- 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...
- prin****l
ultrafilter on X {\displaystyle X} . Moreover,
every prin****l
ultrafilter on X {\displaystyle X} is
necessarily of this form. The
ultrafilter lemma...
- 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...
-
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:...
- ideals. A
variation of this
statement for
filters on sets is
known as the
ultrafilter lemma.
Other theorems are
obtained by
considering different mathematical...
-
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...
-
characterizations of "
ultrafilter" and "ultra prefilter,"
which are
listed in the
article on
ultrafilters.
Important properties of
ultrafilters are also described...
- set of all
ultrafilters on X, with the
elements of X
corresponding to the prin****l
ultrafilters. The
topology on the set of
ultrafilters,
known as the...
- to a
point or a set, and many others.
Special types of
filters called ultrafilters have many
useful technical properties and they may
often be used in place...
