, T1 space with a countable base which is not the continuous image of any compact Hausdorff space.