# Platonian solids and homeomorphism

• May 27th 2011, 10:52 PM
rayman
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
• May 28th 2011, 01:14 AM
Tinyboss
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.
• May 28th 2011, 01:23 AM
rayman
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