<-- Go Back Last Updated: 11/01/2025

Manifold

A topological space \((X,\mathcal{T})\) is called a manifold of dimension \(n\) if there exist open sets \(\{U_i|i\in I\}\subseteq \mathcal{T}\) and maps \(\{\phi_i:U_i\to\mathbb{R}^n|i\in I\}\) such that the \((U_i)\) form a open cover of \(X\), each \(\phi_i(U_i)\) is open in the metric topology of \(\mathbb{R}^n\), and each \(\phi_i\) is a homeomorphism onto its image. Often one also requires that \(X\) be Hausdorff and second countable or paracompact.