My book says: Let $X$ and $Y$ be varieties (so i suppose they can be affine, quasi-affine, projective or quasi,projective; since they are the only that it introduces). Let $f\colon X\to Y$ be a morphism of varieties. Let $Z:= \overline{f(X)}$ i.e. the closure of $f(X)$ in $Y$.
Then $\bar{f}\colon X\to Z$ is a morphism of varieties.
Now, i understand that $\bar{f}$ is again contunuous as a map of topological spaces, since $Z$ is closed in $Y$. Moreover, $X$ is irreducible so it also is $f(X)$ because $f$ si continuous. Moreover $\overline{f(X)}=Z$ is irreducible since it is the closure of an irreducible subset of $Y$.
But, my question is: what sort of variety is $Z$? For example, let suppose that $Y$ is a quasi-affine variety, i.e. $Y$ is a non empty open subset of an affine variety $W$, and $W$ is an irreducible closed subset of $A_K^n$. Now $Z$ is an affine variety or a quasi-affine variety? Or nothing? Or just an irreducible topological space? According to me it isn't an irreducible closed subset of some affine variety, (so it is not an affine variety) and it is not a non empty open subset of an affine variety (so it isn't a quasi affine-variety). So what is it ? Why it is a variety? why the irreducible closed subset of quasi-affine varieties are "varieties"?
• A closed subset of a quasi-affine variety is quasi-affine, because the topology on the ambient quasi-affine variety is the subspace topology: So if we take the closure $Z'$ of $Z$ in $Y'$, where $Y'$ is whatever affine variety we are thinking of $Y$ lying inside of as an open set, then $Z'$ is affine and $Z$ is an open set of that. – John Brevik Nov 24 '17 at 22:23
• @JohnBrevik Thank you! I didn't see that. Can you also help clarify me this topic: my book give me the definition of a regular function in an open subset of an affine variety, say $X$. So if $U$ is an open subset of $X$ and $X$ is an affine variety, then i know who is $O_X(U)$ (the ring of regular function on U). But then he uses , for example , $O_Y(U)$ but whit $Y$ be a quasi-affine variety. But i haven't the definition of a regular function in an open subset of a quasi-affine variety. Have i to interpret it as $O_X(U)$, where $X$ is the affine variety of wich $Y$ is an open subset? – Minato Nov 25 '17 at 10:43