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Topological characterizations and properties of the spaces      ( 2 α  ) α  {({2^\alpha })_\alpha }      , where      α  \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 α  ) α  {({2^\alpha })_\alpha }       in terms of a simple set of conditions, and (b) the      α  \alpha      -Baire category property of      ( 2 α  ) α  {({2^\alpha })_\alpha }       and the stability of the class of spaces homeomorphic to      ( 2 α  ) α  {({2^\alpha })_\alpha }       (or to      ( α  α  ) α  {({\alpha ^\alpha })_\alpha }      ) when taking intersections of at most      α  \alpha       open and dense subsets of      ( 2 α  ) α  {({2^\alpha })_\alpha }      . Among the applications of these results are the following. Assuming      α  + = 2 α  {\alpha ^ + } = {2^\alpha }      , the class of spaces homeomorphic to      ( 2 ( α  + ) ) α  + {({2^{({\alpha ^ + })}})_{{\alpha ^ + }}}       includes the space of uniform ultrafilters on      α  \alpha       with the      P α  + {P_{{\alpha ^ + }}}      -topology      ( U ( α  ) ) α  + {(U(\alpha ))_{{\alpha ^ + }}}      , its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming      ω  + = 2 ω  {\omega ^ + } = {2^\omega }       (or in some cases only Martin’s axiom), the class of spaces homeomorphic to      ( 2 ( ω  + ) ) ω  + {({2^{({\omega ^ + })}})_{{\omega ^ + }}}       includes the following: The space      ( β  X ∖  X ) ω  + {(\beta X\backslash X)_{{\omega ^ + }}}       where  X  is a noncompact locally compact realcompact space such that      | C ( X ) | ≤  2 ω  |C(X)| \leq {2^\omega }       and its subspaces of      P ω  + {P_{{\omega ^ + }}}      -points of      β  X ∖  X \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.

Spaces homeomorphic to $(2\sp{a})\sb{a}$. II