# Compactness

• Aug 15th 2007, 10:55 AM
shilz222
Compactness
Assume $M$ is compact, non-empty, perfect, and homeomorphic to its Cartesian square, $M \simeq M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert Cube, or some combination of them?

Hints are appreciated.

Thanks
• Aug 15th 2007, 12:43 PM
shilz222
I think that $M$ must be homeomorphic to a combination of the two.
• Aug 15th 2007, 08:15 PM
shilz222
Is this a hard problem in general?
• Aug 15th 2007, 09:12 PM
JakeD
Quote:

Originally Posted by shilz222
Is this a hard problem in general?

It demands some level of mathematical sophistication.

If you want to try out the proof of the answer you gave, I think there are people here who will be able to critique it.
• Aug 16th 2007, 07:00 AM
topsquark
To be honest, I know I'm curious as to how shilz222 is going to combine the Cantor set and the Hilbert cube...

-Dan