152 Topological Manifold
A topological manifold is a Hausdorff space that is locally homeomorphic to Euclidean space of fixed dimension that is used to introduce local coordinates.
Definition (Topological Manifold) A topological manifold is a topological space, \(M\), that satisfies the following conditions:
- Locally Euclidean of Dimension \(n\)
- Hausdorff Space
- Second Countable
152.1 Elementary Example
152.1.1 Simple
A topological manifold is locally Euclidean, Hausdorff, and second countable. A line locally looks like \(\mathbb{R}^{1}\).
\[ M = \mathbb{R} \]
\[ n = 1 \]
\[ \text{each } p \in M \text{ has a neighborhood homeomorphic to an open interval in }\mathbb{R} \]
where
- \(M\) is the manifold.
- \(n\) is the dimension.
152.1.2 General
The plane is a \(2\)-manifold: each point has a neighborhood homeomorphic to an open disk in \(\mathbb{R}^{2}\).
\[ M = \mathbb{R}^{2},\quad n = 2 \]
\[ U = \{ (x,y) : x^{2} + y^{2} < 1 \} \]
where
- \(U\) is an open neighborhood of the origin.
- \(U\) is homeomorphic to an open set in \(\mathbb{R}^{2}\).