Manifold with boundary

Let \(n\geq 1\) and let \(\mathbb H^n\coloneqq\ts{x\in\RR^n \st x_n\geq 0}\) be the closed upper half-space, with the subspace topology. A topological \(n\)-manifold with boundary is a Hausdorff space \(M\) in which every point has an open neighborhood homeomorphic to \(\RR^n\) or to \(\mathbb H^n\).