Ultrafilter

Filter vs ultrafilter 210div

Ultrafilter is a concept in the field of mathematics, specifically within topology and order theory. An ultrafilter on a set X is a type of filter that is maximal among all filters on X with respect to inclusion. This means that an ultrafilter on X is a collection of subsets of X that satisfies certain properties, and there is no other filter on X that contains it as a proper subset.

Definition

Formally, a non-empty collection F of subsets of a set X is called an ultrafilter if it satisfies the following conditions:

1. If A and B are elements of F, then so is their intersection A ∩ B.
2. If A is an element of F, and A ⊆ B ⊆ X, then B is also an element of F.
3. For every subset A of X, either A or its complement X\A is in F, but not both.
4. The empty set is not an element of F.

The third condition distinguishes ultrafilters from other filters. It implies that ultrafilters are "decisive" in the sense that for any subset of X, they either contain that subset or its complement, but never both.

Types of Ultrafilters

Ultrafilters can be classified into two main types: principal and non-principal.

Principal Ultrafilters

A principal ultrafilter on a set X is generated by a single element x of X. It consists of all subsets of X that contain x. Principal ultrafilters are relatively easy to understand and construct.

Non-Principal Ultrafilters

Non-principal ultrafilters are more complex and cannot be generated by a single element of X. Their existence in any infinite set X relies on the Axiom of Choice, a fundamental principle in set theory that allows for the selection of elements from an infinite collection of sets.

Applications

Ultrafilters have various applications in mathematics:

• In topology, ultrafilters are used to construct the Stone–Čech compactification, a technique for constructing a compact space from a given topological space.
• In model theory, ultrafilters are employed in the construction of ultraproducts, which are used to prove important results such as Łoś's theorem.
• In set theory and order theory, ultrafilters are used to study the properties of partially ordered sets (posets) and to construct various types of lattices.

Properties

Ultrafilters have several interesting properties:

• Every filter on a set X is contained in some ultrafilter on X.
• The intersection of a family of ultrafilters on X, if non-empty, is an ultrafilter on X.
• Ultrafilters on a finite set X are always principal. The existence of non-principal ultrafilters requires the set to be infinite.

See Also

• Filter (mathematics)
• Compact space
• Axiom of Choice
• Stone–Čech compactification