# Can the Noetherian topological spaces be distinguished by looking at the category of topological spaces only?

A Noetherian topological space is one where every infinite descending sequence of closed sets $$X_0 \supset X_1 \supset X_2 \cdots$$ is eventually constant (paraphrased from Hartshorne page 5).

Assessing the Noetherianness of a given space $$(X, \tau)$$ using this definition, though, requires us to peer inside and look at $$\tau$$ specifically.

I'm curious whether we can identify which topological spaces are Noetherian by examining just the categorical structure of $$\mathsf{Top}$$.

I don't know very much about category theory or topology; the following is just an idea I had about how one might answer this.

I can sort of see the extremely vague beginnings of an argument where we look at ascending chains of open sets $$B_0 \subset B_1 \cdots$$ and we note that each open set can be thought of as a topological space $$(B_i, \{z \cap B_i : z \in \tau\})$$ ... and then we can talk about ascending chains of topological spaces that are subspaces of the original space $$(X, \tau)$$. I think the language of category theory gives me enough tools to talk about being a subspace, since I can talk about whether morphisms compose to the identity morphism and therefore I can talk about an inclusion map.

However, I don't know enough about category theory to know whether this idea will eventually succeed or whether it's inconsistent with the spirit of "examining a property of an individual thing by looking at the category it's in". Also, there might be a simpler way to characterize Noetherianness categorically.

• An idea would be to look at increasing sequences of open sets as maps from the space to $\{0,1,2,\dots\}$ where the opens are the subsets of the form $\{0,1,2,\dots,n\}$. Every map to this space must be bounded, so it's a form of compactness. Also, it might be interesting to know that a space is Noetherian if and only if all its subspaces are quasi-compact. But I don't know if one can characterize quasi-compactness in "purely categorical" terms in an interesting way... Commented Dec 30, 2021 at 2:45
• Could try the analogue for Noetherian commutative rings, in the category of commutative rings... Commented Dec 30, 2021 at 3:05
• We know that closed sets of $X$ are in bijection with continuous maps $f : X \to \mathbb{S}$, where $\mathbb{S}$ is the sierpinski space. So we can categorically get our hands on closed subsets (as pullbacks of $f : X \to \mathbb{S}$ along the map $1 \overset{0}{\to} \mathbb{S}$) and we can categorically talk about one subspace being smaller than another (we ask for the relevant monos to factor in the obvious way), so putting these together we can categorically express a witness to non-noetherianity. Commented Dec 30, 2021 at 3:17
• I'd be curious to know if there's a more "natural" way to do this than by naively translating the classical definition as I've done. In particular "hereditarily compact" sounds like something which might work nicely, but I don't actually know of a categorical characterization of compact (possibly non-hausdorff) spaces... I wouldn't be surprised to hear there's a nicer characterization that works for sober spaces, though. Commented Dec 30, 2021 at 3:19