Semiflows on Metric
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.
(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.