---
tag: V1TBNP8R3S
kind: concept
name: G-delta set
msc: [54H05, "54E52"]
sources:
  - url: https://en.wikipedia.org/wiki/Gδ_set
same_as:
  wikidata: Q127244
  wikipedia: Gδ set
---

## Summary

A $G_\delta$ set in a [[topological space|0CQHFPGJ45]] is an intersection of countably many [[open sets|02BH3MCBQB]].
The name comes from the German words for open set and intersection.
In a [[metric space|0VKD1TP7F3]] every closed set is $G_\delta$, and $G_\delta$ sets are the next step after open and closed sets in the Borel hierarchy.
The complements of $G_\delta$ sets are the [[F-sigma sets|8MGC7M64GW]].

## Explanations

### L1 human

One open set may be large, but intersecting countably many open sets can carve out a much thinner set.
The irrational numbers form a $G_\delta$ set: for each rational $q$, remove the single point $q$ from $\R$, which leaves an open set, and intersect all these sets.
On the other hand, the rationals are not $G_\delta$ in $\R$, which follows from the [[Baire category theorem|0A7KGAYHGV]].