Let $(X,\mathcal{T})$ be a \reftext{def:topological-space-2026a}{topological space}, and let $n\in\mathbb{N}$. A smooth atlas of dimension $n$ on $X$, modeled on the closed upper half-space, is a family of charts

for some set $A$ such that the following conditions hold.

1. The chart domains cover $X$, that is,

2. For every $\alpha,\beta\in A$, the charts $(U_\alpha,\varphi_\alpha)$ and $(U_\beta,\varphi_\beta)$ are smoothly compatible in the sense of \ref{def:smooth-compatible-charts-upper-half-space-2026a}.

A smooth manifold with boundary of dimension $n$ is a topological space together with such a smooth atlas, under the additional assumptions that the underlying topological space is \reftext{def:hausdorff-topological-space-2026a}{Hausdorff} and \reftext{def:second-countable-topological-space-2026a}{second countable}. If the underlying topological space is \reftext{def:compact-space-and-subset-2026a}{compact}, then one says that the manifold with boundary is compact.