---
tag: GRH8MCVN3N
kind: concept
name: Residual set
msc: ["54E52"]
sources:
  - url: https://en.wikipedia.org/wiki/Meagre_set
---

## Summary

A residual set, or comeagre set, in a [[topological space|0CQHFPGJ45]] is a set whose complement is [[meagre|0XGCZ4GDZ6]].
Equivalently, it contains a countable intersection of [[dense|0BSJSX7ME2]] open sets.
In a [[Baire space|0ZH5AJ2ED7]], residual sets are dense, so they are considered "large" sets.
A property is called generic when it holds on a residual set.

## Explanations

### L1 human

A meagre set is a small set, built from countably many thin pieces.
A residual set is what remains after removing such a small set.
For example, the irrational numbers form a residual subset of the real line.
The [[Baire category theorem|0A7KGAYHGV]] says that in a nonempty complete metric space, residual sets are never empty and are in fact dense.