Jan van Mill

An Introduction to βω

Publisher Summary This chapter presents an introduction to the space βω, that is, the Stone space of the Boolean algebra P(ω) of subsets of ω. The results about βω usually have wide applications in various parts of mathematics. The following facts are the most important results obtained in βω in recent years: (1) it is consistent that P-points do not exist in βω/ω, (2) some but not all points in βω/ω are weak P-points, and (3) every point in βω/ω is a c-point. The space βω is a monster having three heads. If one works in a model in which the Continuum Hypothesis (CH) holds, then one will see only the first head. This head is smiling, friendly, and makes one feel comfortable working with βω. Because of the presence of the CH, transfinite inductions have length ω1, and because of the special properties of the Boolean algebras under consideration, one can always continue the transfinite inductions until stage ω1.

Selections and orderability

0. Introduction. Let X be a compact Hausdorff space and let 2x denote the hyperspace of nonempty closed subsets of X. A selection for X is a continuous map F: 2X X such that F(A) E A for all A E 2X. Let X(2) denote the 2-fold symmetric product of X, i.e. the subspace of 2x consisting of all nonempty closed subspaces of X containing at most two points. A weak selection for X is a continuous map s: X(2) -* X such that s(A) E A for all A E X(2). It is easy to see that X has a weak selection if and only if there is a continuous map s: x2 X such that for all x, y C X, (1) s(x, y) = s(y, x), and (2) s(x,y) &{x,y}. Such a map s: X X will also be called a weak selection. Michael [M] showed that for a continuum X the following statements are equivalent: (a) X has a selection, (b) X has a weak selection, and (c) X is orderable. In [Y], Young claims, without giving a proof, that statements (a), (b), and (c) are also equivalent for compact zero-dimensional spaces X. In this paper we will show that, for compacta, statements (a), (b), and (c) are always equivalent.

CONCERNING CONNECTED, PSEUDOCOMPACT ABELIAN GROUPS

Abstract It is known that if P is either the property ω-bounded or countably compact, then for every cardinal α ⩾ ω there is a P-group G such that wG = α and no proper, dense subgroup of G is a P-group. What happens when P is the property pseudocompact? The first-listed author and Robertson have shown that every zero-dimensional Abelian P-group G with wG >ω has a proper, dense, P-group. Turning to the case of connected P-groups, the present authors show the following results: Let G be a connected, pseudocompact, Abelian group with wG = α >ω. If any one of the following conditions holds, then G has a proper, dense (necessarily connected) pseudocompact subgroup: (a) wG ⩽ c ; (b) |G| ⩾ αω; (c) α is a strong limit cardinal and cf(α) >ω; (d) | tor G| > c (e) G is not divisible.

Absorbing systems in infinite-dimensional manifolds

Abstract The aim of this paper is to define absorbing systems in infinite-dimensional manifolds and to derive some basic properties of them along the lines of Chapman. As an application we prove that for a countable nondiscrete Tychonov space X, if Cp(X) is and Fσδ subset of R X then it is an Fσδ-absorber, and hence homeomorphic to the countable infinite product of copies of l2ƒ. This generalizes a result of Dobrowolski, Marciszewski and Mogilski.

On the existence of free topological groups

Abstract Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G ( X , U , V ) is a free ( U , V )-group over X if (a) X is a subspace of G , (b) G ϵ U , and (c) every continuous f : X → H with H ϵ V extends uniquely to a continuous homomorphism f: G → H . For certain classes U and V , we consider the question of the existence of free ( U , V )- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then 1. (a) there is a free ( PA , PA )-group over X iff; X = O and 2. (b) there is for each X a free ( PA , CA )-group over X in which X is closed.

Discrete sets and the maximal totally bounded group topology

Hart, K.P. and J. van Mill, Discrete sets and the maximal totally bounded group topology, Journal of Pure and Applied Algebra 70 (1991) 73-80. If G is an Abelian group, then G # is G with its maximal totally bounded group topology. We prove that every A c G# contains a closed (in G#) and discrete subset B such that lB1 = IAl. This answers a question posed by Eric van Douwen. We also present an example of a countable G’ having an infinite relatively discrete subset that is not closed.

A topological group having no homeomorphisms other than translations

We give an example of a (separable metric) connected and locally connected topological group, the only autohomeomorphisms of which are group translations.

THE BAIRE CATEGORY THEOREM IN PRODUCTS OF LINEAR SPACES AND TOPOLOGICAL GROUPS

Abstract A space is a Baire space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely metrizable linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E × F is not. If X is a completely metrizable linear space of weight ℵ 1 then X is the direct sum E ⊕ F of two linear subspaces E and F such that both E and F are Baire but E × F is not.

Sixteen topological types in βω−ω

Abstract In this paper we describe sixteen topological types in βω−ω. Among others, we show that there is a weak P-point x ∈ βω−ω which is a limit point of some ccc subset of βω−ω−{x} and that there is a point y ∈ βω−ω which is a limit point of some countable subset of βω−ω−{y} but not of any countable discrete subset of βω−ω−{y}.

On the character and π-weight of homogeneous compacta

UnderGCH, χ(X)≤π(X) for every homogenous compactumX.CH implies that a homogeneous compactum of countable π-weight is first countable. There is a compact space of countable π-weight and uncountable character which is homogeneous underMA+GCH, but not underCH.