Manifold

Reference: Introduction to Smooth Manifolds by John M. Lee. A topological space $M$ is a topological manifold of dimension $n$ if $M$ is locally Euclidean of dimension $n$, where locally Euclidean means that for every point $p \in M$, there exists an open neighborhood $U$ of $p$ in $M$ and a homeomorphism $\varphi: U \to B^n$, then we obtain a set of such charts $\{(U_\alpha, \varphi_\alpha)\}$ called a topological atlas on $M$. ...