Eric K. van Douwen

The Integers and Topology

Publisher Summary This chapter discusses integers and topology. Role in topology of certain cardinals is associated with ω. This chapter discusses the problems on first countability, convergence, and separable metrizable spaces. A typical use of these set theoretic cardinals associated with ω involves topologically defined cardinals. Another use of these set theoretic cardinals associated with ω is that certain topological results hold if one of these cardinals equals ω1. The chapter also discusses the set theory, which states an ordinal is the set of smaller ordinals, and a cardinal is an initial ordinal. ω is ω0, and c is 2ω. The chapter also describes sequential and countable compactness. A countable set A of a space X is said to cluster at x ∈ X if each neighborhood of x contains infinitely many points of A, and it is said to converge to x ∈ X if each neighborhood of x contains all but finitely many points of A. A space is called countably compact if each countably infinite set clusters at some point, and it is called sequentially compact if each countably infinite set has an infinite subset that converges somewhere. Moreover, a space X is called subsequential if for every countably infinite A ⊆ X and for every cluster point x of A, there is an infinite subset of A that converges to x.

Applications of maximal topologies

Abstract We construct some unusual spaces by considering maximal members of suitable families of topologies. For example, we construct a countable regular crowded space no point of which is a limit point of two disjoint sets. An application to ω ∗ is that there is a separable space which is a continuous image of ω * under a ⩽ two-to-one map. We also show that for each k ϵ [2, ω), there is a k -irresolvable space.

The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G

Abstract For an Abelian (abstract) group G let bG denote the Bohr-compactification of G and let G ∗ denote G as topological subgroup of bG , or, equivalently, let G ∗ be G with its maximal totally bounded group topology. Our main result says that G ∗ has “many” discrete C -embedded subspaces which are C ∗ -embedded in bG . A consequence is that no sequence in G ∗ convergences to a point of bG . As an application we get information about BG , the unique (up to isomorphism) biggest G -space, for (abstract) Abelian G . We show that π( BG ) > | G | and | BG | = exp 2 | G |. For countable Abelian G this tells us BG is not the absolute of ω 2.

The product of two countably compact topological groups

We use MA (= Martins Axiom) to construct two countably compact topological groups whose product is not countably compact. To this end we first use MA to construct an infinite countably compact topological group which has no nontrivial convergent sequences.

THE PIXLEY-ROY TOPOLOGY ON SPACES OF SUBSETS

Publisher Summary This chapter discusses the Pixley–Roy topology on spaces of subsets. In [PR], Pixley and Roy construct a nonseparable ccc {=countable chain condition = no disjoint collection of open sets is uncountable} Moore space by topologizing F(R) in a suitable fashion {R = reals}. When studying spaces of subsets of a space X, it is usually quite natural to confine attention to nonempty open sets: no basic open set of the Vietoris topology contains φ, and φ would be isolated in the Pixley–Roy topology A {φ} = [φ,φ]. Cospaces were invented to study their associated spaces and more generally, it is not unusual in topology to study a space by looking at a nice weaker topology.

Some questions about Boolean algebras

Very recently there has been much progress on some fundamental settheoretic problems concerning Boolean algebras. The purpose of this article is to indicate some problems still left open, put in perspective by what has been shown recently. We have made no a t tempt to completely cover the field with these questions, but hope that for the problems ment ioned here the picture we give is fairly complete. To some extent this is a survey of recent set-theoretical results on Boolean algebras. In particular, part of the information we give here answers questions f rom earlier informal versions of this paper and has been included so as to make clear what no longer is an open problem. We are grateful to R. Bonnet , S. Koppelberg, K. Kunen, R. Laver, R. McKenzie, P. Nyikos, S. Shelah and M. Weese for comments on earlier versions of this article.

βX and fixed-point free maps

Abstract We prove that if X is a finite-dimensional paracompact space then for every fixed-point free homeomorphism ƒ: X → X its Stone extension βƒ: β X → β X is also fixed-point free. In addition, we construct an example of a (necessarily infinite-dimensional) locally compact separable metrizable space X having a fixed-point free homeomorphism ƒ such that βƒ has a fixed-point.

A technique for constructing honest locally compact submetrizable examples

Abstract We introduce a technique for constructing honest (= not requiring additional axioms) locally compact locally countable topologies on the real line (and on other spaces).

COVERING AND SEPARATION PROPERTIES OF BOX PRODUCTS

ABSTRACT We give a fairly complete survey, including proofs, of what is known about the question of when a box product of compact spaces is paracompact, and show how badly a box product of compact or metrizable spaces can fail to be normal. A side result is that the Tychonoff product of uncountably many infinite discrete spaces is not countably orthocompact. 1980 Math. Subj. Class.: 54B10, 54D15, 54D18, 54A35; 54E35, 54E65

A compact space with a measure that knows which sets are homeomorphic

Abstract We construct a compact homogeneous space bH which has a Borel measure μ which knows which sets are homeomorphic: if X and Y are homeomorphic Borel sets then μ (X) = μ (Y) , and, as a partical converse, if X and Y are open and μ (X) = μ (Y) and X and Y are both compact or both noncompact, then X and Y are homeomorphic. In particular, μ is nonzero and invariant under all autohomeomorphisms; it turns out that up to a multiplicative constant μ is unique with respect to these properties. bH is constructed as an easy to visualize compactification of a very special sub group H of the circle group T; the Haar measure μ on T induces μ and also induces a measure μ on H which knows which subsets of H are homeomorphic.