Ultrafilter on the grid - This is the maximum own filter . The concept of ultrafilter appeared in the general topology , where it is used to generalize the concept of convergence on spaces with uncountable base.
Definition
Own filter on the grill is an ultrafilter if it is not contained in any of its own (that is, different from ) filter.
Set subsets of the set called ultrafilter on , if a
- for any two items their intersection also lies in
- for any item , all its supersets lie in
- for any subset or either
In other words, if we consider a function on sets given as , if a and otherwise then is a finitely additive probability measure on .
Ultrafilters in Boolean Algebras
If the lattice is Boolean algebra , then the following characterization of ultrafilters is possible: filter is an ultrafilter if and only if for any element or either
This characterization makes ultrafilters look like complete theories .
Examples
- any main filter is an ultrafilter
- a subset of the Lindenbaum – Tarski algebra of the complete theory consisting of theorems
Properties
- an ultrafilter on a finite set is always prime .
- Any ultrafilter on an infinite set contains a finite filter .
- if a - the main ultrafilter on the set then its main element is the intersection of all elements of the ultrafilter.
- if a - non-main ultrafilter on set , then the intersection of all its elements is empty.
- Each filter is contained in the ultrafilter.
- This statement cannot be proved without using the axiom of choice .
- Also this statement is equivalent to the theorem on Boolean prime ideals .
- An important consequence of this theorem is the existence of non-principal ultrafilters on infinite sets.
- Stone - Cech compactification of discrete space - this is a set of ultrafilters on a lattice of subsets endowed with Stone's topology . As the base of the open sets of the Stone topology on the set of ultrafilters can take sets for all kinds of
Applications
- Ultrafilters are used in a number of constructions of the theory of models , namely, to formulate the concept of ultra-production .
- Ultrafilters also appear in the formulation of Stone's theorem on the representation of Boolean algebras and in the explicit construction of the Stone – Cech compactification .
- Ultra-limit for metric spaces - generalization of Gromov – Hausdorf convergence