The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.
2.
:( after edit conflict ) If by " circle " you mean the topological concept of a circle " up to homeomorphism ", also known as the 1-sphere " S " 1, and by " product " you mean the topological product, the concept of curvature simply does not apply.
3.
We start with a continuous function \ scriptstyle h from the Cantor space \ scriptstyle \ mathcal { C } onto the entire unit interval \ scriptstyle [ 0, \, 1 ] . ( The restriction of the Cantor function to the Cantor set is an example of such a function . ) From it, we get a continuous function \ scriptstyle H from the topological product \ scriptstyle \ mathcal { C } \; \ times \; \ mathcal { C } onto the entire unit square \ scriptstyle [ 0, \, 1 ] \; \ times \; [ 0, \, 1 ] by setting
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