Topological manifold

An $n$-dimensional topological manifold is a Hausdorff, second-countable topological space $M$ such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb R^n$. The local homeomorphisms are coordinate charts; their collection covers $M$.

The integer $n$ is the manifold's dimension and is fixed on each connected component. No differentiability is assumed. A smooth manifold is a topological manifold equipped with charts whose transition maps are smooth, so every smooth manifold has an underlying topological manifold but not conversely.