Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class �5 (Friday 01/23/98)

SUMMARY Point Sets in the Complex Plane CURRENT READING Brown & Churchill, pages 23-25 NEXT READING Brown & Churchill, pages 26-33

Any collection of points in the complex plane is called a two-dimensional point set, and each point is called a member or element of the set. Here are some fundamental de�nitions describing point sets. \underline{De�nitions} NEIGHBORHOOD A delta or δ neighborhood of a point z_{0} is the set of all points z such that |z − z_{0}| < δ where δ is any given positive (real) number. DELETED NEIGHBORHOOD A deleted δ neighborhood of z_{0} is a neighborhood of z_{0} in which the point z_{0} is omitted, i.e. 0 < |z − z_{0}| < δ LIMIT POINT A point z_{0} is called a limit point, cluster point or a point of accumulation of a point set S if every deleted δ neighborhood of z_{0} contains points of S. Since δ can be any positive number, it follows that S must have in�nitely many points. Note that z_{0} may or may not belong to the set S. INTERIOR POINT A point z_{0} is called an interior point of a set S if we can �nd a neighborhood of z_{0} all of whose points belong to S. BOUNDARY POINT If every δ neighborhood of z_{0} conrains points belonging to S and also points not belonging to S, then z_{0} is called a boundary point. EXTERIOR POINT If a point is not a an interior point or a boundary point of S then it is called an exterior point of S. OPEN SET An open set is a set which consists only of interior points. For example, the set of points |z| < 1 is an open set. CLOSED SET A set S is said to be closed if every limit point of S belongs to S, i.e. if S contains all of its limit points. For example, the set of all points z such that |z| ≤ 1 is a closed set. BOUNDED SET A set S is called bounded if we can �nd a constant M such that z < M for every point in S. An unbounded set is one which is not bounded. A set which is both closed and bounded is sometimes called compact. CONNECTED SET An open set S is said to be connected if any two points of the set can be joined by a path consisting of straight line segments (i.e. a polygonal path) all points which are in S. DOMAIN or OPEN REGION An open connected set is called an open region or domain.

CLOSURE If to a set S we add all the limit points of S, the new set is called the closure of S and is a closed set. CLOSED REGION The closure of an open region or domain is called a closed region. REGION If to an open region we add some, all or none of its limit points we obtain a set called a region. If all the limit points are added the region is closed; if none are added the region is open. Usually if the word region is used without qualifying it with an adjective, it is referring to an open region or domain.

Examples

Consider the following point sets and, using as many of the previous de�nitions as you can, fully describe these examples.