---
tag: 8NS0116H9P
kind: definition
name: G-delta set
msc: [54H05, "54E52"]
defines: V1TBNP8R3S
uses: [0CQHFPGJ45, 02BH3MCBQB, 01TQB88H4C]
sources:
  - url: https://en.wikipedia.org/wiki/Gδ_set
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $A \subseteq X$ is a *$G_\delta$ set* if there are [[open sets|02BH3MCBQB]] $U_1, U_2, U_3, \dots$ of $X$ with
$$A = \bigcap_{n=1}^{\infty} U_n.$$

## Remarks

Every open set is $G_\delta$: take $U_n = A$ for all $n$.
In a [[metric space|0VKD1TP7F3]] with metric $d$, every [[closed set|0ZCK8DQYN1]] $F$ is $G_\delta$, since $F = \bigcap_n \set{x : d(x, F) < 1/n}$.

Example: the set of irrational numbers is $\bigcap_{q \in \Q} (\R \setminus \set{q})$.
Each $\R \setminus \set{q}$ is open and $\Q$ is [[countable|01TQB88H4C]], so the irrationals form a $G_\delta$ set.

Non-example: $\Q$ is not a $G_\delta$ subset of $\R$.
If $\Q = \bigcap_n U_n$ with $U_n$ open, each $U_n$ is open and dense, and so are the sets $\R \setminus \set{q}$ for $q \in \Q$.
These countably many dense open sets have empty intersection, which contradicts the [[Baire category theorem|0A7KGAYHGV]].

A set is $G_\delta$ if and only if its complement is an [[F-sigma set|8MGC7M64GW]].