Cantor set on the circle Draw a Cantor set C on the circle and consider the set A of all the chords between points of C. Prove that A is compact.

$C$ is compact as it's closed and bounded.  Then, $A$ is compact as it's the image of the compact set $C\times C\times [0,1]$ under the continuous map $\phi: {\Bbb R}^2\times {\Bbb R}^2\times [0,1]\to {\Bbb R}^2$ given by $\phi(x,y,\lambda)= \lambda x + (1-\lambda )y$.
