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Topological manifold

Definition

An n-dimensional manifold is a Hausdorff space such that every point lies in an open subset that is homeomorphic to an open subset of $\mathbb{R}^n$.

Types

A compact manifold is called a closed manifold.

A 2-manifold is called a surface.

Examples

Any open subspace of an n-manifold is an n-manifold.

$\mathbb{R}^n$ is a manifold, trivially

$S^n$ is an n-manifold; removing a single point leaves a set that is homeomorphic to $mathbb{R}^n$ by stereographic projection.

All geometric manifolds are topological manifolds.

all these pages adapted with probably insufficient credit from my university's lecture notes