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Are all Platonic solids homeormorphic to one another? Since they all have the same Euler characteristicIt 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...
Please can someone give me a good and easy explanation of it?
Thank you
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.
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