Manifolds

Overview

Manifolds are the abstract mathematical concept that roughly corresponds to a surface in space. The intuition is that while the surface may curve and twist, when you get close enough to it, it looks flat.

Locally Euclidian Space

A topological space is a locally Euclidian manifold if there is a neighborhood {% U %} of every point {% p %} in the space such that there is a homeomorphism {% \phi %} from {% U %} onto an open subset of {% \mathbb{R}^n %}.

The pair {% (U, \phi) %} is known as a chart.

Manifold

A (topological) manifold is a Hausdorff second countable locally Euclidian space.