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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The growth of subuniform ultrafilters
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by S. Negrepontis PDF
Trans. Amer. Math. Soc. 175 (1973), 155-165 Request permission

Abstract:

Some of the results on the topology of spaces of uniform ultrafilters are applied to the space $\Omega ({\alpha ^ + })$ of subuniform ultrafilters (i.e., the set of ultrafilters which are $\alpha$-uniform but not ${\alpha ^ + }$-uniform) on ${\alpha ^ + }$ when $\alpha$ is a regular cardinal. The main object is to find for infinite cardinals $\alpha$, such that $\alpha = {\alpha ^{\underline {a}}}$, a topological property that separates the space $\beta (\Omega ({\alpha ^ + }))\backslash \Omega ({\alpha ^ + })$ (the growth of $\Omega ({\alpha ^ + })$) from the space $U({\alpha ^ + })$ of uniform ultrafilters on ${\alpha ^ + }$. Property ${\Phi _\alpha }$ fulfils this rôle defined for a zero-dimensional space X by the following condition: every nonempty closed subset of X of type at most $\alpha$ is not contained in the uniform closure of a family of $\alpha$ pairwise disjoint nonempty open-and-closed subsets of X. The “infinitary” properties of $\Omega ({\alpha ^ + })$, as they are measured by ${\Phi _\alpha }$, are more closely related to those of $U(\alpha )$ than to those of $U({\alpha ^ + })$. A consequence of this topological separation is that the growth of $\Omega ({\alpha ^ + })$ is not homeomorphic to $U({\alpha ^ + })$ and, in particular, that $\Omega ({\alpha ^ + })$ is not ${C^ \ast }$-embedded in the space $\Sigma ({\alpha ^ + })$ of $\alpha$-uniform ultrafilters on ${\alpha ^ + }$. These results are related to, and imply easily, the Aronszajn-Specker theorem: if $\alpha = \alpha ^{\underline {a}}$ then ${\alpha ^ + }$ is not a ramifiable cardinal. It seems possible that similar questions on the ${C^ \ast }$-embedding of certain spaces of ultrafilters depend on (and imply) results in partition calculus.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • 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