The growth of subuniform ultrafilters
Author:
S. Negrepontis
Journal:
Trans. Amer. Math. Soc. 175 (1973), 155165
MSC:
Primary 04A20; Secondary 02K35, 54C45
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 zerodimensional 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 openandclosed 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 AronszajnSpecker 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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 P. Erdös and A. Tarski, On some problems involving inaccessible cardinals, Essays on the Foundation of Mathematics, Magnes Press, Jerusalem, 1967, pp. 5082. MR 0167422 (29:4695)
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 , Properties of the StoneČech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599606. MR 0292035 (45:1123)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303543804
PII:
S 00029947(1973)03543804
Keywords:
Space of uniform,
of subuniform ultrafilters,
growth,
property ,
spaces,
pigeonhole principle,
diagonal argument,
normality,
embedding,
ramifiable cardinal,
AronszajnSpecker theorem
Article copyright:
© Copyright 1973
American Mathematical Society
