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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 |

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\textit{J. Baker} and \textit{K. Kunen}, Trans. Am. Math. Soc. 353, No. 10, 4083--4093 (2001; Zbl 0972.54019)

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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 · doi:10.2307/1970549 |

[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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