Chart Modeled on the Closed Upper Half-Space

Let $(X,\mathcal{T})$ be a 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 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 bijection.
4. The map $\varphi:U\to\Omega$ is continuous, where $U$ carries the 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.