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Math Help - Platonian solids and homeomorphism

  1. #1
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    Platonian solids and homeomorphism

    Are all Platonic solids homeormorphic to one another? Since they all have the same Euler characteristic  \chi=2 It is commonly known that if topological spaces have the same Eulers characteristic then they are homeomorphic but on the other hand this sometimes does not work....for example a point is not homeomorphic to a line even if their Euler characteristics are equal...
     \chi(point)=\chi(line)=1

    Please can someone give me a good and easy explanation of it?
    Thank you
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  2. #2
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    A point is homotopic to a line (also to any platonic solid or any simply connected space), but not homeomorphic.

    Each platonic solid is homeomorphic to D^3 (the closed unit "disk" in R^3) , and therefore to each other. To write the homeomorphism explicitly, think about rays from the origin.
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  3. #3
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    Well I am not an expert in topology, I study it on my own and try to understand some of its concepts. This example actually comes from my book...

    Unfortunately I have not studied homotopy yet...but I will try to process what you have just written
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