Chapter 1

Semiflows on Metric

Spaces

Since metric spaces are natural state-spaces of semiflows, a brief introduction

of metric spaces is given below, mainly to fix notation. Otherwise the reader

is supposed to be familiar with metric spaces and continuity of functions as

they are taught in an introductory graduate course in analysis.

1.1. Metric spaces

Definition 1.1. A metric space (X, d) is a nonempty set X together with

a function d : X × X → R which satisfies the following axioms:

(1) d(x, y) = d(y, x) for all x, y ∈ X. [symmetry]

(2) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.

[triangle inequality]

(3) d(x, x) = 0 for all x ∈ X.

(4) d(x, y) = 0 for all x, y ∈ X with x = y.

d(x, y) measures the distance between x and y. d is called a metric on

X. If (4) does not hold, d is called a semimetric.

Example 1.2 (trivial metric). Let X be an arbitrary nonempty set. Define

d(x, y) =

0, x = y,

1, x = y.

Then d is a metric on X, called the trivial metric or the discrete metric.

Proof. The only nontrivial property is the triangle inequality.

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http://dx.doi.org/10.1090/gsm/118/02