# Manifold

##### Updated: October 22, 2020

A manifold is a topological space that *locally* resembles Euclidean space near each point.

## As defined in Physics #

An N-dimensional manifold of points is one for which N independent *real* coordinates \((x^{1}, x^{2},…,x^{N})\) are required to specify a point completely. These N coordinates are denoted collectively by \(x^{a}\), where it is understood that \(a\,=\,1,2,…,N\).

As an example, in \(\mathbb{R}^{2}\) we have a 2-dimensional manifold of points descrived by the real coordinates \((x^{1}, x^{2})\).