Exercise 7 Show that if C0 the punctured plane and U C are c

Exercise 7. Show that if C/(0) (the punctured plane) and U C are conformally equivalent, then C/U has no interior, i.e. the complement of U contains no open sets.

Solution

Let P be the punctured plane and U be conformally equivalent.

Then the complements P\' (of P) and U\'( of U) will be topologically equivalent.

But P\' ={0} is a singleton...has no interior points..(hence contains no open sets)

So the same it true of U\'.

Hence the result