Abstract

A P-point ultrafilter over $\omega$ is called an interval P-point if for every function from $\omega$ to $\omega$ there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under $\textsf{CH}$ or $\textsf{MA}$. (2) We identify a cardinal invariant $\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})$ such that every filter base of size less than continuum can be extended to an interval P-point if and only if $\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}$. (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption ${\mathfrak {d}}={\mathfrak {c}}$ or $\textbf{cov}({\mathcal {B}})={\mathfrak {c}}$.