Past exam of the metric spaces and complex analysis course of the University of Oxford / 2015 / 1 / c / iii

The unit circle is the inverse image of the closed set $\{1\}$ under the continuous map $(x,y)\mapsto x^2+y^2$, so it is closed. It is also bounded, since every point has Euclidean norm one. The Heine-Borel theorem gives compactness.