---
tag: CRGZZKQ7Z9
kind: concept
name: Minimal set
msc: [37B05]
specializes: [WWZTW11664]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_dynamics
---

## Summary

A minimal set of a continuous map is a nonempty closed [[invariant set|WWZTW11664]] that contains no smaller nonempty closed invariant set.
Equivalently, the forward orbit of each of its points is dense in it.
Minimal sets are the smallest building blocks of [[topological dynamics|5X6E97EMFR]].
On a nonempty compact space every continuous map has at least one minimal set, by Zorn's lemma.

## Explanations

### L1 human

A minimal set is a piece of the space that the motion cannot split further.
Start anywhere in it, and the orbit comes as close as you like to every point of it.
A periodic orbit is a minimal set: it is finite, closed and invariant, and any invariant subset contains a whole orbit.
For an irrational rotation of the circle, the whole circle is a minimal set, because every orbit is dense.