---
tag: 15PF73BKY6
kind: concept
name: Filter
msc: [54A20, "03E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Filter_(mathematics)
---

## Summary

A filter on a set $X$ is a nonempty family of subsets of $X$ that does not contain $\emptyset$.
It contains every superset of its members and the intersection of any two of its members.
One thinks of the members of a filter as the "large" sets, or as the sets that contain "almost all" points in some sense.
Filters give a theory of [[convergence|70M1VAPFN0]] that works in every [[topological space|0CQHFPGJ45]], where sequences are not enough.
Maximal filters are the [[ultrafilters|J64Y3JC0RE]].

## Explanations

### L1 human

A filter is a way to say which sets are "big enough".
For a point $x$ of a topological space, the sets that contain a [[neighbourhood|0ZCY0Z80B6]] of $x$ form a filter: they are the sets that contain everything close enough to $x$.
On the natural numbers, the sets whose complement is finite form a filter: they contain all numbers from some point on.
Two big sets always overlap in a big set, and a set larger than a big set is big too.