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

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

For every point

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

and a smooth map

such that

for every point

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