---
tag: 8MGC7M64GW
kind: concept
name: F-sigma set
msc: [54H05, "54E52"]
sources:
  - url: https://en.wikipedia.org/wiki/Fσ_set
same_as:
  wikidata: Q4392123
  wikipedia: Fσ set
---

## Summary

An $F_\sigma$ set in a [[topological space|0CQHFPGJ45]] is a union of countably many [[closed sets|0ZCK8DQYN1]].
The name comes from the French words for closed set and sum.
The complements of $F_\sigma$ sets are the [[G-delta sets|V1TBNP8R3S]], and in a [[metric space|0VKD1TP7F3]] every open set is $F_\sigma$.

## Explanations

### L1 human

A single point is closed, so any countable set is a countable union of closed sets.
For example, the set $\Q$ of rational numbers is $F_\sigma$ in $\R$.
An open interval is also $F_\sigma$, since $(0, 1)$ is the union of the closed intervals $[1/n, 1 - 1/n]$.
The irrational numbers, however, are not $F_\sigma$ in $\R$.