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.
