The following is an exercise from page 32 of the 1952 edition of the book Analytic Functions by Stanislaw Saks and Antoni Zygmund (it is freely available online at
matwbn.icm.edu.pl/kstresc.php?tom=28&wyd=10&jez=pl):

If E is the extended complex plane (with the topology induced by the chordal metric) and if P, Q are closed sets in E, then the union of those components of the set P which have points in common with Q is also a closed set.

Every closed set in the extended complex plane is compact and if a sequence of points in the union converges to a point x in P then, for each point in the sequence one can find a point in Q in the same component of P. Since Q is compact there exists a convergent subsequence which converges to a point y in the intersection of P and Q since both are closed. It seems that y and x should be in the same component of P. Can this be proven? If so, how? If not: is there another proof for the above?

Thank you