R. M. Stephenson

Initially κ-Compact and Related Spaces

Publisher Summary This chapter discusses initially κ-compact and related spaces. A main reason for studying initial κ and, more generally, interval compactness is that compactness, countable compactness, and the Lindelof property are special cases of one or both of these concepts. Another reason is that the theory of initially κ-compact and related spaces provides a means for answering fundamental questions that arise in other areas of topology. A third reason is that results in this area illustrate the usefulness of the close relationship that exists between the set theory of uncountable cardinals and properties of topological spaces. A space is countably compact if and only if it is initially ω-compact, and it is Lindelof if and only if it is finally ω1-compact. Like compactness, initial κ-compactness is preserved by continuous mappings, perfect pre-images, and closed subsets.

Products of initially

The main purpose of this paper is to give several theorems and exanples which we hope will be of use in the solution of the following problem. For an infinite cardinal number m, is initial rn-compactness prtserved by products? We also give some results concerning properties of Stone-Cech compactifications of discrete spaces.

m

A study is made of a family of spaces which contains the symmetrizable spaces as well as many of the well-known examples of perfectly

-compact spaces

Abstract Maximal feebly compact spaces (i.e., feebly compact spaces possessing no strictly stronger feebly compact topology) are characterized, as are special classes (countably compact, semiregular, regular) of maximal feebly compact spaces.

Symmetrizable and related spaces

Abstract Several embedding theorems are obtained, such as the following: Let Y be a first countable regular space in which the set B ={ y ϵ Y : y does not have a neighborhood base consisting of feebly compact open subsets of Y } is a countable nowhere dense set. Then Y has a regular(1)-closed (≡ first countable, regular, and feebly compact) extension space. A number of examples are given, including one of a separable Moore space Y such that the set B has a cardinality c , and no extension space of Y can be Moore-closed, regular(1)-closed, or Urysohn(1)-closed.

Maximal feebly compact spaces

Abstract A space X is said to be completely Hausdorff if C ( X ), the set of bounded continuous real valued functions defined on X , is point separating. A completely Hausdorff space X is called an SW space if every point separating subalgebra of C ( X ) which contains the constants is uniformly dense in C ( X ). The purpose of this paper is to show that there exists an SW space whose product with itself is not an SW space. Since a topological space is SW if and only if it is completely Hausdorff-closed, this result will solve Problem 4 of a recent survey of P-closed spaces by Berri et al. [2]. We will also answer a question of Scarborough and Stone [5] by showing that their property R( i ) is not productive.

Moore-closed and first countable feebly compact extension spaces

ABSTRACT: Conditions under which a given topological space does, or does not, possess a stronger maximal feebly compact topology are given. Procedures are developed which can be used, for certain feebly compact spaces, to obtain stronger maximal feebly compact spaces. Several examples are constructed.