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

The union R = P∩Q is a closed set. Would it help if we assumed Q to be connected? Then the union S of all components of P with points common with Q is a connected set. Is S a component of R?

I think I have found a proof:

If {x_n} is a sequence in the union of the components which converges to some point x, then x is in P since P is closed. For each n choose a point y_n in Q which lies in the same component as x_n. Since Q is compact there is a subsequence {y_n_i} which converges to some point y in the intersection of P and Q. Suppose x and y lie in different components of P, then there exists a number c>0 such that every sequence of points joining the points a and b in the set P has a characteristic number greater than or equal to c (this is proven on page 30 of the book by Saks and Zygmund mentioned above: "By the characteristic number of a finite sequence a_1,a_2,...,a_n of points of the plane we shall mean the largest of the distances
d(a_k, a_(k+1)) between consecutive points of this sequence. If all points of a finite sequence a=a_1,a_2,...,a_n=b belong to a certain set A, we say that this sequence joins the points a and b in A") But one can derive a contradiction since a closed set in the plane is connected if and only if for every c>0 it is possible for every two points a and b of this set to be joined in it by a finite sequence of points with characteristic number < c (Cantor's condition) (page 25 of the book) and for each i, x_n_i and y_n_i lie in the same component so, for every c>0, a finite sequence of points in P with characteristic < c and joining x and y can be found