Given the historical development of modern mathematics, everything is ultimately encoded as a set (possibly with some additional structure, also encoded as set(s) ). For example, a topological space is an ordered pair $(X, \tau)$ where $X$ is the underlying point-set and $\tau$ is a set of subsets on $X$ called it's topology, obeying some axioms. But what if I don't care about sets in particular, especially just the sets of ZFC, what if I want to study mathematical structures with notions of nearness, connectedness, continuity, etc. and do not want to be constrained to an underlying set theoretic encoding? Does category theory offer me a way to study the "essence" of a topological space that will hold no matter what encoding (i.e. foundational theory) I use?
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3$\begingroup$ You might want to look at Peter Johnstone's paper on "The point of pointless topology". $\endgroup$– Steven LandsburgCommented Jul 16, 2017 at 17:38
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9$\begingroup$ I somehow feel that this question is the embodiment of the whole "scared of ZFC" philosophy that we are seeing more and more by people who were not exposed to set theory or classical foundations properly in their studies. This is very much how I read the remark "especially just the sets of ZFC". It's as though sets form some psychological barrier in some people. And that is truly a shame. I'm not saying that everyone should be a set theorist or even know set theory. But it seems like a particularly ignorant comment to make, in a setting that feels more and more "normal" in today's mathematics. $\endgroup$– Asaf Karagila ♦Commented Jul 16, 2017 at 17:58
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3$\begingroup$ Perhaps you want to look at locales? $\endgroup$– Benjamin SteinbergCommented Jul 16, 2017 at 19:35
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4$\begingroup$ Then what does set theory even has to do with it? Just study topological structures. People think that ZFC is this sort of rigid and concrete thing that is completely inflexible on how you code things. It's not. In fact, it's even more coding agnostic than category theory in some extent, since the Replacement schema is exactly equivalent to saying that any coding that satisfies the ordered pair property can be used to code ordered pairs (and Replacement is absent from ETCS, for example). So yeah, pretty much go Nike and "Just Do It". Why even bother about your underlying set theory? $\endgroup$– Asaf Karagila ♦Commented Jul 16, 2017 at 20:19
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6$\begingroup$ @AsafKaragila While I agree it's not fair to disparage ZFC, it's disingenuous to suggest that the absence of replacement in ETCS somehow makes it less coding agnostic than ZFC. In fact coding agnosticism is built into ETCS at a much more basic level, so there is no need for a complicated argument using a powerful axiom such as replacement. $\endgroup$– Mike ShulmanCommented Jul 16, 2017 at 21:41
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2 Answers
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There are several ways to understand this question, leading to different answers.
If the question is about studying topological spaces, in the sense of points equipped with open sets, independent of a particular axiom system for set theory, then the commentators are right: ordinary point-set topology is (like nearly all of mathematics) already independent of any particular axiom system for set theory.
If the question is about studying things that behave like topological spaces but don't have a "set of points", then the other commentators are right that you should look into locales and topoi.
If the question is about "notions of nearness, connectedness, continuity" in a more general setting that are not necessarily defined in terms of a notion of "open set", then you may be interested in categorical topology (e.g. topological categories) or axiomatic cohesion.
answered Jul 16, 2017 at 21:46
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3$\begingroup$ For statement 1, could you clarify what you mean by saying "ordinary point-set topology is..already independent of any particular axiom system for set theory", in light of the fact that so much of the (serious) theory of point-set topology makes use of extra-ZFC principles such as diamond, CH, MA or large cardinals. We now know that many questions, concerning whether every such-and-such kind of space is also that kind and so on, are independent of ZFC and require these extra axioms to settle them. So even the ZFC-style versions of point-set topology seem to depend on the set-theoretic system. $\endgroup$ Commented Jul 17, 2017 at 10:12
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2$\begingroup$ In other words, what we've learned in point-set topology is that many fundamental topological questions do depend on one's set theoretic background. $\endgroup$ Commented Jul 17, 2017 at 10:57
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2$\begingroup$ @JoelDavidHamkins Two answers. The first is that I was not thinking about such questions at all, which I regard as rather esoteric. Certainly I believe that if you spend enough time studying point-set topology as a field in its own right you will run into them soon enough, but for someone who primarily wants a tool to study "notions of nearness, connectedness, continuity" in "naturally occurring" spaces (which is what I assumed the OP wanted) I think they will not play much role. $\endgroup$ Commented Jul 17, 2017 at 11:09
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3$\begingroup$ I like your second answer much better, and that is one I can agree with. My point wasn't about ZFC specifically, but rather with the claim you seemed to make that topological questions don't depend on one's foundation, since we know that they definitely do. See en.wikipedia.org/wiki/Set-theoretic_topology for a start. This is not esoteric, but rather an active field of mathematical research, with its own conferences and so on and connected with other fields. As you say, the extra hypotheses can of course be expressed in any sufficiently robust foundational system. $\endgroup$ Commented Jul 17, 2017 at 11:20
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4$\begingroup$ @JoelDavidHamkins By "esoteric" I didn't mean to be disparaging, or imply that it isn't interesting or doesn't have lots of people studying it. All of pure mathematics is pretty esoteric to the man on the street. (-: I just meant that point-set topology is used by lots of people doing algebraic topology, low-dimensional topology, differential geometry, etc., and I don't think any of them have ever had any occasion to worry about large cardinal axioms. $\endgroup$ Commented Jul 17, 2017 at 11:28
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Some categories are not concrete, i.e. one cannot think to objects as sets and to morphisms as function. It's the case of homotopy category of topological space.
Sober spaces are equivalent to localic toposes with enough points but honestly, I am not sure this is so far from a set theoretic approach.
answered Jul 16, 2017 at 19:15