W. W. Comfort

Resolvability: A selective survey and some new results

Abstract Following guidance from the Organizing Committee, the authors give a brief introduction to the theory of spaces which are resolvable in the sense introduced by Hewitt (1943) . The new results presented here are these. (A) A countably compact regular Hausdorff space without isolated points is ω -resolvable—that is, it admits an infinite family of pairwise disjoint dense subsets. (B) Among Tychonoff topologies without isolated points on a fixed set, no pseudocompact topology is maximal.

Estimates for the number of real-valued continuous functions

It is a familiar fact that IC(X)1<2?2x, where IC(X)J is the cardinal number of the set of real-valued continuous functions on the infinite topological space X, and AX is the least cardinal of a dense subset of X. While for metrizable spaces equality obtains, for some familiar spaces-e.g., the one-point compactification of the discrete space of cardinal 2Ko-the inequality can be strict, and the problem of more delicate estimates arises. It is hard to conceive of a general upper bound for I C(X) I which does not involve a cardinal property of X as an exponent, and therefore we consider exponential combinations of certain natural cardinal numbers associated with X. Among the numbers are wX, the least cardinal of an open basis, and wcX, the least m for which each open cover of Xhas a subfamily with m or fewer elements whose union is dense. We show that I C(X)I < (wX)wcx, and that this estimate is best possible among the numbers in question. (In particular, (wX)wcx < 26x always holds.) In fact, it is only with the use of a version of the generalized continuum hypothesis that we succeed in finding an X for which I C(X)J <(wX)WcX. Three further points warrant mention. (a) As a corollary of the result discussed above, we find that (wX)wcx = (wpX)wcx, where /X denotes the Stone-Cech compactification of X; this equation can be taken as a means of estimating w/X in terms of X. (b) For one of the examples indicating the delicacy of our result, we use a product space whose salient features are isolated via the theorem: if 3X, _ m for each a, then in Hta Xa each family of pairwise disjoint open sets has m or fewer members. This is proved by applying to the general cardinal m the ideas used by Marczewski in [M] for m = o. (c) The main theorem, 2.2, and its lemma, 2.1, require no separation axioms whatever. We take as a standing hypothesis elsewhere throughout this paper that each of the spaces considered is completely regular and Hausdorff; furthermore, each example we construct is completely regular and Hausdorff. The case of finite spaces is disposed of in ?1, and thereafter all spaces are infinite. Presented to the Society, August 27, 1969; received by the editors August 6, 1969. AMS Subject Classifications. Primary 5428, 5440; Secondary 0430, 4625, 5452, 5453.

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.

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.

The Dual Group of a Dense Subgroup

AbstractThroughout this abstract, G is a topological Abelian group and

Proper pseudocompact extensions of compact abelian group topologies

\hat G

Proper Pseudocompact Subgroups of Pseudocompact Abelian Groups

is the space of continuous homomorphisms from G into the circle group