Topological Manifolds

Topological manifolds are spaces that locally resemble Euclidean space.

A topological space $(X, \mathcal{T})$ is called a manifold if and only if:

• $\forall x \in \mathcal{T}, U \in \mathcal{N}_x$
  $\exists$ a homeomorphism $\phi_x : U \to V \Subset \mathbb{R}^n$ for some $n \in \mathbb{N}$

Lee, Introduction to Topological Manifolds, Chapter 1
Munkres, Topology, Section 25