---
tag: 09YMV8H4X5
kind: statement
role: proposition
name: Intersection of closed sets is closed
msc: ["54E35"]
uses: [0HEF46RPEK, 0G5AV9Q4RR, 0839XNQXDK, 0VK0W4D5Y5]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_set
    locator: More about closed sets
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(F_i)_{i \in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$, indexed by a nonempty set $I$.
Then the [[intersection|0839XNQXDK]] $\bigcap_{i \in I} F_i$ is closed in $X$.

## Remarks

Proof: by [[De Morgan's laws for sets|0VK0W4D5Y5]], $X \setminus \bigcap_i F_i = \bigcup_i (X \setminus F_i)$. This is a [[union of open sets|0HEF46RPEK]], so it is open.
The same proof works in each [[topological space|0CNQXVBHCX]].
A union of finitely many closed sets is closed, but an infinite union need not be. For example, the union of the sets $[1/n, 1]$ for $n \ge 1$ is $(0, 1]$.

## Explanations

### L1 human

Closed sets behave like open sets with the roles of union and intersection exchanged.
Any intersection of closed sets is closed, and finite unions of closed sets are closed.