Page 37 - Vector Analysis

§2.1 Functions of Several Variables 33

The map f : Aztx P A | h(x) = 0u Ñ V is defined by
h

( f ) f (x) @ x P Aztx P A | h(x) = 0u .
(x) =

h h(x)

Definition 2.3. A set U Ď Rn is said to be open in Rn if for each x P U , there exists r ą 0
such that B(x, r), the ball centered at x with radius r given by

B(x, r) = ␣y P Rn ˇ }x ´ y}Rn ă r( ,
ˇ

is contained in U . A set F Ď Rn is said to be closed in Rn if F A, the complement of F , is
open in Rn.

Let A Ď Rn be a set. A point x0 is said to be

1. an interior point of A if there exists r ą 0 such that B(x0, r) Ď A;

2. an isolated point of A if there exists r ą 0 such that B(x0, r) X A = tx0u;

3. an exterior point of A if there exists r ą 0 such that B(x0, r) Ď AA;

4. a boundary point of A if for each r ą 0, B(x0, r) X A ‰ H and B(x0, r) X AA ‰ H.

The collection of all interior points of A is called the interior of A and is denoted by A˚.
The collection of all exterior points of A is called the exterior of A, and the collection of all
boundary point of A is called the boundary of A. The boundary of A is denoted by B A.
The closure of A is defined as A Y B A and is denoted by A. The derived set of A, denoted
by A1, is the collection of all points in A that are not isolated points.

A is said to be bounded in Rn if there exists a constant M ą 0 such that

}x}Rn ă M @ x P A ()
ô A Ď B(0, M ) .

A is said to be unbounded if A is not bounded.

The following theorem is a fundamental result in point-set topology. We omit the proof
since it is not the main concern in vector analysis; however, the result should look intuitive
and the proof of this theorem is not difficult. Interested readers can try to establish this
result by yourselves.

Theorem 2.4. Let A Ď Rn be a set. Then