---
tag: QQQ9B1DK4P
kind: concept
name: Topological manifold
msc: [57N99]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_manifold
same_as:
  wikidata: Q2721559
  wikipedia: Topological manifold
---

## Summary

A topological manifold of dimension $n$ is a [[topological space|0CQHFPGJ45]] that looks like the Euclidean space $\R^n$ near each of its points.
It is also required to be Hausdorff and second-countable.
It is the basic object of geometric topology.
Curves, surfaces and the space-times of physics are manifolds, and smooth and piecewise linear manifolds are topological manifolds with extra structure.

## Explanations

### L1 human

Imagine a tiny ant on a surface: around it, the world looks like a flat plane, even though the whole surface may be a sphere or a doughnut.
A manifold is a space with this property in some fixed dimension $n$.
The circle is a $1$-dimensional manifold, because a short arc of a circle looks like a short open interval.
The surface of the Earth is a $2$-dimensional manifold, which is why flat maps of small regions work.
A figure eight is not a manifold, because near the crossing point four arcs meet, which no open interval can imitate.