Mathematics of Physics and Engineering

122 Functions of Several Variables

at A and radius b. A point in a set is called interior if there exists a

neighborhood of the point that lies entirely in the set.

A point P is called a boundary point of the set G if every neigh-

borhood of the point contains at least one point that is not in G, and at

least one point that belongs to G and is different from P. The collection of

all boundary points of G is called the boundary of G and is denoted by

dG. FOR EXAMPLE, the boundary points of the set G = {B : \AB\ < 1}

are exactly the points P satisfying \AP\ = 1, so that dG — {P : \AP\ = 1}

is the circle with center at A and radius 1.

A point P in a set is called isolated if there exists a neighborhood of

P in which P is the only point belonging to the set.

A set is called

• Bounded, if it lies entirely inside an open ball of sufficiently large radius.


• Closed, if it contains all its boundary points.


• Connected, if every two points in the set can be connected by a con-
  tinuous curve lying completely in the set.


• Open, if for every point in the set there exists a neighborhood of this
  point that is contained in the set. 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 curve
  that lies entirely in the set and assume that this curve is simple, closed,
  and continuous (see page 25). The set is simply connected if there is
  a surface that lies entirely in the set and has this curve as the bound-
  ary. For a planar set, this condition means that every simple closed
  continuous curve in the set encloses a domain that lies entirely in the
  set.

The closure of a set is the set together with all its boundary points.
The complement of a set is the set of all points that are not in the set.
FOR EXAMPLE, the closure of the open set {P : \OP\ < 1} is {P :
\OP\ < 1}, and the complement of that open set is the closed set {P :
\OP\ > 1}. Every disk or ball is both connected and simply connected,
while the set {P : 0 < \OP\ < 1} is connected, and is simply connected in
M^3 , but not simply connected in E.^2 (make sure you understand why).

EXERCISE 3.I.I.'^4 Give an example of a set in R^3 that is not bounded, is
neither open nor closed and is neither connected nor simply connected.

Intuitively, it is obvious that a simple, closed, continuous curve (also