Skip to main content Accessibility help
×
Home

Ultrafilters generated by a closed set of functions

  • Greg Bishop (a1)

Abstract

Let κ and λ be infinite cardinals, a filter on κ and a set of functions from κ to κ. The filter is generated by if consists of those subsets of κ which contain the range of some element of . The set is -closed if it is closed in the <λ-topology on κκ. (In general, the -topology on IA has basic open sets all such that, for all iI, UiA and ∣{i ∈ I: UiA} ∣.) The primary question considered in this paper asks “Is there a uniform ultrafilter on κ which is generated by a closed set of functions?” (Closed means -closed.) We also establish the independence of two related questions. One is due to Carlson: “Does there exist a regular cardinal κ and a subtree T of κ such that the set of branches of T generates a uniform ultrafilter on κ?”; and the other is due to Pouzet: “For all regular cardinals κ, is it true that no uniform ultrafilter on κ is it true that no uniform ultrafilter on κ analytic?”

We show that if κ is a singular, strong limit cardinal, then there is a uniform ultrafilter on κ which is generated by a closed set of increasing functions. Also, from the consistency of an almost huge cardinal, we get the consistency of CH + “There is a uniform ultrafilter on 1 which is generated by a closed set of increasing functions”. In contrast with the above results, we show that if Κ is any cardinal, λ is a regular cardinal less than or equal to κ and ℙ is the forcing notion for adding at least (κ)+ generic subsets of λ, then in VP, no uniform ultrafilter on κ is generated by a closed set of functions.

Copyright

References

Hide All
[1] Balcar, B., Simon, P., and Vojtáš, P., Refinement properties and extensions of filters in Boolean algebras, Transactions of the American Mathematical Society, vol. 267 (1981), pp. 265283.
[2] Comfort, W. W. and Negrepontis, S., The theory of ultrafilters, Springer-Verlag, Berlin, 1974.
[3] Engelking, R. and Karlowicz, M., Some theorems of set theory and their topological consequences, Fundamenta Mathematicae, vol. 57 (1965), pp. 275285.
[4] Erdös, P., Hajnal, A., and Rado, R., Partition relations for cardinal numbers, Acta Mathematica Academiae Scientiarum Hungarica, vol. 16 (1965), pp. 93196.
[5] Jech, T., Set theory, Academic Press, New York, 1978.
[6] Kunen, K., Set theory: an introducton to independence proofs, North-Holland, Amsterdam, 1980.
[7] Laver, R., Saturated ideals and nonregular ultrafilters, Patras logic symposion, North-Holland, Amsterdam, 1982, pp. 297305.
[8] Pouzet, M., Relations impartibles, Dissertationes Mathematicae, vol. 193 (1981).
[9] Todorčević, S., Partitioning pairs of countable ordinals, Acta Mathematka, vol. 159 (1987), pp. 261294.

Ultrafilters generated by a closed set of functions

  • Greg Bishop (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed