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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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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. – Ali Caglayan 7 hours ago
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It's not important. It is interesting. – Mike Miller 7 hours ago
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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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It is perhaps worth noting that while homeomorphism may seem as being a geometric property, it is not. It deals with identifications which preserve continuous structures but not size and distance (not isometry) – Theo 7 hours ago
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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? – lithium123 7 hours ago
    
@lithium123 As you can tell from theh upvotes on your comment, that's absolutely correct and very well-put. – 6005 2 hours ago
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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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Thanks! What exact topological properties are there? Can you give me some examples? – lithium123 7 hours ago
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@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). – Daniel Robert-Nicoud 7 hours ago
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Connectedness, compactness, separability, being second-countable, among several others. – Ivo Terek 7 hours ago
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I see. So knowing that two surfaces are homeomorphic let's us know a whole lot more other characteristics that these surfaces share? – lithium123 7 hours ago
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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. – Theo 7 hours ago
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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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Actually, that applies to "homeomorphic" rather than "homomorphic" but you are probably right that that was what the OP meant. – user247327 7 hours ago
    
@user247327 I think that homeomorphic is the right word here (or in a broader sense isomorphic). I never encountered the word homomorphic in a context like this. – drhab 7 hours ago
    
Of course, typo on my part, I meant homeomorphic. I just noticed that my phone knows homomorphic but not homeomorphic (as autocomplete) – Theo 7 hours ago

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