---
tag: 2HJGRABSPZ
kind: concept
name: Closure operator
msc: [06A15]
sources:
  - url: https://en.wikipedia.org/wiki/Closure_operator
same_as:
  wikidata: Q10564851
  wikipedia: Closure operator
---

## Summary

A closure operator on a [[partially ordered set|011C29Q1S1]] is a map $c$ that is extensive, monotone and idempotent.
It sends each element to the least "closed" element above it.
Examples on power sets are topological [[closure|0DRKT1QJR7]], the [[linear span|0HR573CHNM]] of vectors and the [[subgroup generated by|0RG9SPQSWM]] a set.
Closure operators on a power set correspond to [[Moore families|9ZVCJWYPB2]].

## Explanations

### L1 human

A closure operator completes an object to the smallest object of a special kind that contains it.
In a vector space, take a set of vectors and replace it by all their linear combinations.
Doing it again adds nothing new, a bigger set gives a bigger span, and the span always contains the original set.
These three features, growing, keeping order, and stopping after one step, define a closure operator.