Ultrafilter

Definition

A ultrafilter on a $E$ unit is a filter such as it does not exist any filter on $E$ strictly finer than him. That means that ultrafilter is a maximum element of the whole of the ordered filters by inclusion.

One can as show as a filter on $E$ is an ultrafilter if and only so for any subset $A$ of $E$, $A$ or its Complémentaire belongs to the ultrafilter. The passage from one definition to another, although easy to show, is fundamental.

An ultrafilter is a filter equal to its Grill.

Example of ultrafilter

The whole of the parts of a $E$ unit which contain an element chosen in this unit is an ultrafilter (such an ultrafilter is known as principal or centered).

Either $E\; =\; \backslash \; \{has,\; B,\; C\; \backslash \}$. $F\; =\; \backslash \; \{\backslash \; \{has\; \backslash \},\; \backslash \; \{has,\; B\; \backslash \},\; \backslash \; \{has,\; C\; \backslash \},\; \backslash \; \{has,\; B,\; C\; \backslash \}\; \backslash \}$ is a principal ultrafilter.

Applications

The use of ultrafilters makes it possible to relatively easily show the Théorème of Tychonov.

The existence of nonprincipal ultrafilters on ℕ makes it possible to build a model of the Analyze nonstandard (see Introduction to the nonstandard analysis via the ultrafilters on ℕ).

The existence of nonprincipal ultrafilters is a consequence of the Axiome of the choice (and more precisely of the Lemme of Zorn).

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