Homeomorphism of Unit Circle and XxX Product Space

hedipaldi
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Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane
 
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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.
 
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You could also look at the fundamental groups.
 

Thread '2-sphere intrinsic definition by gluing disks' boundaries'

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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