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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)

##### MSC:
 54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.) 54D40 Remainders in general topology
##### Keywords:
weak P-point; good ultrafilter; mediocre point
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##### References:
 [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
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