---
tag: GT56RTG9X5
kind: concept
name: Gromov–Hausdorff convergence
msc: [53C23, "54E35"]
sources:
  - url: https://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence
same_as:
  wikidata: Q929777
  wikipedia: Gromov–Hausdorff convergence
---

## Summary

Gromov–Hausdorff convergence is a notion of convergence for sequences of metric spaces, not of points inside one space.
A sequence of compact metric spaces converges when its distance to the limit in the [[Gromov–Hausdorff space|ETRF2FHAEE]] tends to zero.
It lets one take limits of Riemannian manifolds, and such limits are often singular spaces.
It is a basic tool of metric and synthetic methods in global geometry.

## Explanations

### L1 human

Think of shapes seen from far away, where only distances matter.
Two shapes are close when they can be laid in a common space so that each lies near the other.
A sequence of shapes converges when this closeness tends to zero.
For example, the points $0, \tfrac{1}{n}, \tfrac{2}{n}, \ldots, 1$ form finite spaces that converge to the segment $[0, 1]$.
A thin tube around a circle also converges to the circle when the tube gets thinner.