I am trying to make sense of the definition of a Noetherian Topological Space. It is a topological space that satisfies the descending chain condition for closed subsets. Here are my questions:
- Why 'closed' subsets?
- Is there an example of a Noetherian topological space where one has a sequence of the form $Y_1 \supseteq Y_2 \supseteq \ldots$ which doesn't stabilize, where the $Y_i$ are not necessarily closed?
- Is my understanding right that $\mathbb{R}$ with the standard topology is not Noetherian: take $Y_n = [0, \frac{1}{2} + \frac{1}{n}]$, all of these are closed, and $Y_n \supseteq Y_{n+1}$, and they don't stabilize.
- Take affine-n space with the Zariski topology. Is there a sequence of non-closed sets here that do not satisfy the descending chain condition?
I will answer these out of order. I'll assume $k$ is algebraically closed.
(3) Correct, $\mathbb{R}$ in the euclidean topology is not Noetherian since e.g., the sequence $X_n = [0, 1/n]$ is a descending chain which does not stablise.
(4) Affine $n$-space in the Zariski topology is Noetherian, this follows from the Nullstallensatz (there is an inclusion reversing correspondence between closed subsets correspond to radical ideals) and the Hilbert basis theorem (which shows that $k[x_1,...x_n]$ is Noetherian, hence ascending chains of ideals eventually stabilise).
(2) Yes, take $\mathbb{A}^1_{\mathbb{C}}$, by (4) this is Noetherian in the Zariski topology, but there is a the sequence I gave in (3) which does not stablise (it just has nothing to do with the topology). What you're asking is equivalent to asking for a finite set (just remove one point, then 2 points etc).
(1) You want your definition to have something to do with the universe you are working in. If the question is just subsets, you're just asking for finite sets which is neither very interesting, nor enlightening from a geometric perspective. The main reason for this definition is the observation in (4) above that closed subsets and (radical) ideals correspond, so you want your notions of Noetherian to line up. For affine space you can think that you can insert a hypersurface, then a codimension 1 subvariety into that, and so on... but this process will have to terminate at a point.
Edit: I misread your question (4). But see my answer to (2) - just take some balls that would be closed in the Euclidean topology and shrink them (they'll just have nothing to do with the Zariski topology).