Subsets of the Complex Plane Terminology
A setis said to be open if every member of the set is contained in some open disc wholly contained inMore 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 setis 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 ifis a limit point of a setthenis closed iffor all suchObviously this is true of any point in the interior, but will be true for a closed set ifis on the boundary since any open set centred on the boundary must contain infinitely many points interior toso is a limit point ofThis is illustrated below.
A setis compact if it is closed and bounded. Bounded here means that there exists such thatfor 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, sofor 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.
andabove are disjoint except possibly for a pointIfand thenis not connected. Ifororthenis connected.