Vol. 95, No. 2, 1981

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ISSN: 0030-8730
A class of Wallman-type extensions

Ellen Elizabeth Reed

Vol. 95 (1981), No. 2, 443–459

This paper grew out of an attempt to determine what T2-compactifications, the Wallman compactification, and the one-point compactification have in common. It turns out that in each case the associated nearness is generated by those grills which contain ultraclosed filters and are clans with respect to the associated proximity. Such a nearness will be called a Wallman nearness, and this paper is a study of the properties of Wallman nearnesses and their extensions.

A mild “covering” condition on a proximity guarantees that it contains a Wallman nearness. Each covered proximity contains exactly one Wallman nearness. This sets up a 1–1 correspondence between covered proximities and extensions obtained from Wallman nearnesses. The latter will be called Wallman-type extensions. These can be characterized by the fact that they are covered extensions and satisfy a certain completeness property; namely, the duals of certain clans must converge. This summarizes the first two sections.

The last section is a study of compact Wallman-type extensions. A condition on the proximity is obtained which guarantees that the associated Wallman-type extension is compact. The condition states that certain very large grills containing ultraclosed filters must be clans with respect to the given proximity. Such a proximity will be called a compactification proximity. It turns out that compactification proximities give rise to weakly regular compactifications. The paper ends with a study of the relation between weak regularity and Wallman-type extensions.

Mathematical Subject Classification 2000
Primary: 54D35
Secondary: 54E17
Received: 21 December 1979
Revised: 14 March 1980
Published: 1 August 1981
Ellen Elizabeth Reed