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 \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\).

在\(\omega \)上的 P 点超滤器称为区间 P 点，如果对于从\(\omega \)到\(\omega \)的每个函数，在这个超滤器中存在一个集合A，使得限制A的函数要么是一个常数函数，要么是一个区间对一的函数。在本文中，我们证明了以下结果。(1) 区间 P 点在\(\textsf{CH}\)或\(\textsf{MA}\)下不是同构不变的。(2) 我们确定一个基数不变量\(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\)使得每个过滤器基的大小小于连续统可以扩展到区间 P 点当且仅当\(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\)。(3) 在假设\({\mathfrak {d}}={\ mathfrak {c}}\)或\(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\)。