The growth of subuniform ultrafilters

Author:
S. Negrepontis

Journal:
Trans. Amer. Math. Soc. **175** (1973), 155-165

MSC:
Primary 04A20; Secondary 02K35, 54C45

DOI:
https://doi.org/10.1090/S0002-9947-1973-0354380-4

MathSciNet review:
0354380

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Abstract: Some of the results on the topology of spaces of uniform ultrafilters are applied to the space of subuniform ultrafilters (i.e., the set of ultrafilters which are -uniform but not -uniform) on when is a regular cardinal. The main object is to find for infinite cardinals , such that , a topological property that separates the space (*the growth of* ) from the space of uniform ultrafilters on . Property fulfils this rôle defined for a zero-dimensional space *X* by the following condition: every nonempty closed subset of *X* of type at most is not contained in the uniform closure of a family of pairwise disjoint nonempty open-and-closed subsets of *X*. The ``infinitary'' properties of , as they are measured by , are more closely related to those of than to those of . A consequence of this topological separation is that the growth of is not homeomorphic to and, in particular, that is not -embedded in the space of -uniform ultrafilters on . These results are related to, and imply easily, the Aronszajn-Specker theorem: *if* *then* *is not a ramifiable cardinal*. It seems possible that similar questions on the -embedding of certain spaces of ultrafilters depend on (and imply) results in partition calculus.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0354380-4

Keywords:
Space of uniform,
of subuniform ultrafilters,
growth,
property ,
-spaces,
pigeon-hole principle,
diagonal argument,
normality,
-embedding,
ramifiable cardinal,
Aronszajn-Specker theorem

Article copyright:
© Copyright 1973
American Mathematical Society