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Is there any efficient trick (besides doing exercises) to develop intuition in topology?

The question is general but i would like to add my view of things.

I started to teach myself topology through several books a couple of months ago. I already passed the point of being overwhelmed by the amount definitions. Most of them I remember although sometime i check to remind myself (I have, after all, a terrible memory).

The point is most of the theorems and exercises I prove don't sink in and usually whenever I'm given a statement to prove I start with the definitions and work up from there. My feeling is that it's part of the nature of the subject. Browsing through Counterexamples in Topology really makes my head turn (all the one way implications... what ever happened to "if and only if"?)

This is in contrast to when I’m doing problems in analysis where i have a visual picture which tells me usually straight away if a given statement is true or false even before i start proving it.

I think that time here is a key element and intuition will inevitably develop at some point and so my question is:

Is there any efficient way to develop intuition in topology?

By intuition I mean a mental model that helps you see things more clearly for example: If you’re given a space with certain properties (say first countable, countably compact hausdorff space) than your intuition tells you it has to have some other properties ($T_3$ in this case).

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There are lots of iffs in topology (e.g., in metrizable spaces separability, second countability, and the Lindelöf property are equivalent). But when you investigate the structure of spaces in great detail, you inevitably get hierarchies of properties, and when you have true hierarchies, you have ifs that aren’t iffs. –  Brian M. Scott Nov 21 '13 at 23:43
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I usually prefer memorizing some lemmas in if-form in contrast to the if and only if-form. For example, in locally compact Hausdorff spaces a subset is locally compact iff it's the intersections of a closed and an open set. Now for some people locally compact includes Hausdorff, so they'll always have an "iff", but this camouflages the fact that Hausdorff gives you the "only if" implication while locally compact gives the "if". On the other hand, we should seek to prove that the reverse implication is also true whenever possible, but without having to add any hypotheses. –  Stefan Hamcke Nov 22 '13 at 0:36
    
Maybe you're just an analyst at heart! ;) –  Grumpy Parsnip Nov 23 '13 at 23:23
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@GrumpyParsnip Don't mourn me just yet. I still have a few years ahead of me... –  Saal Hardali Nov 23 '13 at 23:52
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7 Answers

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In my opinion, one of the best ways of developing intuition in topology is to study other branches of mathematics in which there are topological spaces. Many of these definitions, properties and theorems were imagined by people who were working in related branches of math, mostly analysis and geometry. These people stumbled upon spaces which had remarkable or singular properties, so they studied these properties. The examples came first.

Why does anyone care about compactness, say? The best way to answer this question is to ask the question: who were the first people to care about compactness, and why did they? What kind of spaces were they working with?

If you want to learn a new language, there is no point in reading the dictionary and the thesaurus. There is also not much point in learning all of the bizarre exceptions before you encounter them naturally. Instead, you should learn a few basic principles, and then go out and talk to people. Figure out how they speak, and refer to the thesaurus as you go along.

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There is not much point ...: Unless what you’re interested in are the properties and the exceptions, irrespective of where they came from and why. And I don’t think that looking at other areas is a very good way to develop specifically topological intuition. It’s a good way to develop intuition for how some topological properties are useful in, e.g., real analysis, which is really not the same thing. –  Brian M. Scott Nov 22 '13 at 0:14
    
Dear @Brian: You are right, of course. I tailored my answer according to what I gathered was Shaul's current situation. And I did include a boldface disclaimer. That being said, I personally am unable to develop anything even approaching intuition about something for which I have no context. I'd be curious to read your own answer to this question! –  Bruno Joyal Nov 22 '13 at 0:24
    
I probably won’t write one: it’s not very helpful to say that it came naturally, admittedly partly because I was having so much fun that I spent a great deal of time on it. (I took a substantial Moore method course in abstract general topology for its own sake, i.e., divorced from any other branch, as a freshman, and loved it from the beginning.) –  Brian M. Scott Nov 22 '13 at 0:31
    
@BrunoJoyal: Thanks! It's a nice answer. Although i like topology there are things that interests me more and I'm trying to get a feel for it mainly because I feel it's the basics and I should get them down first. Your answer gave me a bit of perspective. I do have a tendency to dig deep into things and not be satisfied until i get the bigger picture and "really" understand them. I guess sometimes you do need to "lift your head up to see the forest for the trees". I'm gonna leave the question open to more opinions. Thanks! –  Saal Hardali Nov 22 '13 at 8:59
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Read "Explorations in Topology" by David Gay. I'm too in search for understanding in topology and this was recommended by my adviser (topologist). So far the book is more than expected. It's amazing.

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This is both a difficult and interesting question (for instance because it is difficult to say what intuition really is, among other things), and the only thing that I can offer is merely my opinion. I should like to point out from the beginning that I do not think that there is a very systematic way of obtaining intuition in general, only that there are certain ways to facilitate getting intuition. First of all, in order to be able to develop some intuition at all, one should know the basic examples and counterexamples of the subject (this is particularly important in topology); these are very important in the sense that they allow you to check 'bad' intuition against 'good' intuition, and to get rid of the former. I would like to emphasize that one should make sure that one has mastered the elementary formal apparatus (such as the concepts of initial and final topologies), and that in cases of conflict between intuition and logic, the latter must take precedence.

The concepts of initial and final topologies are extremely important, and it seems to me that it is convenient, say, not to think about the product (subspace, disjoint union, quotient and whatever) topology in terms of basis elements but in terms of its properties, in particular the defining property that it is the weakest topology making the projection maps continuous (the initial topology with respect to the projections). As for your bad memory, for me it is much easier to remember commutative diagrams, and they certainly won't hurt you (I promise); they will lead to a much more conceptual understanding. As Bruno, I too think that is very important to consider the applications to other fields of mathematics; functional analysis, say. There initial topologies (or weak topologies, as they are traditionally called in this field) arise in the most natural fashion conceivable; final topologies arise in the theory of distributions, the topology on the linear space $\mathscr{D}$ of test functions is given by a special kind of a final topology (a so called LF topology).

And I feel that intuition and logic certainly need not be in conflict, and that such a conflict is probably a sign of faulty intuition (there are of course exceptions, and in some cases I even think that there is no point in following one's intuition), provided the logic is flawless. I should also say something about pictures, as they often lead us into temptation. Sometimes they are helpful and sometimes they are not; I think that they should not be taken too seriously altogether, they may stimulate intuition but I don't think that they can guide you (me neither).

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I found this idea of a "Lattice of Topologies" and it really helps me to visualize the relationships between different properties. I now think of different topologies on set as living on this lattice somewhere.

This MO question is where I found the term. The question is interesting and helpful in it's own right.

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Though I don't have any authority on this specific subject matter, I would say the thought of looking for a "trick" or "shortcut" for anything is quite dangerous. Just put the head down and work laboriously is the best approach to everything. "A couple of months" is obviously extremely short time. To really gain any basic understanding about anything you would take at least 3 years or even more. To master one? Probably 10 years or more. In everything I learned, "exercises" are probably exactly the thing that helped me gain "intuition", if any. If you always think for "shortcut" or "trick" you can very well be unintentionally slowing yourself down by distracting yourself too much. This is just a reminder; hopefully I won't be bashed too much for this unanswerlike answer :P

I also don't agree with Joyal's perspective on learning languages. Of course you go and talk to people. But in fact mastering the usage of words and special instances through enough exercises alone will immensely speed up the process. That's "achieve multiple times of effect in half the time" in Chinese. If you don't have a decent command of quite a few words and phrases you're never gonna understand anything no matter how much you "talk". That's why paying unbalanced attention at any of those both aspects will lead to disasters.

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The way I developed intuition in topology is from advanced multivariable calculus. All the concepts of "open", "closed", "compact", "convergence", "connected", and so forth, have a very visual meaning behind them when you study these concepts from advanced calculus. So when you see the abstract definitions in topology they are similar to how it comes up in advanced calculus.

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I'm not sure how useful this approach is for anything but the most bog-standard of spaces (Haussdorf second countable manifolds I guess). –  Daniel Rust Nov 24 '13 at 22:53
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The best apparatus for use in investigating topology are the definitions of various terms in the statement of a theorem, previous results, and the power to discover their natural logical connections. To gain intuition is simply a dream that will never come true.

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