Exercise 7. Show that if C(0) (the punctured plane) and U C are conf.pdf

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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.

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

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