In point set topology, we have the following result, which is easily proved.
Theorem. Let $Y$ be Hausdorff space and $f,g:X \to Y$ be continuous functions. If there exists a set $A\subset X$ such that $\bar{A} = X$ and $f|_A = g|_A$, then $f=g$.
I was trying to understand what would be the natural generalization of this fact in the category of schemes. We know that the correct analogous of a Hausdorff space is a separated scheme. So I was thinking in a statement like this:
"Let $Y$ be a separated scheme and $f,g:X \to Y$ be morphisms of schemes. If there exists a set $A\subset X$ such that $\bar{A} = X$ and $f|_A = g|_A$, then $f=g$."
The first problem is that we must be careful with this restriction "$f|_A$", since $f$ is a morphism and I want to consider the morphism of sheaves also. Then I saw that Liu's book on Algebraic Geometry has the following statement:
"Let $Y$ be a separated scheme, $X$ a reduced scheme, and $f,g:X \to Y$ morphisms of schemes. If there exists a dense open subset $U$ such that $f|_U=g|_U$, then $f=g$."
Now this makes sense, since we are dealing with open subsets now. But I still find this result too restrictive. So I came up with this:
"Let $Y$ be a separated scheme, $X$ a reduced scheme, and $f,g:X \to Y$ morphisms of schemes. If there exists a morphism $\varphi:S \to X$ such that $\varphi(S)$ is dense in $X$ and $f\circ \varphi = g\circ \varphi$, then $f=g$."
It's easy to see that Liu's proof of the result concerning only the open set also applies to this context. Finally, let's go to the questions:
Is this really the best generalization? Is there any other results in this direction that are at least slightly different?
I can see where the "reduced" hypothesis enters in the proof, but I found it a little strange. Is it just a technical point or can be "understanded" in some sense? Maybe counterexamples of this fact when this hypothesis isn't valid would help to clarify , but I didn't think of any.
P.S. Sorry for the bad english.