Jerry E. Vaughan

Countably Compact and Sequentially Compact Spaces

Publisher Summary The chapter presents the basic theory, examples, and techniques of countable compactness and sequential compactness. A space X is called sequentially compact if every sequence in X has a convergent subsequence. A space X is called countably compact if sequence in X has a cluster point. A space X is called totally countably compact if every sequence f in X has a subsequence f | A whose range is contained in a compact subset of X. A space X is called ω-bounded if for every sequence f in X, the range of f is contained in a compact subset of X. The properties of filter bases can be defined as follows: (1) they can be sequentially compact, wherein every countably filter base has a finer countable filter base that is convergent; (2) they can be countably compact, wherein every countable filter base has an adherent point, (3) they can be ω-bounded, in which every filter base on a countable set has an adherent point, and (4) they can be totally countably compact, wherein every countable filter base has a finer countable filter base which is total.

Products of topological spaces

Abstract The main purpose of this paper is to unify a number of theorems in topology whose conclusions state that a product of topological spaces has a compactness-like property.Three such theorems are (1) the Tychonoff theorem: Every product of compact spaces is compact, (2) the theorem of C.T. Scarborough and A.H. Stone: Every product of at most N 1 sequentially compact spaces is countably compact, and (3) the theorem of N. Noble: A countable product of Lindelof P -spaces is Lindelof.

The Scarborough-Stone problem for Hausdorff spaces

Abstract We give two methods of constructing families of sequentially compact Hausdorff spaces whose product is not countably compact.

Sequentially compact, Franklin-Rajagopalan spaces

A locally compact T2-space is called a Franklin-Rajagopalan space (or FR-space) provided it has a countable discrete dense subset whose complement is homeomorphic to an ordinal with the order topology. We show that (1) every sequentially compact FR-space X can be identified with a space constructed from a tower T on w (X = X(T)), and (2) for an ultrafilter u on w, a sequentially compact FR-space X(T) is not u-compact if and only if there exists an ultrafilter v on w such that v D T, and v is below u in the RudinKeisler order on w*. As one application of these results we show that in certain models of set theory there exists a family T of towers such that I T I < 2W, and nH{X(T): T E T } is a product of sequentially compact FR-spaces which is not countably compact (a new solution to the Scarborough-Stone problem). As further applications of these results, we give consistent answers to questions of van Douwen, Stephenson, and Vaughan concerning initially m-chain compact and totally initially m-compact spaces.

A note on [a, b]-compactness

Abstract In this note we study the relationship between [a, b]-compactness in the sense of open covers, and [a, b]-compactness in the sense of complete accumulation points.

From countable compactness to absolute countable compactness

We show that every countably compact space which is monotonically normal, almost 2-fully normal, radial T2, or T3 with countable spread is absolutely countably compact. For the first two mentioned properties, we prove more general results not requiring countable compactness. We also prove that every monotonically normal, orthocompact space is finitely fully normal.

On the Product of a Compact Space with an Absolutely Countably Compact Space

We show that the product of a compact sequential T2‐space, with an absolutely countably compact T3‐space, is absolutely countably compact, and give several related results. For example, we show that every countably compact GO‐space is absolutely countably compact, and that the product of a compact T2‐space of countable tightness with an absolutely countably compact, ω‐bounded T3‐space (in particular a countably compact GO‐space) is absolutely countably compact.

Accessible and Biaccessible Points in Contrasequential Spacesa

ABSTRACT. Two countable spaces having no nontrivial convergent sequences are constructed. One space has every point biaccessible (by a countable discrete set), and the other has every point accessible but not biaccessible (by a countable discrete set). It is shown that in a compact T2‐space if a set A is not closed, then there exists a free sequence contained in A whose closure is not contained in A. It follows that in a compact T2‐space of countable tightness, every nonisolated point is biaccessible.

Nearly metacompact spaces

Abstract A space X is called nearly metacompact (meta-Lindelof) provided that if U is an open cover of X then there is a dense set D⫅X and an open refinement V of U that is point-finite (point-countable) on D. We show that countably compact, nearly meta-Lindelof T 3 -spaces are compact. That nearly metacompact radial spaces are meta-Lindelof. Every space can be embedded as a closed subspace of a nearly metacompact space. We give an example of a countably compact, nearly meta-Lindelof T 2 -space that is not compact and a nearly metacompact T 2 -space which is not irreducible.