# Definition:Topological Manifold

## 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.