# Category:Axioms/Filter Theory

This category contains axioms related to Filter Theory.

Let $\struct {S, \preccurlyeq}$ be an ordered set.

A subset $\FF \subseteq S$ is called a filter of $\struct {S, \preccurlyeq}$ (or a filter on $\struct {S, \preccurlyeq}$) if and only if $\FF$ satisfies the filter axioms:

 $(1)$ $:$ $\ds \FF \ne \O$ $(2)$ $:$ $\ds x, y \in \FF \implies \exists z \in \FF: z \preccurlyeq x, z \preccurlyeq y$ $(3)$ $:$ $\ds \forall x \in \FF: \forall y \in S: x \preccurlyeq y \implies y \in \FF$

## Subcategories

This category has only the following subcategory.

## Pages in category "Axioms/Filter Theory"

The following 2 pages are in this category, out of 2 total.