---
tag: 80GQXWM7BV
kind: concept
name: Moore family
msc: [06A15]
sources:
  - url: https://en.wikipedia.org/wiki/Closure_operator
---

## Summary

A Moore family on a set $X$ is a collection of subsets of $X$ that contains $X$ and is closed under intersections of nonempty subfamilies.
Such a collection is also called a closure system.
Moore families on $X$ are exactly the families of closed sets of [[closure operators|17FVAN6HXX]] on the subsets of $X$.
Ordered by inclusion, a Moore family is a complete lattice.

## Explanations

### L1 human

Many kinds of substructures are stable under intersection.
The intersection of any number of subgroups of a group is a subgroup, and the whole group is a subgroup.
So the subgroups of a group form a Moore family.
The same holds for the subspaces of a vector space and the closed sets of a topological space.