Let $R$ be a commutative ring, $spec(R)$ be the set of all prime ideals on $R$. For any ideal $I$ on $R$, we define the $V_I$ to be the set of all prime ideals containing $I$. We define the Zariski topology on $spec(R)$ via the closed sets $\{V_I:I\textrm{ is an ideal of }R\}$.
I am still wrapping my mind around this topology. Can someone tell me if it is a cofinite topology, i.e. the open sets are complements of finite sets, or not?
Best Answer
There can be infinite closed sets besides the whole space.
In $\Bbb Q[x,y]$, for example, $(y)$ is contained in infinitely many prime ideals, so $V_{(y)}$ is a closed, but not finite set.
Another way: the cofinite topology always makes its space $T_1$, but the spectrum is $T_1$ in the Zariski topology iff prime ideals are maximal, and that only occurs for certain rings.
Another way: The cofinite topology on a finite space is the discrete topology, but the Zariski topology on a ring with a finite spectrum need not be discrete. Consider, for example, $Spec(\Bbb Q[[x]])$, where the spectrum is $\{0, (x)\}$ and the closed sets are $\{\emptyset, \{x\}, \{0,(x)\}\}$
I've spent some time in the same boat as you on this topic. It's especially disorienting if most of your topological experience is derived from thinking about metrizable spaces.