Spaces homeomorphic to $(2^{a})_{a}$. II

Abstract:

Topological characterizations and properties of the spaces ${({2^\alpha })_\alpha }$, where $\alpha$ 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-Čech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces ${({2^\alpha })_\alpha }$ in terms of a simple set of conditions, and (b) the $\alpha$-Baire category property of ${({2^\alpha })_\alpha }$ and the stability of the class of spaces homeomorphic to ${({2^\alpha })_\alpha }$ (or to ${({\alpha ^\alpha })_\alpha }$) when taking intersections of at most $\alpha$ open and dense subsets of ${({2^\alpha })_\alpha }$. Among the applications of these results are the following. Assuming ${\alpha ^ + } = {2^\alpha }$, the class of spaces homeomorphic to ${({2^{({\alpha ^ + })}})_{{\alpha ^ + }}}$ includes the space of uniform ultrafilters on $\alpha$ with the ${P_{{\alpha ^ + }}}$-topology ${(U(\alpha ))_{{\alpha ^ + }}}$, its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming ${\omega ^ + } = {2^\omega }$ (or in some cases only Martin’s axiom), the class of spaces homeomorphic to ${({2^{({\omega ^ + })}})_{{\omega ^ + }}}$ includes the following: The space ${(\beta X\backslash X)_{{\omega ^ + }}}$ where X is a noncompact locally compact realcompact space such that $|C(X)| \leq {2^\omega }$ and its subspaces of ${P_{{\omega ^ + }}}$-points of $\beta X\backslash X$ and/or (if X is in addition a metric space without isolated elements) the remote points. In particular the existence of good and/or Rudin-Keisler minimal ultrafilters and the existence of P-points and/or remote points follows always from a Baire category type of argument.