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.