# Definition:Topological Manifold

*This page is about topological spaces with a differentiable structure. For other uses, see Definition:Manifold.*

## Contents

## Definition

### 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 $\mathcal C^k$, where $k \ge 1$.

Then $\left({M, \mathscr F}\right)$ is a **differentiable manifold of class $C^k$ 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 $\left({M, \mathscr F}\right)$ 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 $\left({M, \mathscr F}\right)$ 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**.