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\centerline{\bf Class \#5 (Friday 01/23/98) }
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{\bf SUMMARY} Point Sets in the Complex Plane\\ 
{\bf CURRENT READING} Brown \& Churchill, pages 23-25\\
{\bf NEXT READING} Brown \& Churchill, pages 26-33\\
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Any collection of points in the complex plane is called a {\it two-dimensional} point set, and each point is called a {\it member} or {\it element} of the set.
Here are some fundamental definitions describing point sets.\\
\underline{\bf Definitions}\\
{\bf NEIGHBORHOOD}\\
A {\it delta} or $\delta$ {\it neighborhood} of a point $z_0$ is the set of all points $z$ such that $|z-z_0| < \delta$ where $\delta$ is any given positive (real) number.

{\bf DELETED NEIGHBORHOOD}\\
A {\it deleted } $\delta$ {\it neighborhood} of $z_0$ is a neighborhood of $z_0$ in which the point $z_0$ is omitted, i.e. $0 < |z-z_0| < \delta$

{\bf LIMIT POINT}\\
A point $z_0$ is called a {\it limit point, cluster point} or a {\it point of accumulation} of a point set $S$ if every deleted $\delta$ neighborhood of $z_0$ contains points of $S$.
Since $\delta$ can be any positive number, it follows that $S$ must have infinitely many points. Note that $z_0$ may or may not belong to the set $S$.

{\bf INTERIOR POINT}\\
A point $z_0$ is called an {\it interior point} of a set $S$ if we can find a neighborhood of $z_0$ all of whose points belong to $S$.


{\bf BOUNDARY POINT}\\
If every $\delta$ neighborhood of $z_0$ conrains points belonging to $S$ and also points not belonging to $S$, then $z_0$ is called a {\it boundary point}.

{\bf EXTERIOR POINT}\\
If a point is not a an interior point or a boundary point of $S$ then it is called an {\it exterior point} of $S$.

{\bf OPEN SET}\\
An {\it open set} is a set which consists only of interior points. For example, the set of points $|z|<1$ is an open set.

{\bf 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 $|z| \le 1$ is a closed set.

{\bf BOUNDED SET}\\
A set $S$ is called {\it bounded} if we can find a constant $M$ such that $z < M$ for every point in $S$. 

An {\it unbounded set} is one which is not bounded. A set which is both closed and bounded is sometimes called {\it compact.}

{\bf CONNECTED SET}\\
An open set $S$ is said to be {\it connected} if any two points  of the set can be joined by a path consisting  of straight line segments (i.e. a {\it polygonal } path) all points  which are in $S$.

{\bf DOMAIN or OPEN REGION}\\
An open connected set is called an {\it open region} or  {\it domain}.

{\bf CLOSURE}\\
If to a set $S$ we add all the limit points of $S$, the new set is called the {\it closure} of $S$ and is a closed set.

{\bf CLOSED REGION}\\
The closure of an open region or domain is called a {\it closed region.}

{\bf REGION}\\
If to an open region we add some, all or none of its limit points we obtain a set called a {\it region}. If all the limit points are added the region is {\it closed}; if none are added the region is {\it open}. Usually if the word {\it region} is used without qualifying it with an adjective, it is referring to an {\it open region} or {\it domain}. \\

{\large Examples}\\
Consider the following point sets and, using as many of the previous definitions as you can, fully describe these examples.

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