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A function $f: X \to Y$ between two topological spaces $(X, T_X)$ and $(Y, T_Y)$ is called a homeomorphism if it has the following properties:

  • $f$ is a bijection (one-to-one and onto)
  • $f$ is continuous
  • the inverse function $f^{-1}$ is continuous ($f$ is an open mapping).

Is homeomorphism a formal definition of stretching and bending of geometrical objects? How can we make sure whether this is the right definition?

Maybe I should ask a different question - why is topology defined as "an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending, but not tearing or gluing."

I belive we cannot prove this statement because the notions of stretching and bendings are informal and relies on our intuitive, informal reasoning.

There are homeomorphisms that cannot be considered as stretching and bending (don't know what the proof of this should look like, as we haven't defined stretching and bending formally, this is my major concern here) - see the bottom of page 22 in "A guide of topology".

Even if we accept that some homeomorphisms cannot be understood as stretching and tearing (whatever stretching and tearing means), we assume that all stretching and tearings that can be represented by homeomorphism - why is that? Has anyone proved it?

These are my major concerns with regard to topology. I've searched through many books about topology written by Munkres (Topology), Janich (Topology), Sidney Morris (Topology without tears), Armstrong (Basic topology). None of those titles contained an explanation of WHY topology is what it is today. They only explain what it is.

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    $\begingroup$ As you say : "the notions of stretching and bending [are] informal and relies on our intuitive, informal reasoning. [...] we haven't defined stretching and bending formally" and thus it is difficult to prove that they are "equivalent" to the precise notion of homeomorphism between topological spaces. The mathematical notions are "abstarctions": the best we can do is try to "elucidate" the intuitive notions, i.e. analyze them deriving from our intuition about them some properties that we agree must be "intrinsic" to our intuition of stretching and bending". 1/2 $\endgroup$ – Mauro ALLEGRANZA Feb 22 '15 at 11:16
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    $\begingroup$ Then we may "compare" them with the properties that topology formally derives form the basic definitions and axioms and see it they "match"... 2/2 $\endgroup$ – Mauro ALLEGRANZA Feb 22 '15 at 11:17
  • $\begingroup$ But there's no guarantee the list of properties we've chosen as intrinsic to our intuition of stretching and bending is complete. Then, even if they will match in billion cases, the billion+1'th case can turn out to be clearly wrong, against our intutition. This is the problem with making the intuitive notions formal. Suppose I changed the definition of homeomorphism a little. Then it would be possible to find an example where certain bending/stretching cannot be represented as homeomorphism. What if our current definiton is similarly flawed? What do you think? $\endgroup$ – user216094 Feb 22 '15 at 13:06
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    $\begingroup$ There's one another concern here. We don't do mathematics to solve real world problems. We take a real-world problem, take the intuitive understanding of the problem, provide a formal definition of the problem in mathematical terms, solve the problem using the appliciable theory, and transform the formal solution back to a real-world, intuitive solution. Now, tis transition to real-world may be problematic, if the formal definitions do not perfectly match our intuition (and we know this is the case here, moreover, we cannot fix it). $\endgroup$ – user216094 Feb 22 '15 at 21:37
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    $\begingroup$ Why do you take the phrase you've quoted as the formal definition of topology? That may be how topologists present their subject to laypeople, but I doubt that's how any topologist would talk about the subject internally. $\endgroup$ – Steven Stadnicki Feb 23 '15 at 16:11
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I’d say just leave it as an informal, but powerful way to intuitively reason about topology.

In an analogous situation, computability and decidability of problems can be defined mathematically, but we never can prove mathematically that these definitions really grasp our intuitive notion of what can be computed. The Church–Turing thesis is a meta-mathematical thesis which says that those definitions really do grasp our intuitive notion of computability.

If viewed as a philosophical statement about the nature of our universe, there really is strong evidence in favor of the Church–Turing thesis: For one, all attempts are formalizing computability have been shown to be equivalent. And on the other hand, as one develops a sense for what is Turing–computable, one finds that anything that looks intuitively computable also looks Turing-computable and vice versa: The intuition for the mathematical concept of computability starts to overlap with the pre-mathematical intuition of computability.

And this is the point of my answer: I think with topology, it’s much the same – only that the thesis in question doesn’t have a name (and is known to be slightly false). If you just start doing topology, you will most probably find that your intuition for bending and stretching and your intuitions for homeomorphisms start to overlap a lot. I think this is the best justification for the cited non-mathematical description of topology as a field.


And now, to address your concerns regarding this:

And as you said, the thesis doesn’t work out completely and I share your concerns about using it to prove stuff. Luckily, I found that most of the stuff I have seen proved using visual bending–stretching arguments can really be proven rigorously using real mathematics. This really helped me a lot to accept the attitude. Maybe this will help you as well.

So my advice to you is to not lose to much sleep over it: Don’t think of it as a definition of the field, but rather as a description and don’t expect stretching and bending to be defined formally.

You may either regard topology as mathematics which formalizes reasoning about things like bending and stretching, or you may regard things like bending and stretching as intuitive notions which help you reason about topology.

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  • $\begingroup$ It's acceptable that homeomorphism occurs even if we cannot think of it in terms of stretching/bending in certain situations. But we cannot tell whether every stretching/bending can be represented by homeomorphism. We've chosen topology to look what it looks like today, I believe the formulation of basic topology definitions was largery based on intuition and informal reasoning. I'd like to see a book/article explaining the development of topology and arguments used in formulating topology in this and not some other way. But they were all informal I'm afraid. $\endgroup$ – user216094 Feb 22 '15 at 13:11
  • $\begingroup$ But I agree with everything you said. The most popular statement is that a coffee mug is homeomorphic to a doughnut, meaning one can be obtained from the other by stretching and bending operations. This example should never be used as an intuitive demostration of homeomorphism for beginners, because one may think every homeomorphism intuitively can be thought of as stretching+bending, which is false. After we've defined a mathematical notion, we should avoid intuitive explanations, as they can be misleading! $\endgroup$ – user216094 Feb 22 '15 at 17:06
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The methodology of mathematics begins with examine some concept by means of presenting formalizations which satisfy the conflicting constraints of (1) faithfully representing the intuition behind the concept and (2) being amenable to mathematical analysis. We judge our formalizations both by these two criteria and by the richness of the pure mathematics that develops out of them.

Topology arises historically as an attempt to understand the notion of continuity, as developed primarily in 19th century analysis. By Cauchy's era continuous real value functions had been defined as we define them today in terms of epsilons and deltas. From here one gets metric space, and their abstraction to topological spaces. The intuition of homeomorphism as stretching and bending comes really from an even more special chintzy than metric spaces, namely manifolds. In this context there's little room for doubt that stretching and bendings can be represented by homeomorphisms-it's absolutely part of our intuition that these map nearby points to nearby points.

It seems to me the richer question is of whether general homeomorphisms have anything to do with stretching and bending. The answer is surely no, so the notion of topology has to defend itself on other grounds. In fact there's not a complete consensus that topological spaces are the right abstraction of continuity-various smaller and later classes of spaces have been proposed. But most mathematicians don't seem much concerned with finding the best possible abstraction of a concept. Topology successfully models the theories of manifolds, algebraic varieties, function spaces, and other important topics, so people learn and use it. Perhaps this attitude is too pragmatic for you, in which case there are plenty of alternatives to investigate, from convergence spaces to toposes.

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I study topology that semester and these questions came to me too. If X,Y are topological spaces and f:X→Y continuous then it can be shown easily that the graph of the function, Gf={(x,f(x):x∈X} with the product topology, is homeomorphic with the domain of f, X. If you see that for functions from real numbers to real numbers (with the ordinary topology) you get a geometric result which is very ok with the intuition about streching. The familiar graphs of functions (of 1 variable) from calculus are homeomorhpic with the real number line. If you ''strech'' the curve of the graph you get a line. Similarly, for function from R^2 to R , you get the image of streching a wrinkled sheet to get a smooth surface. You can also see, why the curve of the circle is not homeomorhpic with the real number line, because if you stretch the circle from two points of it you will get a knot somewhere,(not sure about the last).

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One should be a little wary of thinking too much about stretching and bending because in the real world these are continuous processes (there are lots of steps partway through the process) whereas in a homeomorphism there needn’t be steps in between. For example consider two interlocking (but not touching) rings (circles) in Euclidean 3-space. This subspace is homeomorphic to two separate rings. However you can’t stretch and bend the rings apart. Thus we see that the notion of stretching and bending is a bit wishy-washy and doesn’t work quite like topology. It’s also useless in some weird topologies like the cofinite topology. However that doesn’t mean it’s not useful. Just look at lots of examples and allow the idea to guide your intuition. Two spaces are essentially equal if they are homeomorphic so another way of thinking about a homeomorphism is that it matches up points in one space to points in the other, and does so continuously so patches in one space correspond to patches in the other, and the way patches connect to each other in one space is the same as the way they connect to each other in the other. But I don’t think this is so helpful.

Another notion of stretching and bending in topology is called a homotopy. It’s a way of saying that two maps from $X$ to $Y$ can be continuously transformed into one another. It allows one to define the notion of homotopy equivalence which is bit like stretching and bending but more permissive than homeomorphism.

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