Mathematical Preliminaries - Semantics of Probabilistic Processes: An Operational Approach - page 14

2.4 Metric Spaces

The main subject in probabilistic concurrency theory is the quantitative analysis of
system behaviour. So we meet metric spaces from time to time. This section gives
some background knowledge on them.
Let

R ≥ 0 be the set of all nonnegative real numbers.
Definition 2.12 A metric space is a pair ( X , d ) consisting of a set X and a distance

function d : X × X ₒ R ≥ 0 satisfying:

1. For all x , y ∈ X , d ( x , y ) = 0, iff x = y (isolation);

2. For all x , y ∈ X , d ( x , y ) = d ( y , x ) (symmetry);

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

If we replace the first clause with ∀ x ∈ X : d ( x , x ) = 0, we obtain the definition of

pseudometric space .
A metric d is c -bounded if ∀ x , y ∈ X : d ( x , y ) ≤ c , where c is a positive real

number.

Example 2.12 Let X be a set. The discrete metric d : X × X −ₒ [0, 1] is defined

by

⊧
⊨
⊩

0 if x = y

d ( x , y ) =

1 otherwise.

n denote the product set of n copies of the set

Example 2.13 Let R R of real numbers,

i.e. it consists of all n -tuples a 1 , a 2 , ... , a n of real numbers. The function d is

defined by

( a 1 −

d ( x , y ) = b 1 ) 2 +···+ ( a n − b n ) 2

where x = a 1 , ... , a n and y = b 1 , ... , b n , is a metric, called the Euclidean

n , d ) is called Euclidean n-space .
Let ( X , d ) be a metric space. For any point x

R

n . This metric space (

R

metric on

∈

X in the space and any real number
ʵ> 0, we let S ( x , ʵ ) denote the set of points within a distance of ʵ from x :

S ( x , ʵ ): ={ y | d ( x , y ) <ʵ } .

We call S ( x , ʵ ) the open sphere with centre x and radius ʵ .

Theorem 2.5 Let ( X , d ) be a metric space. The collection of open spheres in X
generates a topology, called the metric topology, whose open sets are those open
spheres in X.

Definition 2.13 A sequence ( x n ) in a metric space ( X , d )is convergent to x ∈ X ,if

for an arbitrary ʵ> 0 there exists N ∈ N , such that d ( x n , x ) <ʵ whenever n>N .