Topological manifold

Note: This page refers to a Topological Manifold

Definition

We say $M$ is a topological manifold of dimension $n$ or simply an $n-$manifold if it has the following properties:

1. $M$ is a Hausdorff space - that is for every pair of distinct points $p,q\in M\ \exists\ U,V\subseteq M\text{ (that are open) }$ such that $U\cap V=\emptyset$ and $p\in U,\ q\in V$
2. $M$ is Second countable - there exists a countable basis for the topology of $M$
3. $M$ is locally Euclidean of dimension $n$ - each point of $M$ has a neighbourhood that his homeomorphic to an open subset of $\mathbb{R}^n$

   This actually means that for each $p\in M$ we can find:

   • an open subset $U\subseteq M$ with $p\in U$
   • an open subset $\hat{U}\subseteq\mathbb{R}^n$
   • and a Homeomorphism $\varphi:U\rightarrow\hat{U}$

Notations

The following are all equivalent (most common first):

1. Let $M$ be a manifold of dimension $n$
2. Let $M$ be an $n-$manifold
3. Let $M^n$ be a manifold