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

Let $(X,\mathcal{T})$ be a topological space, let $n\in\mathbb{N}$, and let $(U,\varphi)$ and $(V,\psi)$ be charts of dimension $n$ on $X$ in the sense of Chart Modeled on the Closed Upper Half-Space. Write

$$\varphi(U)=\Omega_\varphi\subseteq H^n,\qquad \psi(V)=\Omega_\psi\subseteq H^n.$$

We say that these charts are smoothly compatible if either $U\cap V=\varnothing$, or else the following condition holds.

For every point

$$a\in \varphi(U\cap V),$$

there exist open subsets $W,W'\subseteq \mathbb{R}^n$ with

$$a\in W,\qquad (\psi\circ\varphi^{-1})(a)\in W',$$

and a smooth map

$$F:W\to W'$$

such that

$$F(x)=(\psi\circ\varphi^{-1})(x)$$

for every point

$$x\in W\cap \varphi(U\cap V).$$

The analogous extension condition is also required for the transition map $\varphi\circ\psi^{-1}$ at every point of $\psi(U\cap V)$.