Homeomorphism of Unit Circle and XxX Product Space

[hedipaldi]

Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane

 

[jgens]

No. If there were such a homeomorphism, then there would be an open set U in X such that U x U is homeomorphic to an open arc in the 1-sphere. In other words U x U is homeomorphic to the real line. It should be simple enough to see that U is path-connected and an easy argument shows that U x U minus a point is still path-connected. On the other hand the real line minus a point is not path-connected. Contradiction.

 

[micromass]

You could also look at the fundamental groups.