# 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 **topolofical 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 $d$**.

## Also known as

A **topological manifold** of dimension $d$ is often seen referred to as a **$d$-manifold**.

## Also see

## Sources

- 2003: John M. Lee:
*Introduction to Smooth Manifolds*: $\S 1.1$: Smooth Manifolds. Topological Manifolds