## The growth of subuniform ultrafilters

HTML articles powered by AMS MathViewer

- 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.

## References

- W. W. Comfort and S. Negrepontis,
*Homeomorphs of three subspaces of $\beta {\bf {\rm }N}\backslash {\bf {\rm }N}$*, Math. Z.**107**(1968), 53–58. MR**234422**, DOI 10.1007/BF01111048 - Paul Erdős and Alfred Tarski,
*On some problems involving inaccessible cardinals*, Essays on the foundations of mathematics, Magnes Press, Hebrew Univ., Jerusalem, 1961, pp. 50–82. MR**0167422** - N. J. Fine and L. Gillman,
*Extension of continuous functions in $\beta N$*, Bull. Amer. Math. Soc.**66**(1960), 376–381. MR**123291**, DOI 10.1090/S0002-9904-1960-10460-0
G. Kurepa, - S. Negrepontis,
*Extension of continuous functions in $\beta D$*, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math.**30**(1968), 393–400. MR**0240783** - S. Negrepontis,
*The Stone space of the saturated Boolean algebras*, Trans. Amer. Math. Soc.**141**(1969), 515–527. MR**248057**, DOI 10.1090/S0002-9947-1969-0248057-2 - S. Negrepontis,
*The existence of certain uniform ultrafilters*, Ann. of Math. (2)**90**(1969), 23–32. MR**246777**, DOI 10.2307/1970679 - E. Specker,
*Sur un problème de Sikorski*, Colloq. Math.**2**(1949), 9–12 (French). MR**39779**, DOI 10.4064/cm-2-1-9-12
N. M. Warren, - Nancy M. Warren,
*Properties of Stone-Čech compactifications of discrete spaces*, Proc. Amer. Math. Soc.**33**(1972), 599–606. MR**292035**, DOI 10.1090/S0002-9939-1972-0292035-X

*Ensembles linéaires et une classe de tableaux ramifies*(

*tableaux ramifies de M. Aronszajn*), Publ. Math. Univ. Belgrade

**6**(1936), 129-160.

*Extending continuous functions in Stone-Čech compactifications of discrete spaces and in zero-dimensional spaces*, Doctoral Dissertation, University of Wisconsin, Madison, Wis., 1970.

## 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