A smooth topological $m$-dimensional manifold with boundary is a topological m-dimensional manifold with boundary $M$ together with a collection $\mathcal A$ of charts on $M$ such that
- for all $p\in M$, there exists a chart $(U, \phi)\in \mathcal A$ such that $p \in U$;
- for any two charts $(U,\phi)$ and $(V, \psi) \in \mathcal A$, the transition function \(\psi\circ \phi^{-1}:\phi(U\cap V)\to\psi(U\cap V)\) is a smooth diffeomorphism.
- the set $\mathcal A$ (called an atlas) is the maximal set of collections of charts satisfying 1. and 2.
Wikidata ID: Q78338964