Post History

I'm going to focus on the "history-valley" aspect, and to jump to the upshot: No, I don't think we're in a "history-valley" with respect to topology.

I more or less agree with Guilherme Gondin‭ that the significant thing in the ??? period was the foundational "crisis" (spurred on by the kind of issues you mention). The resolution of that "crisis" and the general settling of foundations (~1920s-1930s) is when a *lot* of mathematical concepts would reach their current modern forms, though the formulation of category theory not too long after (~1950s) has definitely led to even more modern reformulations.

It is surprising how much "basic" mathematics only existed in its current form less than 100 years ago. For example, [the modern formulation of a group is a 20th century invention](https://mathshistory.st-andrews.ac.uk/HistTopics/Abstract_groups/), though mathematicians were quite close to it before then. It's a bit less surprising when one realizes that the tools that are necessary for these modern definitions largely did not exist in the 19th century. Many of them were forged during the foundational "crisis". One obvious one was the formulation of set theory, but I'd actually put more significance on the general metamathematical work being done such as the formulation of first-order logic. These are heavily entangled historically though.

Returning specifically to topology, there are plenty substantial areas of mathematics that leverage topological spaces that are quite distinct from the typical motivating examples derived from metric spaces. One of the more significant is the non-Hausdorff [Zariski topology](https://en.wikipedia.org/wiki/Zariski_topology) used extensively in algebraic geometry. There is definitely value in a "general topology". And, of course, one is free to only consider special types of topological spaces, e.g. metric spaces, and many mathematicians do. So the split you describe exists, it just so happens that the general thing is still called "topology" and the more specific thing is called "metric topology". There's tons of non-metric topologies, so we definitely would want a term for the more general theory. (It also doesn't make sense to refer to the objects of a mathematical field by one particular construction.)

Now to actually start justifying my claim that we aren't in a "history-valley". While a notion of "general topology" significantly broader than metric spaces is desirable, there's definitely some feeling that the usual notion of topology is overly broad or otherwise not quite what we want. One manifestation of this is the fact that $\mathbf{Top}$, the category of topological spaces and continuous functions is not cartesian closed. This means that given two topological spaces $X$ and $Y$, the space of functions $Y^X$ cannot always be given a topology that will make it behave as we'd like. Cartesian closure is an extremely nice property to have, and many commonly used specialized subcategories of $\mathbf{Top}$ *do* have this property. As a result, there has been a cottage industry in trying to formulate different notions of "topological space" that capture all the examples we care about while still producing a cartesian closed category. These are called [convenient categories of topological spaces](https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces). A notable example being the [k-spaces](https://en.wikipedia.org/wiki/Compactly_generated_space). More broadly but more vaguely are [nice categories of spaces](https://ncatlab.org/nlab/show/nice+category+of+spaces) where we might care about other properties. This also includes approaches that "solve" the problem by generalizing topological spaces even further such as [equilogical spaces](https://ncatlab.org/nlab/show/equilogical+space) or the even more dramatic departure into [locales](https://ncatlab.org/nlab/show/locales) and pointless topology. An *extremely* new and ambitious program to find a "better" alternative to topological spaces is [condensed mathematics](https://en.wikipedia.org/wiki/Condensed_mathematics).

I think the above illustrates that mathematicians don't feel stuck to this notion of topological space. My impression is that there has been a pretty steady movement away from it in advanced mathematics for the last 60 years or so. However, what definitely *does* have a lot of inertia is the educational infrastructure, especially as you move earlier and earlier into the curriculum. Point-set topology is an undergraduate topic typically. Replacing it with some radically different alternative requires creating a new ecology of textbooks, retraining educators, updating standardized tests, and, most critically, doing all this for the other topics which depend on this topic. I'm not sure this has ever happened in the modern era. The closest I can think of is the slow seeping in of categorical concepts, though that's less of a straight replacement for any one topic and more of a change in style across many topics. And that's certainly not been completed yet. At "advanced" levels (meaning roughly post-undergraduate and certainly post-grad), new formulations of concepts get picked up and dropped all the time.