97
Fixed-Point Theory for Generalized Metric Spaces
(c) The metric d does not in general generate τ , but the iterates (f
n (x)) of
f converge to a both with respect to τ and with respect to d.
Proof: (a) Let x = a, and let ι(l(x)) = 2
−k , say. Then, for any y ∈ X, we have
δ(x, y) ≥ 2
−k , and hence, for each y = x, we have d(x, y) ≥ 2
−k . Therefore,
{y ∈ X | d(x, y) < 2
−k } = {x}, which is consequently open in d. Closedness
is trivial.
(b) In order to see this, it suffices to show that {x} is open with respect to
d for any x = a, which is true by (i) in Step (1) of the proof of Theorem 4.2.5.
(c) Indeed, the topology τ is not in general metrizable. By the proof of
the Banach contraction mapping theorem, (f
n (x)) converges to a with respect
to d. Convergence with respect to τ follows from the hypothesis of Theorem
4.2.5.
•
4.3 Generalized Ultrametrics
The first generalization of the standard notion of metric which we consider
is actually obtained from Definition 4.2.1 by replacing the codomain of �
(the value set of �), namely, the set R
+ of non-negative real numbers, by
0
an arbitrary partially ordered set rather than by relaxing any axioms. This
leads to the notion of “generalized ultrametric” found in parts of algebra
such as valuation theory and first applied to logic programming semantics
by Prieß-Crampe and Ribenboim. Indeed, the main theorem of this section,
Theorem 4.3.6, is due to Prieß-Crampe and Ribenboim.
10
4.3.1 Definition Let X be a set, and let Γ be a partially ordered set with
least element 0. We call (X, � , Γ), or simply (X, �), a generalized ultrametric
space (gum) if � : X × X → Γ is a function such that the following statements
hold for all x, y, z ∈ X and all γ ∈ Γ.
(U1) �(x, x) = 0.
(U2) If �(x, y) = 0, then x = y.
(U3) �(x, y) = �(y, x).
(U4) If �(x, z) ≤ γ and �(z, y) ≤ γ, then �(x, y) ≤ γ.
If � satisfies conditions (U2), (U3), and (U4), but not necessarily (U1), we
call (X, �) a dislocated generalized ultrametric space or simply a d-gum space,
10 The material contained in Section 4.3 up to Theorem 4.3.6 can be found in the following
three papers: [Prieß-Crampe and Ribenboim, 1993, Prieß-Crampe and Ribenboim, 2000a,
Prieß-Crampe and Ribenboim, 2000c].
Fixed-Point Theory for Generalized Metric Spaces
(c) The metric d does not in general generate τ , but the iterates (f
n (x)) of
f converge to a both with respect to τ and with respect to d.
Proof: (a) Let x = a, and let ι(l(x)) = 2
−k , say. Then, for any y ∈ X, we have
δ(x, y) ≥ 2
−k , and hence, for each y = x, we have d(x, y) ≥ 2
−k . Therefore,
{y ∈ X | d(x, y) < 2
−k } = {x}, which is consequently open in d. Closedness
is trivial.
(b) In order to see this, it suffices to show that {x} is open with respect to
d for any x = a, which is true by (i) in Step (1) of the proof of Theorem 4.2.5.
(c) Indeed, the topology τ is not in general metrizable. By the proof of
the Banach contraction mapping theorem, (f
n (x)) converges to a with respect
to d. Convergence with respect to τ follows from the hypothesis of Theorem
4.2.5.
•
4.3 Generalized Ultrametrics
The first generalization of the standard notion of metric which we consider
is actually obtained from Definition 4.2.1 by replacing the codomain of �
(the value set of �), namely, the set R
+ of non-negative real numbers, by
0
an arbitrary partially ordered set rather than by relaxing any axioms. This
leads to the notion of “generalized ultrametric” found in parts of algebra
such as valuation theory and first applied to logic programming semantics
by Prieß-Crampe and Ribenboim. Indeed, the main theorem of this section,
Theorem 4.3.6, is due to Prieß-Crampe and Ribenboim.
10
4.3.1 Definition Let X be a set, and let Γ be a partially ordered set with
least element 0. We call (X, � , Γ), or simply (X, �), a generalized ultrametric
space (gum) if � : X × X → Γ is a function such that the following statements
hold for all x, y, z ∈ X and all γ ∈ Γ.
(U1) �(x, x) = 0.
(U2) If �(x, y) = 0, then x = y.
(U3) �(x, y) = �(y, x).
(U4) If �(x, z) ≤ γ and �(z, y) ≤ γ, then �(x, y) ≤ γ.
If � satisfies conditions (U2), (U3), and (U4), but not necessarily (U1), we
call (X, �) a dislocated generalized ultrametric space or simply a d-gum space,
10 The material contained in Section 4.3 up to Theorem 4.3.6 can be found in the following
three papers: [Prieß-Crampe and Ribenboim, 1993, Prieß-Crampe and Ribenboim, 2000a,
Prieß-Crampe and Ribenboim, 2000c].