Definition of Ultrafilter. Meaning of Ultrafilter. Synonyms of Ultrafilter

- In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle...
- In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words...
- be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added...
- ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical...
- prin****l ultrafilter on X {\displaystyle X} . Moreover, every prin****l ultrafilter on X {\displaystyle X} is necessarily of this form. The ultrafilter lemma...
- property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection F of subsets of X such that:...
- ultrafilter is called the ultrafilter lemma and cannot be proven in Zermelo–Fraenkel set theory (ZF), if ZF is consistent. Within ZF, the ultrafilter...
- element i ∈ I {\displaystyle i\in I} (all of the same signature), and an ultrafilter U {\displaystyle {\mathcal {U}}} on I . {\displaystyle I.} For any two...
- to the following criterion: ****uming the ultrafilter lemma, a space is compact if and only if each ultrafilter on the space converges. With this in hand...
- former is equivalent in ZF to Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter. One of the most interesting aspects of the...