Smooth Atlas and Smooth Manifold with Boundary

Let $(X,\mathcal{T})$ be a 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

$$\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A}$$

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

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

$$X=\bigcup_{\alpha\in A} U_\alpha.$$

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 Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space.

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 Hausdorff and second countable. If the underlying topological space is compact, then one says that the manifold with boundary is compact.