# Is there a formulation of topology which excludes most of the pathological objects?

I have been learning elementary topology and I keep coming across what I take to be a standard phenomenon: that something sounds true but is false because it fails in some pathological way, usually on infinitely-detailed curves.

Examples:

• the Topologist's Sine Curve shows that it's possible for a space to be connected but not path-connected.
• the Peano Space-filling Curves shows that there are e.g. continuous maps from the unit interval to the unit square
• the Jordan Curve Theorem's proof is complicated (as far as I can tell) because it has to deal with infinitely-detailed curves that e.g. fill the plane or spiral infinitely as they approach a point or something.

I'm wondering: is there some alternate treatment of topology that is based on different definitions or different axioms so as to avoid at least some of these pathological cases?

(I'm aware that there are often proofs for piecewise-linear or piecewise-smooth objects that are simpler than the general cases, but they're in the middle of a larger text that is mostly about the general continuous cases that I don't really care about. I'd rather find a text that doesn't even discuss these sorts of examples, if such a thing exists.)

• Or maybe nice categories of spaces. Commented Jun 21 at 15:42
• I do not think your question in the current form is answerable, since it is very unclear what counts as "pathological" (you just gave a couple of examples) and "middle of a larger text." Also, your terminology is highly nonstandard ("infinitely-detailed curves"). My suggestion is to learn basics of topology as it is developed in the last 150 (or so) years. Commented Jun 21 at 15:42
• If you want nicer Theorems, you have to work with nicer spaces. What exactly constitutes a "nice" space depends entirely on what context you are working in. It's not possible to give a more precise feedback without knowing what exactly you are looking to learn. If you merely don't care for pathological examples, you can also skip them while reading. Lastly, a different formalism will not make counter-examples go away unless you (implicitly) restrict yourself to a nicer class of objects, as initially alluded to. Commented Jun 21 at 15:45
• This is not a direct answer to your question, but the model-theoretic setting of o-minimality is one in which the pathologies of general topology and real analysis do not occur. See ACL's answer here for a nice summary. Commented Jun 21 at 16:42
• Why do you think that your examples are pathological? Just because they are in a sense more complicated than other? Commented Jun 21 at 18:40

Reading about the history of topology was enormously clarifying for me here; I can warmly recommend checking out James' History of Topology and/or Dieudonné's History of Algebraic and Differential Topology.

People were trying to do topology long before the modern definition of a topological space; the ideas go back at least as far as Riemann and Poincaré and probably further. When people were trying to clarify exactly what spaces we ought to allow as topological spaces it was not clear exactly how permissive or restrictive we should be (e.g. it was not clear whether we should require spaces to be Hausdorff). The definition of a topological space needs to satisfy several criteria to be useful, some of which trade off against each other:

• It should include the examples people actually want to study, e.g. manifolds, topological vector spaces, topological groups. It should be reasonably easy to verify that these are examples.
• It should straightforwardly allow new spaces to be constructed from old spaces in natural ways, e.g. taking products and quotients.
• It should allow proofs of important general theorems (e.g. the basic results on compactness and connectedness) which are reasonably straightforward.
• It should hopefully not constrain future mathematicians too much if they want to study stranger spaces than we currently do.

Broadly speaking the tradeoff is that if you require more axioms (e.g. the Hausdorff axiom) then you can prove stronger theorems but those theorems apply to fewer spaces. For example if you really dislike the topologist's sine curve then you might decide you want to restrict your attention to topological spaces which are locally path-connected or something like that. This includes all manifolds and CW complexes but it does not include some very important examples such as infinite Galois groups, which are totally disconnected. For another example, you might decide you want to require topological spaces to be Hausdorff because you think non-Hausdorff spaces are weird and poorly behaved, but again this means you exclude very important examples like the Zariski topology, which is almost never Hausdorff.

So, overall the modern definition of a topological space is a reasonable historically contingent compromise between multiple competing considerations, which was settled on from experience trying to use it to do stuff; in particular I want to emphasize that this was a choice we made to use this definition to study topology, and if history had gone another way we could have made a different choice. I personally do not consider topological spaces to be "God-given" the same way I consider, say, the integers to be. And in fact for some applications people have found that topological spaces are not the right tool and they use different tools, for example:

• In some contexts (e.g. constructively) it turns out to be better to isolate the behavior of the lattice of open sets away from the set of points of a topological space, and treat it as an abstract lattice which does not necessarily need to embed into a lattice of subsets. This leads to the study of locales, which loosely speaking are "topological spaces which may not have enough points."
• Topological spaces are not cartesian closed, meaning given two topological spaces $$X, Y$$ we cannot generally assign a meaningful topology to the set of continuous functions $$X \to Y$$ that satisfies various nice properties we might want. This is an impediment to certain constructions in algebraic topology, so algebraic topologists often use slightly different convenient categories of topological spaces, which are cartesian closed.
• Topological abelian groups do not form an abelian category, which makes it difficult to do homological algebra with them, and in some applications people really want to do this. Recently some replacements for topological spaces have appeared, namely condensed sets and pyknotic sets, which fix this issue by, loosely speaking, allowing some stranger "even more non-Hausdorff quotients" than are possible in topological spaces.

However, it's worth noting that none of these examples are motivated by the desire to throw out pathological-looking spaces. Actually some of them allow "even more pathological-looking" spaces! Grothendieck taught us that it is better to have a nice category with nasty-looking objects than to have nice objects but living in a nasty category, and the category of topological spaces is nice but not that nice; the examples above are trying to rectify this, as opposed to trying to remove individual pathological-looking spaces. The individual pathological-looking spaces just turn out not be too much of an impediment, and some of them even end up being surprisingly important.

On the other hand, sometimes people have asked for a simpler kind of topology exactly as you are asking for, which allows e.g. nice spaces like manifolds and not pathological examples like space-filling curves. This goes under the broad heading of "tame topology," and I know nothing about it but you can use that search term to find more. It is not a mainstream subject, and my impression is that it mostly has not seemed worth the effort since topological spaces are mostly adequate for foundations.

For example if you ultimately only care about manifolds (say your interest is in differential geometry) then the way the foundations of manifold theory get set up is that we define manifolds as being certain special topological spaces, namely those that are locally Euclidean and Hausdorff (plus a size axiom). This works fine. Manifolds are very nice, e.g. locally path-connected. So in manifold topology you can mostly ignore stuff like the topologist's sine curve and space-filling curves. I don't know that using a theory of tame topology as the foundations here would meaningfully simplify anything, and it might be annoyingly restrictive.

But also, topology has many more applications than this and in some of those the spaces that occur are strange-looking at first glance (e.g. the Galois groups mentioned above, which are Stone spaces).

• This is a brilliant answer! I learnt a lot. Thanks for taking the effort to write this out in detail. Commented Jun 23 at 9:03
• Agreed. I now intend to read the histories, that's a good idea. It's a bit unfortunate for me though. This is not stuff I want to spend a lot of brain-time on; I wish I would come across a book-length intro to topology that takes a few more axioms and reduces the complexity somewhat. Otherwise I feel I am going to have to (very slowly) construct my own notes that do exactly that. Commented Jun 26 at 20:51
• @Alex: it really depends on why you want to learn topology / what you want to apply it to. For many applications the spaces that show up are quite nice. On a first pass you can safely ignore all the strange counterexamples. Commented Jun 26 at 20:57
• Oh yes, I understand that. I just want to learn from a book that has already done that work for me---like, sorting the subject by complexity rather than by a linearly development of ideas. This question was inspired by wading through proofs for e.g. the JCT that were complex to account for pathology. It ends up being hard to tell, especially as a non-expert, which parts of the proofs and definitions are there to handle the pathological stuff. Commented Jun 26 at 21:00
• @Alex: frankly I don't think it matters, because I don't think the full Jordan curve theorem has any applications. Where do people use it? The only place I can think of is to maybe to prove very general results about contour integrals. But in practice I think you can pretty much always assume that contours of contour integrals are piecewise analytic, there is no need to work with arbitrarily poorly behaved continuous contours. And for those the Jordan curve theorem should be simpler. Commented Jun 26 at 21:04

I suggest an alternate strategy. To paraphrase Stanley Kubrick: stop worrying and learn to love pathological objects.

"Pathological objects" teach you things. For example, they teach you how to appreciate the various special hypotheses and definitions: how those hypotheses and definitions are needed for the statements of various theorems; how they are used in the proofs of those theorems.

Learn about the topologists sine curve. It will help appreciate why path connectivity is needed and how it is used. If you ever get to algebraic topology, it will help you appreciate why Čech cohomology is needed and how it is used.

Learn about space filling curves. It will help you appreciate the various hypotheses that analysists use for paths: smooth; piecewise smooth; Lipschitz; Hölder. It will help you appreciate the Lipschitz hypothesis in the existence/uniqueness theorem for solutions of ODE's, and to appreciate how that hypothesis is used to prove existence and uniqueness of the solution curves.

Learn about the Jordan Curve Theorem. It sounds like you have only seen a terrible early $$20^{\text{th}}$$ century proof. Learn to love that terrible proof. Then you will appreciate all the more the slick and beautiful proof based on homology theory. And again, as you get into algebraic topology, you will learn to appreciate the vast and beautiful generalizations of the Jordan Curve Theorem using algebraic topology: the Jordan separation theorem in higher dimensions; the Alexander Duality Theorem.

And besides the "pathologies" you mentioned, let me throw in a few more (I cannot help myself). Definitely learn to love wild knots and the Alexander horned sphere. They will teach you so many things about topology: an appreciation of the fundamental group; of Van Kampen's theorem; of the distinction between finitely generated and infinitely generated fundamental groups; of the Schönflies Theorem (and, after that, the importance of the Schönflies theorem in the classification of surfaces); the generalized Schönflies Theorem and its beautiful proof by Morton Brown and Barry Mazur.

Okay, I'll stop now.

• (Ah, I have seen the homological proof for JCT and that is definitely preferable! But I needed examples for the question.) I dunno, it's one perspective, but I'm entirely motivated by topology as it applies to physics. The uses in physics (even for function spaces and the like) are constrained. It is almost certainly not going to be useful to have theorems that content with the existence of e.g. the horned sphere. I want to figure out which parts of topology are physical, and I posted this hoping someone had done it for me already :p Commented Jun 26 at 20:53

I think this is a good and natural question, that nevertheless doesn't really have an answer.

With most subjects in math, as you learn the theory, you learn a stable of examples that you can apply the theory to, to help motivate the material and see it in action. In point-set topology, you learn a lot of examples, but they don't really play that same role. The examples often are weird spaces that no one really cares about for their own sake. The purpose of these examples is to help you understand the definitions, what implies what, and why for example connectedness and path-connectedness aren't the same notion.

In this sense, point-set topology is a weird subject, and I can see why you're asking, essentially, why not study a different class of objects that's not as weird? For some purposes, we do, but in general we need topology in all its weirdness. To take an example from my field (analysis), one often needs to think about spaces of functions. You can do a lot in this direction using metric spaces (which have fewer pathologies than general topological spaces) but you can't do everything, because some important function spaces aren't metric spaces. That's one example, but we could find others from almost any branch of math.

Why does the category of topological spaces come up so often? One answer might be, it's the most natural setting in which to study continuity, and continuity is one of the most fundamental ideas in math. But despite the importance of the concept, continuous functions can be very weird. That's just life.

It depends on what you mean by continuous curves. The ingenious and general $$\delta$$-$$\epsilon$$ definition may be the most general definition and therefore includes non-intuitive "pathological" curves. With stronger definitions, there will be less "pathology". For example, it is not counterintuitive to claim that the curves in Jordan curve theorem must be both continuous and rectifiable, which should simplify the proof.

In Lipschitz spaces are all morphisms, Lipschitz continuous functions, rectifiable. So considering $$\mathbb R^2$$ as a Lipschitz space should make the proof of Jordan curve theorem more easy to prove.

The Cantor function and the Peano curve are not Lipschitz continuous.

• Oh that sounds great. Commented Jun 26 at 23:25

Well, I mean in a way, yes? Take a look at this lecture of Frederic P schuller where he defines Space time as "Space time is a four dimensional topological manifold with a smooth atlas carrying a torsion free connection compatible with a Lorentzian metric and a time orientation satisfzing the Einstein equations"

From this, we can imagine that when we actually want to do any sort of physical modelling where pathological examples are of less interest, we must introduce a whole bunch of restrictions on the properties of the toplogy.

For example, even if we go and talk about something like Analysis, then we are dealing with topological spaces which can be induced from the inner product on a vector space, which is really a special case of a topological space and is no way shape or form a good picture on how a general topological space behaves.

For instance, even ideas such as having a unique limits for sequences do not exist for topological spaces. Consider the trivial topology over a set $$X$$ consisting of only itself and the empty set, in this topology all sequences converge to all points in the set.

This is phenomena would be something which would be highly senseless outside of when we speak about convergence in the context of topological spaces.

Another example I can give is from compactness, in analysis would mean that a set is closed and bounded. Bounded meaning that you it is contained entirely in some circle of some radius. This could be loosely thought to mean a set is "not too big".

If we were to equip a set with the power set topology, then it would be trivially so that every set is compact. This breaks the familiar intuition.

Adding additional conditions on the space like Hausdorf, locally compact reduces the pathological cases, and that's why we typically only work with spaces with those properties and even are intereeted in things like compactifications.