## Journal of Symbolic Logic

### Ultrafilters Generated by a Closed Set of Functions

Greg Bishop

#### Abstract

Let $\kappa$ and $\lambda$ be infinite cardinals, $\mathscr{F}$ a filter on $\kappa$, and $\mathscr{G}$ a set of functions from $\kappa$ to $\kappa$. The filter $\mathscr{F}$ is generated by $\mathscr{G}$ if $\mathscr{F}$ consists of those subsets of $\kappa$ which contain the range of some element of $\mathscr{G}$. The set $\mathscr{G}$ is $<\lambda$-closed if it is closed in the $<\lambda$-topology on $^\kappa\kappa$. (In general, the $<\lambda$-topology on $^IA$ has basic open sets all $\Pi_{i\in I} U_i$ such that, for all $i \in I, U_i \subseteq A$ and $|\{i \in I : U_i \neq A\}| < \lambda$.) The primary question considered in this paper asks "Is there a uniform ultrafilter on $\kappa$ which is generated by a closed set of functions?" (Closed means $<\omega$-closed.) We also establish the independence of two related questions. One is due to Carlson: "Does there exist a regular cardinal $\kappa$ and a subtree $T$ of $^{<\kappa}\kappa$ such that the set of branches of $T$ generates a uniform ultrafilter on $\kappa$?"; and the other is due to Pouzet: "For all regular cardinals $\kappa$, is it true that no uniform ultrafilter on $\kappa$ is analytic?" We show that if $\kappa$ is a singular, strong limit cardinal, then there is a uniform ultrafilter on $\kappa$ 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 $\aleph_1$ which is generated by a closed set of increasing functions". In contrast with the above results, we show that if $\kappa$ is any cardinal, $\lambda$ is a regular cardinal less than or equal to $\kappa$ and $\mathbb{P}$ is the forcing notion for adding at least $(\kappa^{<\lambda})^+$ generic subsets of $\lambda$, then in $V^\mathbb{P}$, no uniform ultrafilter on $\kappa$ is generated by a closed set of functions.

#### Article information

Source
J. Symbolic Logic, Volume 60, Issue 2 (1995), 415-430.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744744

Mathematical Reviews number (MathSciNet)
MR1335125

Zentralblatt MATH identifier
0830.04001

JSTOR
links.jstor.org

#### Citation

Bishop, Greg. Ultrafilters Generated by a Closed Set of Functions. J. Symbolic Logic 60 (1995), no. 2, 415--430. https://projecteuclid.org/euclid.jsl/1183744744