Fixed-Point Theory for Generalized Metric Spaces
91
4.1.7 Theorem Given a continuity space X = (X, d, A, P ), the collection
T (X ) of subsets of X is a topology on X. Conversely, given a topology T on
a set X, there is a continuity space X = (X, d, A, P ) with the property that
T = T (X ).
Given a topology T on X, it is worth noting that the continuity space
X = (X, d, A, P ) with the property that T = T (X ) used in the proof of
Theorem 4.1.7 is obtained, see [Kopperman, 1988], by taking A to be the
product of T copies of R and P to be the product of T copies of (0, ∞]. The
continuity function d is defined coordinatewise by d(x, y)(S) = d S (x, y) for
each S ∈ T , where d S (x, y) = 0 if (x ∈ S implies y ∈ S), and d S (x, y) = q
otherwise, where q is an element of (0, ∞] fixed once and for all.
4.2 Metrics and Their Generalizations
As already noted, it is our intention, with applications in mind, to choose
suitable value sets for distance functions and to impose various useful conditions on the distance functions themselves. We begin by considering the most
familiar of these, where the value set is taken to be the set of non-negative
real numbers.
4.2.1 Definition Let X be a set, and let � : X × X → R
+ be a distance
0
function, where R
+ denotes the set of non-negative real numbers. We consider
0
the following conditions on �.
(M1) For all x ∈ X, �(x, x) = 0.
(M2) For all x, y ∈ X, if �(x, y) = �(y, x) = 0, then x = y.
(M3) For all x, y ∈ X, �(x, y) = �(y, x).
(M4) For all x, y, z ∈ X, �(x, y) ≤ �(x, z) + �(z, y).
(M5) For all x, y, z ∈ X, �(x, y) ≤ max{�(x, z), �(z, y)}.
If � satisfies conditions (M1) to (M4), it is called a metric and is called an ultrametric if it also satisfies (M5).
5 If it satisfies conditions (M1), (M3), and (M4),
it is called a pseudometric. If it satisfies (M2), (M3), and (M4), we will call
it a dislocated metric (or simply a d-metric). Finally, if it satisfies conditions
(M1), (M2), and (M4), it is called a quasimetric. Condition (M4) is usually
5 For elementary properties and notions relating to conventional metrics, such as Cauchy
sequences and completeness, we refer to [Willard, 1970]; these notions will, in any case, be
defined later in this chapter in greater generality.
91
4.1.7 Theorem Given a continuity space X = (X, d, A, P ), the collection
T (X ) of subsets of X is a topology on X. Conversely, given a topology T on
a set X, there is a continuity space X = (X, d, A, P ) with the property that
T = T (X ).
Given a topology T on X, it is worth noting that the continuity space
X = (X, d, A, P ) with the property that T = T (X ) used in the proof of
Theorem 4.1.7 is obtained, see [Kopperman, 1988], by taking A to be the
product of T copies of R and P to be the product of T copies of (0, ∞]. The
continuity function d is defined coordinatewise by d(x, y)(S) = d S (x, y) for
each S ∈ T , where d S (x, y) = 0 if (x ∈ S implies y ∈ S), and d S (x, y) = q
otherwise, where q is an element of (0, ∞] fixed once and for all.
4.2 Metrics and Their Generalizations
As already noted, it is our intention, with applications in mind, to choose
suitable value sets for distance functions and to impose various useful conditions on the distance functions themselves. We begin by considering the most
familiar of these, where the value set is taken to be the set of non-negative
real numbers.
4.2.1 Definition Let X be a set, and let � : X × X → R
+ be a distance
0
function, where R
+ denotes the set of non-negative real numbers. We consider
0
the following conditions on �.
(M1) For all x ∈ X, �(x, x) = 0.
(M2) For all x, y ∈ X, if �(x, y) = �(y, x) = 0, then x = y.
(M3) For all x, y ∈ X, �(x, y) = �(y, x).
(M4) For all x, y, z ∈ X, �(x, y) ≤ �(x, z) + �(z, y).
(M5) For all x, y, z ∈ X, �(x, y) ≤ max{�(x, z), �(z, y)}.
If � satisfies conditions (M1) to (M4), it is called a metric and is called an ultrametric if it also satisfies (M5).
5 If it satisfies conditions (M1), (M3), and (M4),
it is called a pseudometric. If it satisfies (M2), (M3), and (M4), we will call
it a dislocated metric (or simply a d-metric). Finally, if it satisfies conditions
(M1), (M2), and (M4), it is called a quasimetric. Condition (M4) is usually
5 For elementary properties and notions relating to conventional metrics, such as Cauchy
sequences and completeness, we refer to [Willard, 1970]; these notions will, in any case, be
defined later in this chapter in greater generality.