Nearly realcompact spaces

Maximal Realcompact Spaces and Measurable Cardinals

Comfort and Hager investigate the notion of a maximal realcompact space and ask about the relationship to the first measurable cardinal m. A space is said to be a P (m) space if the intersection of fewer than m open sets is again open. They ask if each realcompact P (m) space is maximal realcompact. We establish that this question is undecidable.

Locally realcompact and HN-complete spaces

Two classes of spaces are studied, namely locally realcompact spaces and HNcomplete spaces, where the latter class is introduced in the paper. Both of these classes are superclasses of the class of realcompact spaces. Invariance with respect to subspaces and products of these spaces are investigated. It is shown that these two classes can be characterized by demanding that certain equivalences ...

Nearly Countable Dense Homogeneous Spaces

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X \ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of count...

Products of Nearly Compact Spaces

Uniformizable and Realcompact Bornological Universes

Bornological universes were introduced by Hu in [11] and obtained renewed interest in recent articles on convergence in hyperspaces and function spaces and optimization theory. In [11] and [12] Hu gives a necessary and sufficient condition for which a bornological universe is metrizable. In this article we will give a characterization of uniformizable bornological universes. Furthermore, a cons...