Definition:Topological Manifold
This page is about Topological Manifold. For other uses, see Manifold.
Definition
Let $M$ be a Hausdorff second-countable locally Euclidean space of dimension $d$.
Then $M$ is a topological manifold of dimension $d$.
Differentiable Manifold
Let $M$ be a second-countable locally Euclidean space of dimension $d$.
Let $\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\CC^k$, where $k \ge 1$.
Then $\struct {M, \mathscr F}$ is a differentiable manifold of class $\CC^k$ and dimension $d$.
Smooth Manifold
Let $M$ be a second-countable locally Euclidean space of dimension $d$.
Let $\mathscr F$ be a smooth differentiable structure on $M$.
Then $\struct {M, \mathscr F}$ is called a smooth manifold of dimension $d$.
Complex Manifold
Let $M$ be a second-countable, complex locally Euclidean space of dimension $d$.
Let $\mathscr F$ be a complex analytic differentiable structure on $M$.
Then $\struct {M, \mathscr F}$ is called a complex manifold of dimension $d$.
Examples
Real Cartesian Space
The real Cartesian space $\R^n$ is an example of an $n$-manifold.
$n$-Sphere
The $n$-sphere $\Bbb S^n$ is an example of an $n$-manifold.
Torus
The torus is an example of a $2$-manifold.
Projective Plane
The projective plane is an example of a $2$-manifold.
Klein Bottle
The Klein bottle is an example of a $2$-manifold.
Also known as
A topological manifold of dimension $d$ is often seen referred to as a $d$-manifold.
Also see
- Results about topological manifolds can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): manifold
- 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 1.1$: Smooth Manifolds. Topological Manifolds
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): manifold
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): manifold