Spaces homeomorphic to (2^{𝑎})ₐ. II

Abstract

Topological characterizations and properties of the spaces (2a)a, where a is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Cech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces (2a)a in terms of a simple set of conditions, and (b) the a-Baire category property of (2a)a and the stability of the class of spaces homeomorphic to (2a)a (or to (aa)a) when taking intersections of at most a. open and dense sub* sets of (2a)a. Among the applications of these results are the following. Assuming a+ = 2a, the class of spaces homeomorphic to (2(a))a + includes the space of uniform ultrafilters on <x with the Pa+-topology (f(a))a+, its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming at = 2a (or in some cases only Martin's axiom), the class of spaces homeomorphic to (2(<a includes the following: The space (ßX\X) + where X is anoncompact locally compact realcompact space such that |C(A)| s 2*" and its subspaces of /^+-points of ßX\X and/or (if X is in addition a metric space without isolated elements) the remote points.

Special subsets of the generalized Cantor space $2^κ$ and generalized Baire space $κ^κ$

“…is homeomorphic to 2 κ (see [40,Theorem 2.3] and [41]). On the other hand, if κ is weakly compact, then a completely regular κ-additive space X without isolated points is homeomorphic to 2 κ if and only if there exists a family of open sets B in X satisfying conditions (1)-( 3) and also:…”