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I attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of algebraic topology is to classify surfaces in a way that it is easy to tell whether or not surfaces are homeomorphic to each other. I was just wondering now, why are homeomorphisms important? Why is it so important to find out whether two surfaces are homeomorphic to each other or not?

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    $\begingroup$ Topological properties consist of how spaces are connected. For example on a donut, you can go in one direction and end up back where you are. If my donut is big, small, squiggly or randomly pulled out it will not effect this property. Hence a notion of equivalence that doesn't take into account shape or length is good for topologists. $\endgroup$ – Ali Caglayan Oct 2 '16 at 20:07
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    $\begingroup$ It's not important. It is interesting. $\endgroup$ – user98602 Oct 2 '16 at 20:15
  • $\begingroup$ Did the topologist define homeomorphism formally, and if so, was the definition in terms of continuous functions (or continuous 1-1 correspondence) between two surfaces, or in terms of each surface being equipped with some kind of "topological structure? $\endgroup$ – zyx Oct 3 '16 at 16:15
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    $\begingroup$ The definition was a continuous bijective function with a continuous inverse between surfaces. $\endgroup$ – lithium123 Oct 3 '16 at 23:50
  • $\begingroup$ In mathematics, often objects are considered stripped of any specific properties except for a limited set of properties that one wants to study. Thus, such objects can look - naively spoken - very different, but with respect to this constrained set of properties, they are indistinguishable. Typically maps from one object to another which preserve (in one direction) such properties are called homomorphisms. Homeomorphisms are bidirectional, and apply specifically to the property of continuity in topological spaces. The general principles of such mappings are studied in category theory. $\endgroup$ – Captain Emacs Oct 4 '16 at 9:50
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Homeomorphisms are important because they are instances of a more general idea: structure-preserving isomorphisms. You will learn to appreciate this idea as you study more advanced mathematics.

In many domains of mathematical inquiry, the objects of study carry important kinds of "structure," and we don't care to distinguish two objects so long as they have the same structure. We can make the notion of "having the same structure" precise by saying two objects X and Y have the same structure precisely when there is a bijection between them that "preserves the structure" (in a sense that can also be made precise).

Homeomorphisms are precisely those functions for topology. Their cousins are group isomorphisms in group theory, and ring isomorphisms in ring theory, bijective linear transformations in vector space theory, etc.

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    $\begingroup$ So essentially what you're saying is that, in topology, we wish to study surfaces that share common topological properties (connectedness, compactness separability etc.). Surfaces that share all these topological properties can be regarded as "the same" (homeomorphic) and be grouped together. Thus by studying the properties of one member of the group linked together by homeomorphism, we can extend our discoveries to other surfaces in the same group and conclude that they share all the other topological properties. Is my knowledge so far accurate? $\endgroup$ – lithium123 Oct 2 '16 at 20:26
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    $\begingroup$ @lithium123 As you can tell from theh upvotes on your comment, that's absolutely correct and very well-put. $\endgroup$ – 6005 Oct 3 '16 at 1:12
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    $\begingroup$ Dear @lithium123: Have you learned all these topological properties in high school?(just curious).Amazing,bravo! $\endgroup$ – Arpit Kansal Oct 3 '16 at 4:37
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    $\begingroup$ @robertbristow-johnson: with appropriate technical hypotheses, yes, you can view the Fourier transform as a homeomorphism on $L^2(\mathbb{R})$, where the topology is the one induced by the $L^2$ norm. this follows immediately from the fact that the Fourier transform is unitary, hence an isometry, hence a homeomorphism. $\endgroup$ – symplectomorphic Oct 3 '16 at 5:31
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    $\begingroup$ @lithium123: what you say is basically right, but (1) algebraic topology doesn't just study "surfaces" -- that word implies the space is two-dimensional, but topology studies spaces of higher dimension, too (if you think classifying two-dimensional spaces is tough, do a little reading about the wild world of four-dimensional manifolds!), and (2) the notion of homeomorphism is logically prior to the notion of a topological property. we first define a homeomorphism to be a bijection that respects topological structure (which is defined in terms of "open sets"), and then (contd.) $\endgroup$ – symplectomorphic Oct 3 '16 at 5:45
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The notion of homeomorphism is of fundamental importance in topology because it is the correct way to think of equality of topological spaces. That is, if two spaces are homeomorphic, then they are indistinguishable in the sense that they have exactly the same topological properties.

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  • $\begingroup$ Thanks! What exact topological properties are there? Can you give me some examples? $\endgroup$ – lithium123 Oct 2 '16 at 20:06
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    $\begingroup$ @lithium123 Well, two intuitive ones are for example being connected (i.e.: just one piece) and being path connected (i.e.: every two points can be connceted by a curve). $\endgroup$ – Daniel Robert-Nicoud Oct 2 '16 at 20:09
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    $\begingroup$ Connectedness, compactness, separability, being second-countable, among several others. $\endgroup$ – Ivo Terek Oct 2 '16 at 20:10
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    $\begingroup$ I see. So knowing that two surfaces are homeomorphic let's us know a whole lot more other characteristics that these surfaces share? $\endgroup$ – lithium123 Oct 2 '16 at 20:11
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    $\begingroup$ A good example of a topological property is being simply connected. Imagine that you have a loop of string on your surface. Cut the loop, hold on one end of the string, and pull on the other. Will you get stuck, or can you pull it all the way through? If you are on the muffin (sphere) you can, but if you are on the dohnaught (torus) , a loop around the hole cannot be pulled all the way. $\endgroup$ – Theo Oct 2 '16 at 20:13
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If they are homeomorphic, they have the same topological properties. Topologically, they are the same, thus the joke that a topologist cannot tell apart the doughnut from the coffee mug. They are the same. Intuitively, you can transform a surface into another without making any tearings in the surface. If your doughnut is a muffin, without a hole, you cannot transform it into the coffee mug (or the doughnut) without making a hole, thus breaking the structure. The muffin is not homeomorphic to the coffee mug, that's why you should never have muffins with your coffee.

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Just to add to the already-good answers provided so far: Human beings in general, and scientists in particular, are classifiers. We group "like things" in buckets and try to make statements that must be true for everything in the bucket (e.g., how biologists classify kingdoms, phyla, species, etc.) In the case of topology, we can place spaces in buckets according to their homeomorphism type. We could, instead, use other classification schemes (e.g., homotopy type, which is a looser form of equivalence.)

Category theory provides a nice setting for making sense of this general, scientific process: When studying a certain class of objects, the fundamental (e.g., "important"/"interesting"/primary) goal is to classify the objects up to categorical isomorphism. And for the category of topological spaces and continuous maps, isomorphisms are homeomorphisms.

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The specific form of the 20th century definition of homeomorphism, with inverse pairs of continuous functions, is not necessarily important. It was not originally used for the classification of shapes of 2-dimensional surfaces, a problem solved in the middle of the 19th century without a precise definition of "surface" or (topological) "shape". Some other shape classification problems, such as determining which knots can be untied, have never used homeomorphism as the definition of equivalence.

What is more important than the particular definitions, is to understand, classify and inter-relate the shapes of geometric objects, and to have a theory of that can give clear formulations of all those problems and provide tools for solving them.

If "shape" is formalized in terms of the formal data used in topology (open sets, continuous maps, convergence, ...) then homeomorphism is by definition the notion of equivalence associated to that; a one-to-one correspondence that matches the data. When the lecturer equated the goal of studying shapes with the goal of studying spaces up to homeomorphism, that was the same as saying that people have settled on the theory of topological spaces as an adequate formalization of "shape".

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