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• For every sequence $(A_n)_{n\in\mathbb N}$ of sets from $\mathcal I$ there is $A\in\mathcal I$ such that $A_n\subseteq^* A$ for all $n$'s.
• Any sequence $(F_n)_{n\in\mathbb N}$ of sets from $\mathcal F(\mathcal I)$ has a pseudointersection in $\mathcal F(\mathcal I)$. (Pseudointersection of $(F_n)_{n\in\mathbb N}$ is a set $F$ such that $F\subseteq^* F_n$ for each $n\in\mathbb N$.)
• For every sequence $(A_n)_{n\in\mathbb N}$ of sets belonging to $\mathcal I$ there exists a sequence $(B_n)_{n\in\mathbb N}$ of sets from $\mathcal I$ such that $A_j =^* B_j$ for $j\in\mathbb N$ and $B=\bigcup_{j\in\mathbb N} B_j\in\mathcal I$.
• For every sequence of mutually disjoint sets $(A_n)_{n\in\mathbb N}$ belonging to $\mathcal I$ there exists a sequence $(B_n)_{n\in\mathbb N}$ of sets belonging to $\mathcal I$ such that $A_j =^* B_j$ for $j\in\mathbb N$ and $B=\bigcup_{j\in\mathbb N} B_j\in\mathcal I$.
• For every non-decreasing sequence $A_1\subseteq A_2 \subseteq \dots \subseteq A_n \subseteq \dots$ of sets from $\mathcal I$ there exists a sequence $(B_n)_{n\in\mathbb N}$ of sets belonging to $\mathcal I$ such that $A_j =^* B_j$ for $j\in\mathbb N$ and $B=\bigcup_{j\in\mathbb N} B_j\in\mathcal I$.
• In the Boolean algebra $\mathcal P(S)/\mathrm{Fin}$ the ideal $\mathcal I$ corresponds to a $\sigma$-directed subset, that is, every countable subset has an upper bound.

9.24 Remark.
There is a general topological concept of P-point (see for example [95, 50]), namely a point (in a topological space) such that every countable intersection of open neighborhoods of it includes another open neighborhood of it. When applied to the topological space βω − ω, the Stone-Čech remainder of the discrete space ω, whose points are naturally identified with (non-trivial) ultrafilters on ω, this topological notion becomes the concept defined above.
The “P” in “P-point” refers to prime ideals (in rings of functions); see [50, Exercises 4J and 4L].
The “Q” in “Q-point” was chosen because it is next to “P” in the alphabet. Q-points are also called rare ultrafilters.
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[50] Leonard Gillman and Meyer Jerison. Rings of Continuous Functions. Van Nostrand, Princeton, 1960.
[95] Walter Rudin. Homogeneity problems in the theory of Čech compactifications. Duke Mathematical Journal, 23:409–419, 1956. (projecteuclid, DOI: 10.1007/978-3-0348-7524-0_7)

Dalsia ekvivalentna definicia je v spominanom clanku Grekos. Misik, Ziman. Hovori, ze pre kazdy I-rozklad N existuje mnozina z dualneho filtra, ktora ma konecny prienik s kazdou mnozinou v rozklade.

A free ultrafilter $\mathscr U$ is a P-point if for each partition $\{u_n\subseteq\omega; n\in\omega\}$ of $\omega$ either $u_n \in \mathscr U$ for a (unique) $n\in\omega$, or there exists an $x \in \mathscr U$ such that for each $n\in\omega$, $x\cap u_n$ is finite.

Furthermore, a free ultrafilter $\mathscr U$ is a Q-point if for each partition of $\omega$ info finite pieces $\{I_n\subseteq\omega; n\in\omega\}$ (i.e., for each $n\in\omega$, $I_n$ is finite), there exists an $x\in\mathscr{U}$ such that for each $n\in\omega$, $x\cap I_n$ has at most one element.