Q&A

# Are we in a "history-valley" for Topology?

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Here is my current rough timeline of Topology:

• Newton invents the calculus
• People like Riemann and Cauchy make it rigorous, and by this time, we have the $\varepsilon,\delta$ definition of continuity of a real valued function of a real variable.
• We realize that we want a definition of the continuity of vector-valued functions of vector spaces, possibly over complex numbers. So we have to generalize the idea of a continuous function. This makes it clear that we start from an open neighborhood of a point in the codomain and search for an appropriate open neighborhood in the domain.
• We start encountering other weird functions, for which we still want a notion of continuity. From probability theory we want set functions. From Fourier analysis we want maps from function spaces to function spaces. So we need a broader notion of continuity. We hone in on the idea of an open set not determined by any distance metric. In order to do this we need the ability to declare the open sets in a way that replaces and generalizes our notion of an open ball.
• Chaos reigns. Different people start talking about a topology, but define it in different ways. Some people include the Hausdorff axioms, some don't. Some people define it this way, some people define it that way.
• ???
• Now professors yell at you if you don't understand it fast enough because it's just a way of saying what your open sets are, what's so hard about that?!

So a few questions.

1. Am I embarrassingly wrong about any of that?

2. What happened in the "???" phase? In particular, why did we eventually decide that we wanted the Sierpinski topology to be a topology, for instance? Why did we ultimately not pack the Hausdorff axioms into the idea of a topology, and instead left it as a property that some topologies can have but others might not? What is the value added by doing it this way? Or to put it the other way around, would we be any worse off if we required topologies to satisfy properties that make it more intelligible from its growth out of the idea of open balls? We could then give a different name to structures which are merely "subsets of the powerset containing {} and X, closed under unions and finite intersections".

(This is the idea of a "history-valley" in my title: We got to a point where we do things this way, not because it's an optimum but because of a sequence of historically random events. And now that we're here it's too hard to climb out. Or at least the question is whether that's true, or is there a better motivation for why we do things the way we do them?)

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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, 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 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. A notable example being the k-spaces. More broadly but more vaguely are nice categories 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 or the even more dramatic departure into locales and pointless topology. An extremely new and ambitious program to find a "better" alternative to topological spaces is 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.

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First a very important side note, by the chaos period set theory, mathematical logic and all the modern theory of mathematic foundation was happening. With this in mind, by ???, more specificaly in 1914, Hausdorff published a book called Principles of Set Theory,one of the most important books in the history of mathematics, in this book, while researching how set theory could formalize what were know by the time about analysis, Hausdorff used the idea of neighborhood, that were already know to be a valid idea in all approachs of analysis yet studied,for defining the now standard axioms for a topological space, but including the separation axiom as an axiom for any topological spaces. Not much latter mathematicians realized that removing the separation axiom, now know as the Hausdorff axiom, whould be a good idea, not only because this way the notion of topological space includes more spaces, but also because some important properties of real analysis do not hold for Hausdorff spaces alone, but do hold for the more abstract notion tological space defined by the other axioms without the separation axiom.

Edit: A correction, Hausdorff created an equivalent definition of the modern one, but not the actual modern definition, he defined a topological space by neighbours of points, the current definition was given by Alexandrov in 1925.

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