By varieties I mean any of affine varieties, quasi-affine varieties, projective varieties or quasi-projective varieties over an algebraic closed field $k$, where for affine varieties, I mean the classical definition of irreducible Zariski closed subset of $k^n$ as in the first Chapter of Hartshorne.
Is it true that, if $f:X \rightarrow Y$ is a morphism between varieties, then $f(X)$ is a finite union of locally closed sets?
I asked this because Chevalley's theorem suggests that, if $f:X \rightarrow Y$ is a morphism of finite type of noetherian schemes, then $f(X)$ is finite union of locally closed sets. Does this theorem implies that the above question is true?
Also I notice that the main theorem of elimination theory seems to suggest that when $X$ is a projective variety, $f$ is necessarily a closed map and $f(X)$ is closed, is this true?
However, in general, a morphism of varieties does not have to be open or closed map, for example, consider $f:A^1 \rightarrow P^2$, where $x$ is sent to $(x,1,0)$, then $f(X)$ is neither open nor closed. Notice that in this example, $f(X)$ is locally closed.