Chapter 1

Introduction

The purpose of this introductory chapter is to fix some terminology that

will be used throughout the book and to review the results from general

topology and measure theory that will be needed later. It is intended as a

reference chapter that initially may be skipped.

1.1. Topological spaces

Recall that a metric or distance on a nonempty set X is a function

d : X × X → [0, ∞)

with the following properties:

1. d(x, y) = 0 if and only if x = y,

2. d(x, y) = d(y, x) for all x, y ∈ X, and

3. d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X (triangle inequality).

The set X equipped with the distance d is called a metric space. If

x ∈ X and r 0, the open ball of X with center x and radius r is the set

BX(x, r) = {y ∈ X; d(y, x) r}, while

¯

B

X

(x, r) = {y ∈ X; d(y, x) ≤ r}

denotes the corresponding closed ball.

Of course, a first example is the real n-dimensional Euclidean space,

Rn,

with the Euclidean distance between two points x, y ∈

Rn

defined to be

d(x, y) := |x − y| =

n

j=1

(xj − yj)2

1

http://dx.doi.org/10.1090/gsm/116/01