Prove that there is no continuous bijective map from the uni

Prove that there is no continuous bijective map from the unit circle in R^2 to the interval [0,1].

Solution

Assume on the contrary that there exists f: A (Unit Circle) -> B (=the interval [0,1]) , a continuous , bijective map.

Now f being a continuous map between compact Hausdorff spaces is a homeomorphism . PROOF: if f:X->Y is such a map, ,let Z be a closed subset of X. To show that f^-1 is continuous, enough to show (f^-1)^-1 (Z) = f(Z) is closed in Y for every closed subset Z of X. But Z closed in a compact space X implies Z is compact implies f(Z) is compact in Y , which in turn implies f(Z) is closed in Y , as Y is Hausdorff, so we are done.

But the unit circle and the interval [0,1] cannot be homeomorphic : removing a point disconnects [0,1], whereas A-{point} remains connected.

Conclusion: there does not exist f: A (Unit Circle) -> B (=the interval [0,1]), f continuous and bijective