# Topological manifold

From Maths

**Note:** This page refers to a **Topological Manifold** a special kind of Manifold

## Contents

## Definition

We say [ilmath]M[/ilmath] is a *topological manifold of dimension [ilmath]n[/ilmath]* or simply *an [ilmath]n-[/ilmath]manifold* if it has the following properties^{[1]}:

- [ilmath]M[/ilmath] is a Hausdorff space - that is for every pair of distinct points [ilmath]p,q\in M\ \exists\ U,V\subseteq M\text{ (that are open) } [/ilmath] such that [ilmath]U\cap V=\emptyset[/ilmath] and [ilmath]p\in U,\ q\in V[/ilmath]
- [ilmath]M[/ilmath] is Second countable - there exists a countable basis for the topology of [ilmath]M[/ilmath]
- [ilmath]M[/ilmath] is locally Euclidean of dimension [ilmath]n[/ilmath] - each point of [ilmath]M[/ilmath] has a neighbourhood that his homeomorphic to an open subset of [ilmath]\mathbb{R}^n[/ilmath]
- This actually means that for each [ilmath]p\in M[/ilmath] we can find:
- an open subset [ilmath]U\subseteq M[/ilmath] with [ilmath]p\in U[/ilmath]
- an open subset [ilmath]\hat{U}\subseteq\mathbb{R}^n[/ilmath]
- and a Homeomorphism [ilmath]\varphi:U\rightarrow\hat{U} [/ilmath]

- This actually means that for each [ilmath]p\in M[/ilmath] we can find:

## Notations

The following are all equivalent (most common first):

- Let [ilmath]M[/ilmath] be a manifold of dimension [ilmath]n[/ilmath]
- Let [ilmath]M[/ilmath] be an [ilmath]n-[/ilmath]manifold
- Let [ilmath]M^n[/ilmath] be a manifold

## See also

## References

- ↑ John M Lee - Introduction to smooth manifolds - Second Edition