zbMATH — the first resource for mathematics

Limits in the uniform ultrafilters. (English) Zbl 0972.54019
This paper answers a long-standing and important question about the existence of ultrafilters with special topological properties in the space of ultrafilters (over a discrete space). The main result is to prove that for each regular cardinal \(\kappa\), there is a uniform ultrafilter x on \(\kappa\) which is not a limit in \(\beta \kappa\) of any set of fewer than \(\kappa^+\)-many uniform ultrafilters on \(\kappa\), a so-called weak \(P_{\kappa^{++}}\)-point in the subspace \(u(\kappa)\) of uniform ultrafilters on \(\kappa\). The paper has a very informative and interesting introduction which reviews the history of the problem and the relationship to \(\kappa^+\)-good ultrafilters. The key idea of the proof has its origins in an earlier proof that a \(\kappa^{++}\)-good ultrafilter is a weak \(P_{\kappa^{++}}\)-point but it was known to be consistent that there were no \(\kappa^{++}\)-good ultrafilters on \(\kappa\). The first author’s thesis apparently includes the very interesting result that it the existence of \(\kappa^{++}\)-good ultrafilters just follows from some cardinal arithmetic. The authors isolate the key step in the above mentioned proof and introduce a new combinatorial, or basis, property of an ultrafilter, a mediocre point. Finally, a remarkable step is to greatly generalize the notion of independent family and to produce such an independent matrix. The actual construction of a mediocre point from the matrix is similar to earlier constructions (of the second author).
Reviewer: A.Dow (North York)

54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
54D40 Remainders in general topology
Full Text: DOI
[1] J. Baker, Ph.D. Thesis, University of Wisconsin, 2001, to appear.
[2] Alan Dow, Good and OK ultrafilters, Trans. Amer. Math. Soc. 290 (1985), no. 1, 145 – 160. · Zbl 0532.54021
[3] R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275 – 285. · Zbl 0137.41904
[4] H. Jerome Keisler, Good ideals in fields of sets, Ann. of Math. (2) 79 (1964), 338 – 359. · Zbl 0137.00803
[5] Kenneth Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299 – 306. · Zbl 0263.02033
[6] K. Kunen, Weak \?-points in \?*, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 741 – 749.
[7] W. Rudin, Homogeneity problems in the theory of Cech compactifications. Duke Math. J. 23 (1956) 409-419. · Zbl 0073.39602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.