89
Fixed-Point Theory for Generalized Metric Spaces
see Theorem A.2.5 in particular. This last observation connects topology and
distance in full generality, and this setting, while not the most general to
have been found to be of interest in computer science, as already noted in
Chapter 3, is sufficient for our purposes here. In fact, we shall make no actual
use of continuity spaces and present them purely as a framework within which
to work. However, continuity spaces do provide a smooth transition from the
topology presented in Chapter 3 to the work of this chapter, and indeed they
bridge the two chapters.
Before turning to the details of continuity spaces in general, it will be worth
considering first the familiar case of distance functions d which are metrics,
see Definition 4.2.1 and Remark 4.2.2. In this case, the usual value set A of d is
the interval [0, ∞). Given some real number ε > 0, one defines the (open) ball
N ε (x) of radius ε about a point x ∈ X by setting N ε (x) = {y ∈ X | d(x, y) <
ε}. A subset O of X is then declared to be open if, for each x ∈ X, there
is some ε > 0 such that N ε (x) ⊆ O. It is easy to see that the collection of
such open sets O forms a topology on X. Notice that in defining “open” sets
O here, one can equivalently require B ε / (x) ⊆ O for suitable ε
' > 0, where
B ε (x) = {y ∈ X | d(x, y) ≤ ε} denotes the (closed) ball of radius ε about a
point x ∈ X.
However, it is not true that every topology on X arises thus via a metric d, and, for example, this statement applies to the Scott topology since
this topology in not even T 1 in general, see Proposition A.6.5, whereas every
metrizable topology is Hausdorff. Nevertheless, every topology can be generated by means of a suitable distance function, as already noted, and we next
consider briefly the details of one way of establishing this claim, beginning
with several definitions.
4.1.1 Definition A semigroup is a set A together with an (additive) associative binary operation + : A × A → A. If + is also commutative, then the
semigroup is called commutative or Abelian. A semigroup A is called a semigroup with identity if there exists an element 0 ∈ A, called the identity, such
that 0 + a = a + 0 = a for all a ∈ A. We note that an (additive) Abelian semigroup with identity is also called a commutative monoid or Abelian monoid.
By an ordered semigroup with identity we mean a semigroup A with 0, say,
on which there is defined an ordering ≤ satisfying: 0 ≤ a for all a ∈ A, and if
'
'
'
'
'
'
a 1 ≤ a 2 and a 1 ≤ a , then a 1 + a 1 ≤ a 2 + a for all a 1 , a 1 , a 2 , a 2 ∈ A.
2
2
4.1.2 Definition A value semigroup A is an additive Abelian semigroup with
4
identity 0 and absorbing element ∞, where ∞ = 0, satisfying the following
axioms.
(1) For all a, b ∈ A, if a + x = b and b + y = a for some x, y ∈ A, then a = b.
(Note that, using this property, we can define a partial order ≤ on A by
setting a ≤ b if and only if b = a + x for some x ∈ A; we call ≤ the partial
4 An element satisfying a + ∞ = ∞ + a = ∞ for all a ∈ A.
Fixed-Point Theory for Generalized Metric Spaces
see Theorem A.2.5 in particular. This last observation connects topology and
distance in full generality, and this setting, while not the most general to
have been found to be of interest in computer science, as already noted in
Chapter 3, is sufficient for our purposes here. In fact, we shall make no actual
use of continuity spaces and present them purely as a framework within which
to work. However, continuity spaces do provide a smooth transition from the
topology presented in Chapter 3 to the work of this chapter, and indeed they
bridge the two chapters.
Before turning to the details of continuity spaces in general, it will be worth
considering first the familiar case of distance functions d which are metrics,
see Definition 4.2.1 and Remark 4.2.2. In this case, the usual value set A of d is
the interval [0, ∞). Given some real number ε > 0, one defines the (open) ball
N ε (x) of radius ε about a point x ∈ X by setting N ε (x) = {y ∈ X | d(x, y) <
ε}. A subset O of X is then declared to be open if, for each x ∈ X, there
is some ε > 0 such that N ε (x) ⊆ O. It is easy to see that the collection of
such open sets O forms a topology on X. Notice that in defining “open” sets
O here, one can equivalently require B ε / (x) ⊆ O for suitable ε
' > 0, where
B ε (x) = {y ∈ X | d(x, y) ≤ ε} denotes the (closed) ball of radius ε about a
point x ∈ X.
However, it is not true that every topology on X arises thus via a metric d, and, for example, this statement applies to the Scott topology since
this topology in not even T 1 in general, see Proposition A.6.5, whereas every
metrizable topology is Hausdorff. Nevertheless, every topology can be generated by means of a suitable distance function, as already noted, and we next
consider briefly the details of one way of establishing this claim, beginning
with several definitions.
4.1.1 Definition A semigroup is a set A together with an (additive) associative binary operation + : A × A → A. If + is also commutative, then the
semigroup is called commutative or Abelian. A semigroup A is called a semigroup with identity if there exists an element 0 ∈ A, called the identity, such
that 0 + a = a + 0 = a for all a ∈ A. We note that an (additive) Abelian semigroup with identity is also called a commutative monoid or Abelian monoid.
By an ordered semigroup with identity we mean a semigroup A with 0, say,
on which there is defined an ordering ≤ satisfying: 0 ≤ a for all a ∈ A, and if
'
'
'
'
'
'
a 1 ≤ a 2 and a 1 ≤ a , then a 1 + a 1 ≤ a 2 + a for all a 1 , a 1 , a 2 , a 2 ∈ A.
2
2
4.1.2 Definition A value semigroup A is an additive Abelian semigroup with
4
identity 0 and absorbing element ∞, where ∞ = 0, satisfying the following
axioms.
(1) For all a, b ∈ A, if a + x = b and b + y = a for some x, y ∈ A, then a = b.
(Note that, using this property, we can define a partial order ≤ on A by
setting a ≤ b if and only if b = a + x for some x ∈ A; we call ≤ the partial
4 An element satisfying a + ∞ = ∞ + a = ∞ for all a ∈ A.