I have been spending a few days now proving the last bit of the following problem of Atiyah Macdonald:
Prove that $X = \operatorname{Spec}(A)$ as a topological space with the Zariski Topology is $T_0$.
Now since the Zariski Topology is specified in terms of closed sets, I thought it might be easier to prove that given $x,y \in X$ such that $x \neq y$, there exists a closed set $U$ such that $x \in U$, $y \notin U$, or else there exists a closed set $V$ such that $y \in V$, $x \notin V$.
So I tried to follow my nose by explicitly producing such a closed set, namely
$\overline{\{x\}} = V(\mathfrak{p}_x)$
following the notation of Atiyah Macdonald. Now one of the things I tried from here was to use the fact that closed subsets of compact sets are compact. Hence $V(\mathfrak{p}_x)$ is compact but this approach did not work out. I realised while typing this problem that if I were actually able to prove it like this, then switching the roles of $x$ and $y$ I would have proved $T_1 - $ ness. However this cannot be possible for the finite point set $\{x\}$ is not closed when $\mathfrak{p}_x$ is not a maximal ideal.
What should I look at now to try to solve the problem? Please do not post complete solutions. Thanks.