## Subsets of the Complex Plane Terminology

A set is said to be open if every member of the set is contained in some open disc wholly contained in More intuitively, we may say that a set is open if it does not contain its own boundary. Part of the boundary of a set may be a single point. The set is open. The boundary is the circle along with the single point If a set includes its boundary, it is said to be closed. More concisely, we may say that if is a limit point of a set then is closed if for all such Obviously this is true of any point in the interior, but will be true for a closed set if is on the boundary since any open set centred on the boundary must contain infinitely many points interior to so is a limit point of This is illustrated below. A set is compact if it is closed and bounded. Bounded here means that there exists such that for all Example: Every set consisting of a finite set of points is compact. Every element is its own boundary, and since there are a finite number of elements, there must be a largest one, so for all A set is said to be path connected if there is a path from every element of the set to every other element. A set is said to be connected if it cannot be written as the union of two open, disjoint sets. If it cannot be written as such a union, then either the two sets have a point in common, or there exists a point which is a limit point of both sets and belongs to one of the sets, so this set is not open.  and above are disjoint except possibly for a point If and then is not connected. If or or then is connected. 