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would like to know what a homeomorphism means ( how do you geometrically visualize it?)

AND is the symbol 8 homeomorphic to the symbol X? Why or why not?

( from whatever little i know intuitively about homeomorphisms, i think it is not....)

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- Thread starter mansi
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- #1

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would like to know what a homeomorphism means ( how do you geometrically visualize it?)

AND is the symbol 8 homeomorphic to the symbol X? Why or why not?

( from whatever little i know intuitively about homeomorphisms, i think it is not....)

- #2

mathwonk

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So yuo need a continuous, reversable transfomation from on space to the other, that is, if you can "convert one space to the other" by doing reversable maps then the two spaces are homeomorphic.

So as you can map the surface of a sphere with apoint missing onto a plane, the two spaces are homeomorphic.

It is often easier to see when two spaces are not homeomorphic, and as mathwonk uses in your example of 8 and X, conectivness is a good way to see this.

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Hurkyl

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And, of course, there's the classic joke: a topologist is someone who can't tell the difference between a doughnut and a coffee cup! (Exercise: imagine how to deform a doughnut into a coffee cup)

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mathwonk

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In 3 dimensional space, you have to remove a plane to lose connectedness. So your left with showing that a plane can't be the homeomorphic picture (is that the right term in English?) of a line.

But we already did that...

I guess the famous joke about the coffee pot is not entirely untrue... :rofl:

Just had the final exam in topology this morning. Went pretty well.

By the way, one of the questions there where to show that 0,1,4,6,8 are not homeomorphic to one another. (That is, when drawn like my lecturer drew them).

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mathwonk

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good idea on discnnecting the plane and three sp[ace. that is correct of course, in essence, but not in detail. i.e. if a line disconnects the plane then you have to prove no homeomorphic image of a line can disconnect three space. it is not sufficient to prove a line is not homeomorphic to a plane for this.

the problem is with your assertion that it requies a plane to disconnect three space. this is not true. a sphere disconnects three space. what you have to prove rather is that nothing homeomorphic toa line can disconnect three space. this is easier in the previous case because the only thing homeomorphic toa point is a point, but there ise a huge variety of things homeomorphic to a line.

by the way i do not know an elementary proof of this. if one knows the construction of "homology" groups, and the basic tools for calculating them, it can be shown using them that the 2 sphere and the three sphere are not homeomorphic. it follows that a plane and three space are not homeomorphic, since their one point compactifications are those spheres.

the problem is with your assertion that it requies a plane to disconnect three space. this is not true. a sphere disconnects three space. what you have to prove rather is that nothing homeomorphic toa line can disconnect three space. this is easier in the previous case because the only thing homeomorphic toa point is a point, but there ise a huge variety of things homeomorphic to a line.

by the way i do not know an elementary proof of this. if one knows the construction of "homology" groups, and the basic tools for calculating them, it can be shown using them that the 2 sphere and the three sphere are not homeomorphic. it follows that a plane and three space are not homeomorphic, since their one point compactifications are those spheres.

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I'd like to take the more advanced topo. course when someone decides to give it here. My T.A. told me some of the things you just mentioned, but in a very "wavy" way... I don't have the proper tools yet.

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mathwonk

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Heard of it. We mentioned it in the course. But only mentioned.

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mathwonk

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Hint: use polar cordinates. This can also be proved using calculus, if you give yourself a smooth, i.e. continuously differentiable map. Then it all follows from the existence of the angle form, "dtheta", and integration i.e. polar coordinates again.

this generalizes to 2 dimensions, the old intermediate value theorem, that a continuous map from the closed unit interval to the line, which sends the end points 0 and 1, to numbers of opposite sign, must send some point of the open unit interval to zero.

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