Exercise 13.  Let  S  be the open set consisting of all points z such that   or .   Show that  S  is not connected.  

Solution 13.

See text and/or instructor's solution manual.

    Let      be any curve joining    and  .  

Then    and  .  

By the Intermediate Value Theorem, there is some    such that   .  

But this means    is on the imaginary axis and is not in the set  S.     

Therefore  S  is not connected.  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell