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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 $ \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 ^{\underbar{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^{\underbar{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

DOI: https://doi.org/10.1090/S0002-9947-1973-0354380-4
Keywords: Space of uniform, of subuniform ultrafilters, growth, property $ {\Phi _\alpha }$, $ {F_\alpha }$-spaces, pigeon-hole principle, diagonal argument, normality, $ {C^ \ast }$-embedding, ramifiable cardinal, Aronszajn-Specker theorem
Article copyright: © Copyright 1973 American Mathematical Society

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