I thought of this question based on my other question here.
I understand that for a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $X$ and $Y$, both of which are connected but not necessarily compact, if $F$ is defined to have a ramification point at $p \in X$ if for all neighborhoods $U$ of $p$ in $X$, the restriction $F|_U$ is not injective.
Question 1: So if, on the contrary, $F$ were injective, then no $p$ is a ramification point because we could just pick $U=X$?
Question 2: What is the relationship between Question 1 and the fact that injective holomorphic maps between Riemann surfaces are isomorphisms onto their images (Proposition II.3.9 in Rick Miranda - Algebraic curves and Riemann surfaces; also see here: Whether or not riemann surfaces can be singletons (and consequences and generalisation))?
