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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**W. W. Comfort and S. Negrepontis,*Homeomorphs of three subspaces of*, Math. Z.**107**(1968), 53-58. MR**38**#2739. MR**0234422 (38:2739)****[2]**P. Erdös and A. Tarski,*On some problems involving inaccessible cardinals*, Essays on the Foundation of Mathematics, Magnes Press, Jerusalem, 1967, pp. 50-82. MR**0167422 (29:4695)****[3]**N. J. Fine and L. Gillman,*Extension of continuous functions in*, Bull. Amer. Math. Soc.**66**(1960), 376-381. MR**23**#A619. MR**0123291 (23:A619)****[4]**G. Kurepa,*Ensembles linéaires et une classe de tableaux ramifies*(*tableaux ramifies de M. Aronszajn*), Publ. Math. Univ. Belgrade**6**(1936), 129-160.**[5]**S. Negrepontis,*Extension of continuous functions in*, Nederl. Akad. Wetensch. Proc. Ser. A**71**= Indag. Math.**30**(1968), 393-400. MR**39**#2128. MR**0240783 (39:2128)****[6]**-,*The Stone space of the saturated Boolean algebras*, Trans. Amer. Math. Soc.**141**(1969), 515-527. MR**40**#1311. MR**0248057 (40:1311)****[7]**-,*The existence of certain uniform ultrafilters*, Ann. of Math. (2)**90**(1969), 23-32. MR**40**#46. MR**0246777 (40:46)****[8]**E. Specker,*Sur un problème de Sikorski*, Colloq. Math.**2**(1949), 9-12. MR**12**, 597. MR**0039779 (12:597b)****[9]**N. M. Warren,*Extending continuous functions in Stone-Čech compactifications of discrete spaces and in zero-dimensional spaces*, Doctoral Dissertation, University of Wisconsin, Madison, Wis., 1970.**[10]**-,*Properties of the Stone-Čech compactifications of discrete spaces*, Proc. Amer. Math. Soc.**33**(1972), 599-606. MR**0292035 (45:1123)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
04A20,
02K35,
54C45

Retrieve articles in all journals with MSC: 04A20, 02K35, 54C45

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