The image of a morphism between two varieties

Question

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.

Answer 1

If you're interested in these statements in the context where "variety" means what it does in the first chapter of Hartshorne (this is how I'll use "variety" for the rest of the post), then all of these statements are true and the easiest way to see that is to upgrade our varieties to schemes. Given any variety, there's a fully faithful functor $t$ from varieties over $k$ to quasiprojective integral schemes of finite type over $k$ constructed in proposition II.2.6 which has the useful property that $V\subset t(V)$ with the subspace topology. In particular, if $\varphi:t(V)\to t(W)$ is a morphism of schemes over $k$, then it corresponds uniquely to a morphism $f:V\to W$ of varieties over $k$, and $\varphi(t(V))\cap W = f(V)$ as subsets of $W$. Thus if we know the results you mention for schemes of finite type over $k$, we know the results for varieties.