pi-base
Space Property Suggestion: The Hilbert Cube is homogeneous
(not sure if this is a "theorem" in the same way as the properties=>properties theorems)
Theorem Suggestion
The Hilbert Cube S32 is homogeneous P86.
Proof/References
The result is nontrivial and most proofs don't generalize well to properties=>properties theorems. Also, any way to prove a more general statement for infinite-dimensional compact convex subsets of \ell^2 are useless, as these are all homeomorphic to the Hilbert Cube anyway.
There is a nice discussion on MSE and in particular a link to the elementary proof The Homogeneous Property of the Hilbert Cube, by Denise M. Halverson, David G. Wright, 2012.
Great suggestion; I've been meaning to add this for a while since it would give an example of a nontrivial homogeneous space with the fixed point property. (The real projective plane would be another nice one to add, but that's worthy of a separate issue.)
