186 Algebra of Complex Numbers
there exists a neighborhood of the point that lies entirely in the set. FOR
EXAMPLE, the center of the disk (open or closed) is an interior point of the
disk.
A point P is called a boundary point of a set if every neighborhood of
the point contains at least one point that is not in the set, and at least one
point that belongs to the set and is different from P. FOR EXAMPLE, the
boundary points of the set {z : [^r — 2r 0 | < r} are exactly the points of the
circle {z :\z — ZQ\ = r}.
A point P is called an isolated point of a set if there exists a neighbor-
hood of P in which P is the only point belonging to the set. FOR EXAMPLE,
the set {z : z = a + ib} in the complex plane, where a, b are real integers,
consists entirely of isolated points.
A set is called
- Bounded, if it lies entirely inside an open disk of sufficiently large radius.
- Closed, if it contains all its boundary points.
- Connected, if every two points in the set can be connected with a
continuous curve lying completely in the set. - Open, if every point belongs to the set together with some neighborhood.
In other words, all points of an open set are interior points. - Domain, if it is open and connected.
- Simply connected, if it has no holes. More precisely, consider a simple,
closed, continuous curve that lies entirely in the set (see page 25); such
a curve encloses a domain (recall that we take for granted the Jordan
curve theorem, see page 123). The set is simply connected if this domain
lies entirely in the set.
The closure of a set is the set together with all its boundary points.
The complement of a set are all the points that are not in the set.
FOR EXAMPLE, the closure of an open disk {z : \z\ < 1} is the closed disk
{z : \z\ < 1}, and the complement of that open disk is the set {z : \z\ > 1}.
Every disk, open or closed, is both connected and simply connected, while
the set {z : 0 < \z\ < 1} is open, connected, but not simply connected
because the point z = 0 is missing from the set.
EXERCISE 4.1.8.c Give an example of a set in C that is not bounded, is
neither open nor closed, is not connected, but is simply connected.