---
tag: BJVRFFJTV6
kind: concept
name: Vietoris topology
msc: [54B20]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Hypertopology
---

## Summary

The Vietoris topology is a topology on a set of subsets of a topological space, usually the closed subsets or the compact subsets.
A set is near a set $A$ when it lies inside a given open neighbourhood of $A$.
It must also meet some given open sets that $A$ meets.
It turns a family of subsets into a space, called a hyperspace.
On the nonempty compact subsets of a metric space, it is the topology of the Hausdorff metric.

## Explanations

### L1 human

Sometimes the objects you want to move around are sets, not points.
The Vietoris topology says when two sets are close.
A set $B$ is close to $A$ when $B$ stays inside a thin open region around $A$, and $B$ reaches every part of $A$ you chose to watch.
For example, in the plane, a fine grid of points inside a disc is close to the disc in this sense.