---
tag: 70M1VAPFN0
kind: concept
name: Convergence of filters
msc: [54A20]
sources:
  - url: https://en.wikipedia.org/wiki/Filters_in_topology
---

## Summary

A [[filter|15PF73BKY6]] on a [[topological space|0CQHFPGJ45]] converges to a point $x$ when it contains every [[neighbourhood|0ZCY0Z80B6]] of $x$.
This notion extends the [[convergence of sequences|0W4CN9E2W5]] to all spaces.
Closures, [[continuity|0VGFHTM75Y]], [[compactness|0TVJZYB6SH]] and the [[Hausdorff property|0S1FJ8VEX9]] can all be described by convergent filters.

## Explanations

### L1 human

A filter is a family of "big" sets, and it converges to $x$ when every set around $x$, however small, counts as big.
For a sequence $x_1, x_2, \dots$, take as big sets those that contain all terms from some index on.
This filter converges to $x$ exactly when the sequence converges to $x$.
In spaces where sequences do not see the whole topology, filters still do.