Some questions about Hartshorne's exercise II.3.17 (b)~(c) Zariski space I have some questions while trying to solve 'Exercise II 3.17(a)~(c), Hartshorne.
Zariski Space: A topological space $X$ is called a Zariski space if it is Noetherian and every (nonempty) closed irreducible subsets has a unique generic point(Ex2.9)...
$~~~~~~~$(b) Show that any minimal nonempty closed subset of a Zariski space consists of one point.
$~~~~~~~$(c) Show that a Zariski space $X$ satisfies the axiom $T_0$: given any two distinct points of X, there is an open set containing $~~~~~~~$one but not the other.
I roughly sketched the proof of (b) and (c). I wonder whether my sketch is right or not...
Exercise (b) : Since $X$ is a Zariski space, $X$ has notherian, thus for every sequence $Y_1 \supset Y_2 \supset.... \supset $ for closed subsets , there exists closed an integer $r$ s.t $Y_r=Y_{r+1}=...$
Claim  : $Y_r$ should be a singleton set [?]
If my claim is right, for any closed subset $K$ is given, by letting $K=K_1$, by the definition of Noetherian, we always make a descending chain and the smallest closed set of the given chain is a singleton set.
Exercise (c) : If the statement (b) is true, since a singleton set is closed, choose different points $p,q \in X$, then $X-\left \{ p \right \}$  is an open subset which contains $q$ but does not contain $p$.
 A: For part (b), as KReiser said, you don't explain how $Y_r$ is a singleton, which you must do! In fact, there are Noetherian topological spaces which are not $T_0,$ so your strategy will need more juice to be able to work! Consider for example the space $X = \{a,b\}$ with the indiscrete topology. This is clearly Noetherian, as there are only two closed subsets, but there is no open set containing exactly one of $a$ and $b$.
Hint for (b): Instead of using the Noetherian condition here, try using the other condition, that every nonempty irreducible closed subset has a unique generic point. Start by letting $Y$ be a minimal nonempty closed subset, and let $y\in Y.$ Now, what can you say about $\overline{\{y\}}$ and what does this imply about $Y$?
Again, as KReiser says, singleton sets need not be closed. An example of a Zariski space is $\{x,\eta\}$, with open sets $\emptyset,\{\eta\},$ and $\{x,\eta\}.$ Then you can see that $\{\eta\}$ is not closed!
Hint for (c): Again, try using the generic point condition. Let $x,y\in X$ be distinct points. What happens if $y\in\overline{\{x\}}$? What happens if $y\not\in\overline{\{x\}}$?
Bigger hint: Don't look until you're thought hard about the above!

 For any point $x,$ $\overline{\{x\}}$ is irreducible.

