some Euclidean space ${\mathbb{R}}^N$ and then to proceed similarly.

We can assume that the holomorphic isomorphisms $\phi_j : U_j \to {\sf
A}_j \subset \widetilde{ \sf B}_j \subset {\mathbb{C}}^{ N_j}$ are defined in slightly larger open sets $U_j' \Supset U_j$, for all $j \in J$. Pick $\mathcal{ C}^\infty$ functions $\lambda_j$ having compact support in $U_j'$ and satisfying $\lambda_j = 1$ on $\overline{
U}_j$; prolong them to be $0$ on $X$ outside $U_j'$. By compactness, there is a finite open cover: $$\overline{\Omega}
\subset
U_{j_1}\cup\cdots\cup 
U_{j_m}.$$ Consider the $\mathcal{ C}^\infty$ map, valued in ${\mathbb{R}}^N$ with $N:= 2(
N_{j_1} + \cdots + N_{ j_m}) + m$, which is defined by: $$\Psi
:=
\big(
\lambda_{j_1}\cdot\phi_{j_1},\dots,
\lambda_{j_m}\cdot\phi_{j_m},\ \
\lambda_{j_1},\dots,\lambda_{j_m}
\big).$$ It is an immersion at every point $x$ of $U_{ j_1} \cup \cdots \cup
U_{ j_m}$, because $x$ belongs to some $U_{ j_k}$, whence the $j_k$-th component $\lambda_{ j_k} \cdot \phi_{ j_k} \equiv \phi_{ j_k}$ of $\Psi$ is even an embedding of $U_k \ni x$. Furthermore, we claim that $\Psi$ separates points. Indeed, if we set: $$W_{j_k}
:=
\big\{
z\in X:\ 
\lambda_{j_k}(z)=1
\big\},$$ then clearly $U_{ j_k} \subset W_{ j_k} \subset U_{ j_k}'$. Pick two distinct points $x, y \in U_{ j_1} \cup \cdots \cup U_{ 