Hartshorne Exercise 1.1.10: Give an example of a noetherian topological space of infinite dimensions

Question

Hartshorne Exercise 1.1.10: Give an example of a noetherian topological space of infinite dimensions.

I'm baffled by why such space can exist. Instincts told me that I shouldn't take the Spec of any Noetherian ring because then prime ideals have finite height, which makes the dimension of the Spec to be finite as well. But this still doesn't make sense for the following reason: by part $(a)$, we know that $X$ is a topological space which is covered by a family of open subsets $\{U_i\}$, then $\dim X = \sup \dim U_i$. $X$ by being a Noetherian topological space, is campact and thus we can have a finite sub-cover of the cover, and $\dim X = \max U_j$, $j \in J$ where $J$ is a finite index set. Since each $U_j$ is then finite dimensional space, we thus have $\dim X$ in finite.

Am I missing something? What would be an example?

Answer 1

An example of such a space would be the unit interval where you make the closed intervals $\left[ \frac{1}{n}, 1 \right]$ the closed sets.

Answer 2

Similarly take $\mathbb{N}$ with the topology where the closed sets are {1,2,...,n} for any n.

Answer 3

prime ideals have finite height, which makes the dimension of the Spec to be finite as well

This is not true, because you could find longer and longer chains of prime ideals, each becoming stationary after finitely many steps but there is no global bound before which each and every one of them becomes stationary.

For the example see Stackexchange question