1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 47
- Let S = { z1, z2, ... , Zn} be a finite set of points. Show that S is a bounded set.
- Prove that the boundary of the neighborhood D.(zo) is the circle C,(zo).
- Let S be the open set consisting of all points z such that Jz + 21 < 1 or Jz - 21 < 1.
Show that S is not connected.
- Prove that the only accumulation point of { ~ : n = 1, 2, ... } is the point O.
- Regarding the relation between closed sets and accumulation points,
(a) prove that if a set is closed, then it contains all its accumulations points.
(b) prove that if a set contains all its accumulation points, t hen it is closed.
- Prove that D1 (0) is the set of accumulation points of
(a) the set D 1 (0).
(b) the set Di (0).
17. Memorize and be prepared to illustrate all the terms in bold in t his section.
