Let $(X,\mathcal{T})$ be a \reftext{def:topological-space-2026a}{topological space}, let $n\in\mathbb{N}$, and let $U\subseteq X$. A chart of dimension $n$ on $X$, modeled on the closed upper half-space, is a pair $(U,\varphi)$ with the following properties.

1. $U\in\mathcal{T}$.
2. If $H^n$ denotes the half-space from \reftext{def:closed-upper-half-space-euclidean-2026a}{the definition of the closed upper half-space}, then there exists a subset $\Omega\subseteq H^n$ that is open in $H^n$.
3. $\varphi:U\to\Omega$ is a \reftext{def:bijection-sets-2026a}{bijection}.
4. The map $\varphi:U\to\Omega$ is \reftext{def:continuous-map-topological-spaces-2026a}{continuous}, where $U$ carries the \reftext{def:subspace-topology-2026a}{subspace topology} from $X$ and $\Omega$ carries the subspace topology from $H^n$.
5. The inverse map $\varphi^{-1}:\Omega\to U$ is continuous for the same topologies.

The set $U$ is called the chart domain, and $\varphi$ is called the chart map.