�
121
Fixed-Point Theory for Generalized Metric Spaces
4.8.3 GUMS and Chain Complete Posets
In this section, we will invert the point of view of the previous one by
associating a chain-complete partial order with any generalized ultrametric
space (X, �, Γ) whose distance set Γ is an ordinal endowed with, essentially,
the dual ordering as considered in the previous section. Thus, for the duration
of this section, Γ is the set Γ γ+1 for some ordinal γ with the ordering described
in Remark 4.3.2. For convenience, we will henceforth call such a generalized
ultrametric space a gum with ordinal distances; recall that we denote 2
−γ by
0.
The motivation for adopting our current point of view is to provide a
domain-theoretic proof of the Prieß-Crampe and Ribenboim theorem.
24 In
fact, we will prove the Prieß-Crampe and Ribenboim theorem using the
Knaster-Tarski theorem in this special case of gums with ordinal distances.
As a matter of fact, this special case will suffice for all our purposes since, in
applications, all the gums we encounter have ordinal distances, simply because
they arise from level mappings.
Our main technical tool is the space of formal balls associated with a given
metric space, see [Edalat and Heckmann, 1998]. Our first task is to extend this
notion to generalized ultrametrics.
25
Let (X, �, Γ) be a generalized ultrametric space with ordinal distances,
and let B
' X be the set of all pairs (x, α) with x ∈ X and α ∈ Γ. We define
an equivalence relation ∼ on B
' X by setting (x 1 , α 1 ) ∼ (x 2 , α 2 ) if and only
if α 1 = α 2 and �(x 1 , x 2 ) ≤ α 1 . The quotient space BX = B
' X/ ∼ will be
called the space of formal balls associated with (X, �, Γ), and it carries an
ordering [ which is well-defined (on representatives of equivalence classes) by
(x, α) [ (y, β) if and only if �(x, y) ≤ α and β ≤ α. We denote the equivalence
class of (x, α) by [(x, α)], and note of course that the use of the same symbol
[ between equivalence classes and their representatives should not cause any
confusion.
4.8.15 Proposition The set BX is partially ordered by [. Moreover, X is
spherically complete if and only if BX is chain complete.
Proof: That BX is partially ordered by [ is clear.
Let X be spherically complete, and let [(x β , β)] be an ascending chain in
BX. Then B β (x β ) is a chain of balls in X with non-empty intersection; let
x ∈ B β (x β ). Then �(x β , x) ≤ β for all β. Hence, the chain [(x β , β)] in BX
has [(x, 0)] as an upper bound. Now consider the set A of all α ∈ Γ such
that [(x, α)] is an upper bound of [(x β , β)]. Since we are working with ordinal
distances only, the set A has a supremum γ, and hence [(x, γ)] is the least
upper bound of the chain [(x β , β)].
Now suppose BX is chain complete, and let (B β (x β ))
be a chain of
β∈Λ
24 This approach is inspired by [Edalat and Heckmann, 1998], where the Banach contraction mapping theorem is derived from Kleene’s theorem.
25 For more details, see [Hitzler and Seda, 2003].
121
Fixed-Point Theory for Generalized Metric Spaces
4.8.3 GUMS and Chain Complete Posets
In this section, we will invert the point of view of the previous one by
associating a chain-complete partial order with any generalized ultrametric
space (X, �, Γ) whose distance set Γ is an ordinal endowed with, essentially,
the dual ordering as considered in the previous section. Thus, for the duration
of this section, Γ is the set Γ γ+1 for some ordinal γ with the ordering described
in Remark 4.3.2. For convenience, we will henceforth call such a generalized
ultrametric space a gum with ordinal distances; recall that we denote 2
−γ by
0.
The motivation for adopting our current point of view is to provide a
domain-theoretic proof of the Prieß-Crampe and Ribenboim theorem.
24 In
fact, we will prove the Prieß-Crampe and Ribenboim theorem using the
Knaster-Tarski theorem in this special case of gums with ordinal distances.
As a matter of fact, this special case will suffice for all our purposes since, in
applications, all the gums we encounter have ordinal distances, simply because
they arise from level mappings.
Our main technical tool is the space of formal balls associated with a given
metric space, see [Edalat and Heckmann, 1998]. Our first task is to extend this
notion to generalized ultrametrics.
25
Let (X, �, Γ) be a generalized ultrametric space with ordinal distances,
and let B
' X be the set of all pairs (x, α) with x ∈ X and α ∈ Γ. We define
an equivalence relation ∼ on B
' X by setting (x 1 , α 1 ) ∼ (x 2 , α 2 ) if and only
if α 1 = α 2 and �(x 1 , x 2 ) ≤ α 1 . The quotient space BX = B
' X/ ∼ will be
called the space of formal balls associated with (X, �, Γ), and it carries an
ordering [ which is well-defined (on representatives of equivalence classes) by
(x, α) [ (y, β) if and only if �(x, y) ≤ α and β ≤ α. We denote the equivalence
class of (x, α) by [(x, α)], and note of course that the use of the same symbol
[ between equivalence classes and their representatives should not cause any
confusion.
4.8.15 Proposition The set BX is partially ordered by [. Moreover, X is
spherically complete if and only if BX is chain complete.
Proof: That BX is partially ordered by [ is clear.
Let X be spherically complete, and let [(x β , β)] be an ascending chain in
BX. Then B β (x β ) is a chain of balls in X with non-empty intersection; let
x ∈ B β (x β ). Then �(x β , x) ≤ β for all β. Hence, the chain [(x β , β)] in BX
has [(x, 0)] as an upper bound. Now consider the set A of all α ∈ Γ such
that [(x, α)] is an upper bound of [(x β , β)]. Since we are working with ordinal
distances only, the set A has a supremum γ, and hence [(x, γ)] is the least
upper bound of the chain [(x β , β)].
Now suppose BX is chain complete, and let (B β (x β ))
be a chain of
β∈Λ
24 This approach is inspired by [Edalat and Heckmann, 1998], where the Banach contraction mapping theorem is derived from Kleene’s theorem.
25 For more details, see [Hitzler and Seda, 2003].