Difference between revisions of "Topological manifold"

Revision as of 00:51, 6 April 2015

Note: This page refers to a Topological Manifold

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]:

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]
2. [ilmath]M[/ilmath] is Second countable - there exists a countable basis for the topology of [ilmath]M[/ilmath]
3. [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]

Notations

The following are all equivalent (most common first):

1. Let [ilmath]M[/ilmath] be a manifold of dimension [ilmath]n[/ilmath]
2. Let [ilmath]M[/ilmath] be an [ilmath]n-[/ilmath]manifold
3. Let [ilmath]M^n[/ilmath] be a manifold

References

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