@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 1, 2003

pp. 47–70

The local triangle axiom in topology and
domain theory

Pawe lWaszkiewicz

Abstract. We introduce a general notion of distance in weakly
separated topological spaces. Our approach differs from existing ones
since we do not assume the reflexivity axiom in general. We demon-
strate that our partial semimetric spaces provide a common general-
ization of semimetrics known from Topology and both partial metrics
and measurements studied in Quantitative Domain Theory. In the pa-
per, we focus on the local triangle axiom, which is a substitute for the
triangle inequality in our distance spaces. We use it to prove a coun-
terpart of the famous Archangelskij Metrization Theorem in the more
general context of partial semimetric spaces. Finally, we consider the
framework of algebraic domains and employ Lebesgue measurements to
obtain a complete characterization of partial metrizability of the Scott
topology.

2000 AMS Classification: 54E99, 54E35, 06A06.
Keywords: Partial semimetric, partial metric, measurement, Lebesgue mea-

surement, local triangle axiom, continuous poset, algebraic dcpo.

1. Introduction

Over recent years, a number of attempts have been made to equip semantic
domains with a notion of distance between points in order to provide a mech-
anism for making quantitative statements about programs, such as speed of
convergence or complexity of algorithms. This particular branch of research is
now known as Quantitative Domain Theory. The work in this area has led to a
number of concepts which generalise the classical notion of a metric space to an
ordered setting. For example, Smyth [21] introduced quasi-metrics to Domain
Theory, which by virtue of being non-symmetric encode topology and order on
a domain at the same time. Partial metrics were defined by Matthews [14] and
studied further by O’Neill [16, 17]. Although they are symmetric, they are still
capable of capturing order. This work was further extended by Heckmann in
[9], leading to a concept of weak partial metrics.



48 P. Waszkiewicz

More recently, Keye Martin introduced the idea of a measurement on a
domain [13]. At first glance, measurements are quite different from distance
functions because they take only one argument. Nevertheless, they allow for
making precise quantitative statements about domains.

The desire to understand the interplay between partial metrics and measure-
ments, discovered by the author in [24], is a major motivation for the present
work. Research in this direction led to a number of general question about
the nature of distance in weakly separated spaces in Topology (continuous do-
mains in their Scott topology are examples of such spaces). Our approach is
quite distinct from existing theories of generalized metric spaces in Quantita-
tive Domain Theory, e.g. [18, 5, 6, 23] and Topology [8], since we do not in
general assume either the reflexivity axiom or the triangle inequality. Instead,
we introduce partial semimetric spaces, which generalize both partial metrics
and semimetrics. The theory of partial semimetrics coincides with the theory
of measurements in the framework of continuous domains.

In the paper we focus on a certain condition on sequences in partial semimet-
ric spaces substituting the triangle inequality, which we call the local triangle
axiom. It was recognised as early as 1910 in the work of Fréchet [7] and in-
vestigated in [3, 4]. The name local axiom of triangle was used by Niemytzki
[15] and the property was further analysed in [27]. On continuous domains in
their Scott topology, stable partial symmetrics (defined in Section 5.1) satisfy-
ing the local triangle axiom correspond to Lebesgue measurements (Definition
6.5) introduced in a more general context by Martin in [13].

There are two major results of the paper. The first one is a metrization
result (Theorem 3.2), which is a twin of the famous Archangelskij Metrization
Theorem working in partial semimetric spaces. We apply it in Section 4 to
prove quasi-developability of our distance spaces, improving a similar result
by Künzi and Vajner [10]. In the other part of the paper we consider partial
semimetric spaces and the local triangle axiom in the framework of Domain
Theory (Section 6) to obtain the second major result of the paper. Our result
provides a solution to an open problem stated in [9] asking for conditions,
which guarantee partial metrizability of continuous domains. We show that an
algebraic domain admits a partial metric for its Scott topology if and only if
it admits a Lebesgue measurement (Theorem 6.12). The Heckmann problem
remains open for arbitrary continuous domains.

For the reader’s convenience we present a summary of symbols and terms
used in the paper in Table 1.



The local triangle axiom 49

Table 1: Symbols and terms used in the paper

symbol or term meaning definition related results

〈X,ρ〉 distance space Def. 2.1.
τρ distance topology Sect. 2.

τopρ dual distance Sect. 2.

topology

(cv) a convergence prop. Sect. 2.1. Prop. 2.5, 2.10.

(symm) symmetry axiom Sect. 2.2.

(ssd) small self-distance Sect. 2.2.

axiom

(r) reflexivity axiom Sect. 2.2.

(∆]) sharp triangle Sect. 2.2.

inequality axiom

partial symmetric Def. 2.6.

partial semimetric Def. 2.7. Cor. 2.9, 2.12. Prop. 2.10,

2.11, 2.14, 2.16. Def. 6.2.

Thm. 6.4.

partial metric Sect. 2.2 Cor. 2.18.

local axiom, Sect. 2.5, Prop. 2.16, 2.17, 2.19,

of triangle (L) Sect. 6.3. 6.6, 6.7. Cor. 2.18.

Thm. 3.2, 4.1, 6.8, 6.12.

induced distance symmetrization Sect. 2.4. Prop. 2.13. Thm. 3.2.

dρ of distance ρ

and its dual.

self-distance µρ Sect. 5. Sect. 5.1, 6.2.

measurement Def. 6.1. Sect. 6.

2. Outline of a general theory of distance

First, let us consider the definition of a distance space, its intrinsic topologies
and some natural axioms that can be introduced in distance spaces.

Definition 2.1. A distance on a set X is a map ρ: X ×X → [0,∞). A pair
〈X,ρ〉 is called a distance space.

A distance function assigns to each element x of X a filter Nx of subsets of
X by taking U ∈ Nx if and only if there exists ε > 0 such that Bρ(x,ε) ⊆ U.
The set

Bρ(x,ε) := {y ∈ X | ρ(x,y) < ρ(x,x) + ε}
is called a ball centered at x with radius ε. In the same way one forms a
collection Nopx by replacing the ball in the definition of Nx by a dual ball

Bopρ (x,ε) := {y ∈ X | ρ(y,x) < ρ(y,y) + ε}.



50 P. Waszkiewicz

The collection
τρ := {U | ∀x ∈ U. U ∈Nx}

is a topology on X called the distance topology on X. Dually, the family of sets

τopρ := {V | ∀x ∈ V. V ∈N
op
x }

constitutes a dual (distance) topology on X. In general the collection Bx of
balls centered at x is not a neighborhood base at x and the balls are not open
themselves.

The set X together with an operation N : X → P(P(X)), which assigns
to each point the filter Nx, is an example of a neighbourhood space in the
sense of [22]. The distance topology just defined is in fact one of the natural
topologies for neighbourhood spaces, studied in more detail by Smyth (cf. [22],
Proposition 2.10).

Example 2.2. To illustrate the difference between arbitrary distance spaces
and metric ones, consider the four-element chain as shown in Figure 1. (Num-
bers denote respective self-distances; we set the distances ρ(x,y) and ρ(y,x)
between points x,y to be max{ρ(x,x),ρ(y,y)}). The distance is symmetric
and satisfies the triangle inequality. Note that open sets are upper; in particu-
lar, the only open set containing the bottom element (denoted ⊥) is the whole
space. Indeed, the distance between ⊥ and any other element of the space is 3
and hence any ball containing ⊥ must be the whole space.

In contrast, any metric on a finite space induces the discrete topology.

Note that in this particular example the balls are open themselves. In arbi-
trary distance spaces it is usually not the case.

3 d
2 d
1 d
0 d

Figure 1: A non-metric distance.

2.1. Basic properties of distance spaces. For a subset A of X and an
element x ∈ X we define:

ρ(x,A) := inf{ρ(x,a) | a ∈ A}.

Proposition 2.3. In a distance space 〈X,ρ〉 a subset H ⊆ X is closed iff for
all x /∈ H we have ρ(x,H) > ρ(x,x).



The local triangle axiom 51

Proof. Straight from the definition of the distance topology we infer that a
subset H of X is closed iff for all x /∈ H there exists ε > 0 such that for all
y ∈ H we have ρ(x,y) ≥ ρ(x,x) + ε. It is however equivalent to say that
ρ(x,H) ≥ ρ(x,x) + ε > ρ(x,x). �

As a corollary we obtain the following interesting characterization of the
specialisation preorder vτρ:

Corollary 2.4. In a distance space 〈X,ρ〉 the following are equivalent:
1. x vτρ y;
2. x = y or ∀ε > 0. ∃z 6= x. (ρ(x,z) < ρ(x,x) + ε and z vτρ y).

Proof. (⇒) For x,y ∈ X,

x v/τρy iff x /∈ cl{y}
iff ∃ε > 0. ρ(x, cl{y}) ≥ ρ(x,x) + ε
iff ∃ε > 0. ∀z ∈ Bρ(x,ε). z /∈ cl{y}
iff ∃ε > 0. ∀z ∈ Bρ(x,ε). z v/τρy.

Note that we have used Proposition 2.3 in the second equivalence.

Conversely, assume x 6= y and let U be an open set in X. Then x ∈ U
implies that there exists ε > 0 such that x ∈ Bρ(x,ε) ⊆ U. By assumption,
there is z ∈ Bρ(x,ε) ⊆ U with z vτρ y. But open sets are upper with respect
to the specialisation preorder and so y ∈ U. �

Proposition 2.5. Let 〈X,ρ〉 be a distance space, (xn) be a sequence of elements
of X and x ∈ X. Then

ρ(x,xn) → ρ(x,x) implies xn →τρ x.

Proof. Let U be any open set around x. Then there exists ε > 0 such that
x ∈ Bρ(x,ε) ⊆ U. Suppose that ρ(x,xn) → ρ(x,x). Then there exists Nε ∈ ω
such that for all n ≥ Nε, |ρ(x,xn) −ρ(x,x)| < ε. It is equivalent to say that
either 0 ≤ ρ(x,xn) −ρ(x,x) < ε or 0 ≥ ρ(x,xn) −ρ(x,x) > −ε for all n ≥ Nε.
In either case, we are able to conclude that xn ∈ Bρ(x,ε) ⊆ U for all n ≥ Nε.
Therefore, xn →τρ x. �

Note that the corresponding equivalence:

(cv) ρ(x,xn) → ρ(x,x) iff xn →τρ x
may not hold in arbitrary partial symmetric spaces. Figure 2 provides a
counterexample: we assume that all of the distances between elements, which
are not given on the picture are 5. One can check that the bottom element
is in every open set and hence every sequence converges to it. On the other
hand, for the sequence (xn) we have ρ(⊥,xi) → 5, for i = 1, 2, 3, . . ., while
ρ(⊥,⊥) = 3.



52 P. Waszkiewicz

⊥
3 d

2 d 2 d 2 d 2 d · · ·
2 2 2 2

33
3 3

H
H
H

H
H
H
H
HH

J
J
J
JJ











�
�
�
�
�
�
�
��

1 d 1 d 1 d 1 d · · ·x1 x2 x3 x4 · · ·

Figure 2: (cv) does not hold in general.

It is known that (cv) holds if the T2 axiom is assumed [8], Lemma 9.3 p.481.

2.2. Distance axioms. On a distance space a number of further axioms can
be introduced. We consider symmetry of the distance map

(symm) ∀x,y ∈ X. ρ(x,y) = ρ(y,x)

and the axiom of small self-distances

(ssd) ∀x,y ∈ X. ρ(x,x) ≤ ρ(x,y) and ρ(x,x) ≤ ρ(y,x),

which obviously reduces to

∀x,y ∈ X. ρ(x,x) ≤ ρ(x,y)

in the presence of symmetry. The latter condition is a generalisation of the
reflexivity axiom

(r) ∀x,y ∈ X. ρ(x,x) = 0

known from the theory of metric spaces.
In Topology, consult for example [8], a symmetric is a distance map, which

satisfies (symm) and (r). On the other hand, a symmetric distance function
satisfying the sharp triangle inequality

(∆]) ∀x,y,z ∈ X. ρ(x,y) ≤ ρ(x,z) + ρ(z,y) −ρ(z,z)

is named a weak partial metric [9]. A partial metric [14] is a weak partial metric
with (ssd). Lastly, one considers mostly distance topologies, which satisfy the
T0 separation axiom. An elegant formulation of the (T0) axiom in terms of the
distance mapping will be given later for a wide class of spaces (cf. Corollary
2.12). The following definition is a compromise on the existing terminology.

Definition 2.6. A distance function ρ: X×X → [0,∞) on a set X is a partial
symmetric whenever it satisfies (symm), (ssd) and the distance topology τρ
is T0.



The local triangle axiom 53

2.3. Partial semimetric spaces. In this section we introduce a particularly
interesting subclass of partial symmetric spaces. Both partial metric spaces
considered by Matthews [14], O’Neill [16, 17], Schellekens [19, 20], Heckmann
[9] and Waszkiewicz [24] and semimetric spaces in Topology are our major
examples of partial semimetric spaces.

Definition 2.7. Let 〈X,ρ〉 be a partial symmetric space. The map ρ is a
partial semimetric if for any x ∈ X, the collection {int(Bρ(x,ε)) | ε > 0} is a
base for the filter Nx.

In other words, we require that the family {Bρ(x,ε) | x ∈ X,ε > 0} forms
a (not necessarily open) neighborhood base at x with respect to the distance
topology τρ on X.

To summarize the key differences between “classical” distance maps and
ours, we collect their properties in the following table. A ‘+’ indicates that the
respective condition is satisfied. All maps in the table satisfy (symm), (ssd)
and (T0).

Distance Balls form a neighbourhood base (r)
partial symmetric
partial semimetric +

symmetric +
semimetric + +

Proposition 2.8. Let 〈X,ρ〉 be a distance space. The following are equivalent:
(1) {Bρ(x,ε) | ε > 0} forms a (not necessarily open) neighborhood base at

x;
(2) ∀x ∈ X. ∀H ⊆ X. ρ(x,H) ≤ ρ(x,x) iff x ∈ cl(H).

Proof. (⇒) Let x ∈ X and ε > 0.
x ∈ cl(H) iff ∀ε > 0. Bρ(x,ε) ∩H 6= ∅

iff ∀ε > 0. ∃z ∈ H. ρ(x,z) < ρ(x,x) + ε
iff ρ(x,H) ≤ ρ(x,x).

Conversely, Since X \Bρ(x,ε) = {z | ρ(x,z) ≥ ρ(x,x) + ε}, we have
ρ(x,X \Bρ(x,ε)) ≥ ρ(x,x) + ε > ρ(x,x).

Therefore, x /∈ cl(X \Bρ(x,ε)) and so x ∈ int(Bρ(x,ε)). �

As an immediate corollary, we note that a partial semimetric can be intro-
duced by an appropriate closure operator familiar from the theory of metric
spaces. This was stated in [8], Theorem 9.7, for semimetrics.

Corollary 2.9. Let 〈X,ρ〉 be a symmetric distance space. The following are
equivalent:



54 P. Waszkiewicz

(1) ρ is a partial semimetric.
(2) ∀x ∈ X. ∀H ⊆ X. ρ(x,H) = ρ(x,x) iff x ∈ cl(H).

One can check that if the distance topology is Hausdorff, then X is partially
semimetrizable iff it is partially symmetrizable and first countable (see the
discussion before Definition 9.5 of [8]). In the absence of separation axioms,
the property (cv) is necessary for this characterization to hold.

Proposition 2.10. Let 〈X,ρ〉 be a partial symmetric space. Then the following
are equivalent:

(1) The map ρ is a partial semimetric;
(2) X is first countable and (cv) holds.

Proof. If ρ is a partial semimetric, then the collection

{int(Bρ(x, 1/2n+1)) | n ∈ ω}

is a neighborhood base at x ∈ X, which amounts to first countability of X.
Now, if for some sequence (xn) of elements of X and some x ∈ X, we have
xn →τρ x, then for every ε > 0, (xn) is cofinally in int(Bρ(x,ε)) and so in
Bρ(x,ε). Therefore, using the (ssd) axiom,

∀ε > 0. ∃N ∈ ω. ∀n ≥ N. |ρ(x,xn) −ρ(x,x)| < ε.

That is, ρ(x,xn) → ρ(x,x).
Conversely, if X is first countable and (cv) holds, then a point x ∈ X is

in int(Bρ(x,ε)) for any ε > 0. Suppose not; then there exists ε0 > 0 and a
sequence (xn) with xn →τρ x such that xn /∈ Bρ(x,ε0). (Choose xn to be in
the nth member of a decreasing countable base for x and not in Bρ(x,ε0)). So
ρ(x,xn) 9 ρ(x,x), which contradicts (cv). Now, if x ∈ int(Bρ(x,ε)), then the
collection {Bρ(x,ε) | ε > 0} forms a neighborhood base at x. �

In a partial semimetric space the specialisation preorder reflects the distance
in a simple way.

Proposition 2.11. Let 〈X,ρ〉 be a partial semimetric space. Then for all
x,y ∈ X,

x vτρ y iff ρ(x,y) = ρ(x,x).

Proof.

x vτρ y iff x ∈ cl({y})
iff x ∈{z | ρ(z,y) = ρ(z,z)}
iff ρ(x,y) = ρ(x,x).

We use Proposition 2.8 (2) and (ssd) in the second equivalence. �

As a result we are able to characterize the (T0) axiom in terms known from
the theory of partial metric spaces.



The local triangle axiom 55

Corollary 2.12. The distance topology in a partial semimetric space 〈X,ρ〉 is
T0 iff

x = y iff ρ(x,y) = ρ(x,x) = ρ(y,y).

�

2.4. Derived distance functions and their topologies.

Convention In the rest of this paper we assume that the (T0)
axiom holds in all distance spaces that we consider.

In a distance space 〈X,ρ〉 one can introduce a number of other maps, which
are derived from the distance. Whenever the map ρ satisfies (ssd), it is con-
venient to form the corresponding quasi-distance qρ : X ×X → [0,∞) by

qρ(x,y) := ρ(x,y) −ρ(x,x).

Whenever possible, we will drop the index ρ from qρ for the sake of clarity.
The (ssd) axiom assures that the function q is well-defined. The usefulness
of the quasi-distance stems from the fact that it satisfies the reflexivity ax-
iom (r) and still induces the same topology as the distance ρ. Whenever the
map ρ satisfies more axioms, the name for the quasi-distance q will change
accordingly. For instance, a quasi-semimetric is a quasi-distance formed from
a partial semimetric.

The quasi-distance has a dual, namely the map qopρ : X×X → [0,∞) defined
by qopρ (x,y) := ρ(x,y) − ρ(y,y). The dual quasi-distance induces the dual
distance topology.

The symmetrization of the quasi-distance and its dual is the map

dρ : X ×X → [0,∞),

∀x,y ∈ X. dρ(x,y) := qρ(x,y) + qopρ (x,y) = 2ρ(x,y) −ρ(x,x) −ρ(y,y).
The function dρ is always symmetric and reflexive and hence is a symmetric.
It is called the induced symmetric. We do not know whether it is true in
general that the induced symmetric derived from a partial semimetric is a
semimetric. However, there are two notable cases when the prefix “semi-” is
retained: firstly, whenever a partial semimetric topology is Hausdorff, secondly,
in the presence of the local triangle axiom studied in the next section (the
latter claim follows from Proposition 2.16 and the observation that a partial
semimetric is a semimetric if and only if it satisfies the reflexivity axiom).

Lastly, for a distance function, the self-distance mapping (called also the
weight function) is non-trivial in general and, as we shall see in Section 5,
exhibits many interesting properties.

Proposition 2.13. For a distance space 〈X,ρ〉 with (ssd), the induced topology
τdρ is the join of the distance topology τρ and its dual τ

op
ρ . Moreover, τdρ is

semimetrizable whenever both τρ and τopρ are partially semimetrizable.



56 P. Waszkiewicz

Proof. Both statements follow from the fact that

Bdρ(x,ε) ⊆ B(x,ε) ∩B
op
ρ (x,ε) ⊆ Bdρ(x, 2ε)

for any x ∈ X and ε > 0. �

Proposition 2.14. For a sequence (xn) and an element z ∈ X in a partial
semimetric space 〈X,ρ〉, the following are equivalent:

(a) lim dρ(xn,z) = 0;
(b) lim ρ(xn,z) = ρ(z,z) and lim ρ(xn,z) = lim ρ(xn,xn);
(c) (xn) →τρ z and lim ρ(xn,z) = lim ρ(xn,xn).

Proof. Note that all limits are taken with respect to the Euclidean topology on
the real line. The equivalence of (a) and (b) follows easily since

dρ(xn,z) = (ρ(xn,z) −ρ(z,z)) + (ρ(xn,z) −ρ(xn,xn))
and both terms on the right-hand side of the equation are non-negative by
(ssd). The equivalence of (b) and (c) is clear by Proposition 2.10. �

2.5. The local triangle axiom. In this subsection we study a certain condi-
tion on the convergence of sequences in a distance space 〈X,ρ〉, namely:
(L) qρ(x,yn) → 0, qρ(yn,zn) → 0 imply qρ(x,zn) → 0,
for any two sequences (yn), (zn) and element x from X. The property (L) of a
distance space is called here the local triangle axiom.

We give several characterization of the local triangle axiom. We show that
the condition implies partial semimetrizability of the underlying topology. We
discuss its dependence on the other axioms and the similarity to the triangle
inequality. For symmetric spaces in Topology a similar condition was recognised
as early as 1910 in the work of Fréchet [7] and investigated in [3, 4]. The
local triangle axiom was defined by Niemytzki [15] and analysed in [27]. A
more modern-style proof of metrizability of a topological space, which admits
a symmetric satisfying the local triangle axiom, is given by Archangelskij [2]
and explained in detail in [8]. The precise formulation of the Archangelskij
Metrization Theorem follows:

Theorem 2.15 (Archangelskij). Let the set X be a T1 topological space sym-
metrizable with respect to the symmetric d. If for all x ∈ X and all sequences
(yn), (zn) ⊆ X we have

d(x,yn) → 0 and d(yn,zn) → 0 imply d(x,zn) → 0,
then X is metrizable. �

In Section 3 we prove a counterpart of Archangelskij’s theorem, which works
in partial symmetric spaces. Here, we demonstrate some useful characteriza-
tions of the local triangle axiom and its basic implications.

In fact any partial symmetric with property (L) is a partial semimetric.

Proposition 2.16. Let 〈X,ρ〉 be a partial symmetric space. If the condition
(L) holds in X, then the map ρ is a partial semimetric.



The local triangle axiom 57

Proof. For any x ∈ X denote by Bx the collection of all ρ-balls centered at x.
For all x ∈ X the family Bx constitutes a neighbourhood base at x if and only
if for all x ∈ X and for all B ∈ Bx there exists C ∈ Bx such that if y ∈ C,
there is some D ∈ By with D ⊆ B. (See for example [26], Theorem 4.5 p.33.)
Equivalently, we have

∀x ∈ X. ∀ε > 0. ∃δ1 > 0. ∀y ∈ X. ∃δ2 > 0. ∀z ∈ X.
qρ(x,y) < δ1, qρ(y,z) < δ2 imply qρ(x,z) < ε,

or in other words,
∀x ∈ X. ∀ε > 0. ∀(yn), (zm) ⊆ X.
limn qρ(x,yn) = 0, limn limm qρ(yn,zm) = 0 imply limm qρ(x,zm) = 0,

which is in general a weaker condition than (L). �

Alternative proof : By Corollary 2.9 it is enough to show that for any subset
H of X, the set H′ := {x | qρ(x,H) = 0} is closed. Suppose not. Then
qρ(x,H′) = 0 for some x /∈ H′. By definition of the distance between a point
and a set, there exist sequences (yn) ⊆ H′ and (zn) ⊆ H such that qρ(x,yn) → 0
and qρ(yn,zn) → 0. By assumption, qρ(x,zn) → 0. Hence qρ(x,H) = 0 and
consequently, x ∈ H′, a contradiction. �

Proposition 2.17. For a distance space 〈X,ρ〉, the following are equivalent:
(1) ρ satisfies (L);
(2) ∀x ∈ X. ∀ε > 0. ∃δ > 0. ∀y,z ∈ X.

ρ(x,z) < ρ(x,x) + δ,ρ(z,y) < ρ(z,z) + δ imply ρ(x,y) < ρ(x,x) + ε.

Proof.

(2) holds
iff
∀x ∈ X. ∀ε > 0.
¬[∃(yk), (zk) ⊆ X. ρ(x,zk) → ρ(x,x),ρ(zk,yk) → ρ(z,z),

ρ(x,yk) ≥ ρ(x,x) + ε],
iff
∀x ∈ X.
¬[∃(yk), (zk) ⊆ X. ρ(x,zk) → ρ(x,x),ρ(zk,yk) → ρ(z,z),

ρ(x,yk) 9 ρ(x,x)],
iff
∀x ∈ X. ∀(yk), (zk) ⊆ X.
ρ(x,zk) → ρ(x,x),ρ(zk,yk) → ρ(z,z) imply ρ(x,yk) → ρ(x,x)

iff
(1) holds.

�

Therefore, we can easily show that every partial metric satisfies the local
triangle axiom. From the proof of the following result it is also obvious that
the converse claim will not hold in general.



58 P. Waszkiewicz

Corollary 2.18. Let 〈X,p〉 be a partial metric space. Then the map p satisfies
(L).

Proof. We will prove a more general statement. We claim that a mapping
ρ: X ×X → [0,∞) satisfies (∆]) if and only if
∀x,y,z ∈ X. ∀ε1,ε2 > 0. ρ(x,z) < ρ(x,x) + ε1 and ρ(z,y) < ρ(z,z) + ε2

imply ρ(x,y) < ρ(x,x) + ε1 + ε2.

The formula above implies (L) (which is easily seen by Proposition 2.17 (2)).
To prove the claim, let x,y,z ∈ X, ε1,ε2 > 0 and assume the hypothesis of

the implication above. Then by the sharp triangle inequality we have

ρ(x,y) ≤ ρ(x,z) + ρ(z,y) −ρ(z,z) < ρ(x,x) + ε1 + ε2,

as required. Conversely, take any ε > 0. We have ρ(x,z) < ρ(x,x) + (ρ(x,z)−
ρ(x,x) + ε) and ρ(z,y) < ρ(z,z) + (ρ(z,y) − ρ(z,z) + ε). By assumption,
ρ(x,y) < ρ(x,x) + (ρ(x,z) − ρ(x,x) + ε) + (ρ(z,y) − ρ(z,z) + ε) = ρ(x,z) +
ρ(z,y) − ρ(z,z) + 2ε. Hence the sharp triangle inequality follows from the
arbitrariness of ε and the claim is now proved. �

The following result generalizes a similar result for symmetric spaces ob-
tained by H. W. Martin in [12].

Proposition 2.19. Let 〈X,ρ〉 be a partial symmetric space. Then the following
are equivalent:

(1) The map ρ satisfies (L).
(2) If K is compact, H is closed and K ∩H = ∅, then qρ(K,H) > 0.

Proof. We will show that qρ(K,H) = 0 implies K ∩H 6= ∅. Let qρ(K,H) =
0. Hence there is a sequence (kn) of elements of K such that qρ(kn,H) <
1/2n+1 for every n ∈ ω. This means that there also exists a sequence (hn) of
elements from H such that qρ(kn,hn) → 0. By compactness of K, there exists
a convergent subsequence (knm) of (kn) with knm → x ∈ K. Proposition 2.16
guarantees that the map ρ is a partial semimetric and hence qρ(x,knm) → 0.
Then for the corresponding subsequence (hnm) of hn we have qρ(knm,hnm) → 0.
By assumption, qρ(x,hnm) → 0 and hence x ∈ cl(H) = H. This means that
K ∩H 6= ∅.

Conversely, for any two sequences (xn), (yn) and an element x of X, suppose
that qρ(x,xn) → 0 and qρ(xn,yn) → 0. Define a compact subset K of X and
a closed subset H of X in the following way:

K := {x}∪{xn | n ∈ ω},
H := clτρ{yn | n ∈ ω}.

There are two cases to consider: either K ∩ H = ∅ or K ∩ H 6= ∅. In the
former case, by assumption (2), we infer that there exists ε > 0 such that
qρ(K,H) > ε. Therefore,

infn{qρ(xn,yn)}≥ qρ(K,H) > ε,



The local triangle axiom 59

which is impossible since qρ(xn,yn) → 0. In the latter case we distinguish three
simple subcases. If no xn’s belong to K∩H, then x ∈ K∩H and hence (yn) →τρ
x, which by Proposition 2.10 is equivalent to saying that qρ(x,yn) → 0. If
infinitely many xn’s belong to K∩H, then since H is closed, also x ∈ K∩H and
then by the same argument as above, qρ(x,yn) → 0. Finally, if a finite number
of xn’s belong to K∩H, say x1, . . . ,xk−1 ⊆ K∩H, then consider the remaining
sequence (xn)n≥k and repeat the proof with K := {x}∪{xn | n ≥ k}. �

3. Metrizability of the induced distance space

In this section we present the first of the two major results of this paper.
We prove a counterpart for the Archangelskij Metrization Theorem (Theorem
2.15) working in partial symmetric spaces.

Lemma 3.1. If 〈X,ρ〉 is a semimetric space that satisfies (L), then the induced
distance space 〈X,dρ〉 satisfies (L).

Proof. Suppose that for some sequences (yn), (zn) ⊆ X and an element x ∈ X
we have lim dρ(x,yn) = 0 and lim dρ(yn,zn) = 0. Then by Proposition 2.14.(b),
one sees that lim ρ(x,yn) = ρ(x,x), lim ρ(yn,zn) = lim ρ(yn,yn) and ρ(x,x) =
lim ρ(zn,zn). The two former equalities imply that lim ρ(x,zn) = ρ(x,x) by
(L) for the map ρ. Together with the third equality, we get lim dρ(x,zn) = 0
by yet another application of Proposition 2.14. �

Theorem 3.2. For any partial symmetric space 〈X,ρ〉 with (L) the induced
distance dρ is metrizable.

Proof. By Lemma 3.1 and Proposition 2.16, the induced distance is a semi-
metric and satisfies (L).

We will show that 〈X,dρ〉 is Hausdorff. Since it is first-countable by Propo-
sition 2.10, it will be enough to demonstrate that limits of sequences are unique
in X. Hence, let (yn) be a sequence of elements of X and suppose that (yn)
has two limits x and z in X. Then again by Proposition 2.10 applied to the
mapping dρ we have dρ(x,yn) → 0 and dρ(z,yn) → 0. Using (L) we con-
clude that dρ(x,z) = 0. The last equality is equivalent to ρ(x,z) = ρ(x,x) and
ρ(x,z) = ρ(z,z). The characterization of the order in partial semimetric spaces
from Proposition 2.11 implies that x vτρ z and z vτρ x. Hence, x = z using
the T0 axiom.

Taking all of the proved properties together, one can see that 〈X,dρ〉 is a
Hausdorff semimetric space, which satisfies the local triangle axiom. Therefore,
Archangelskij’s Theorem applies and we conclude that 〈X,dρ〉 is metrizable.

�

4. Quasi-developability of a distance space

As an application of the metrizability theorem proved in the last section, we
consider the problem of quasi-developability of partial symmetric spaces. Let
us first introduce the necessary terminology.



60 P. Waszkiewicz

Let 〈X,τ〉 be a topological space, x ∈ X and C be any collection of subsets
of X. Denote card{C ∈C | x ∈ C} by ord(x,C).

A sequence G1,G2,G3, . . . of collections of open subsets of a topological space
〈X,τ〉 is called a quasi-development for X provided that if x ∈ U ∈ τ, then there
exists n ∈ ω and G such that x ∈ G ∈Gn and St(x,Gn) ⊆ U, where St(x,Gn) :=⋃
{V ∈Gn | x ∈ V}. A topological space 〈X,τ〉 is quasi-developable if it admits

a quasi-development.
Proposition 1 of [10] states that every partial metric space is quasi-develop-

able. Since we have proved in Corollary 2.18 that every partial metric satisfies
(L) and noted that the converse does not hold in general, the following theorem
improves Künzi and Vajner’s result. We adapt the idea of the proof and the
notation from [10].

Theorem 4.1. Let 〈X,ρ〉 be a partial symmetric space with (L). Then the
topological space 〈X,τρ〉 is quasi-developable.

Proof. For simplicity, denote the induced semimetric by d, instead of the stan-
dard dρ. For each k,n ∈ ω, set

Ank := {x ∈ X | ρ(x,x) ∈ [(k − 1)2−n,k2−n)}.
Let

⋃
t∈ω Bt be a base for the metrizable induced topology τd such that each

collection Bt is discrete. For each t, l ∈ ω, define
Ctl := {x ∈ X | Bd(x, 2−l) hits at most one element of Bt}.

For each k,l,t ∈ ω set
Rklt := {intBρ(B ∩Ctl ∩A(l+1)k, 2−(l+1)) | B ∈Bt}.

We claim that
⋃
k,l,tRklt is a quasi-development for τρ.

For an arbitrary x ∈ X and a natural number h, denote by m := m(x,h)
the natural number chosen according to Proposition 2.17 (that is, for x ∈ X
and ε := 2−h we take δ := 2−m). By Proposition 2.16, the map ρ is a partial
semimetric. Therefore, x ∈ intBρ(x, 2−m). Using the fact that τρ ⊆ τd, there
exists t0 ∈ ω such that x ∈ B0 ⊆ intBρ(x, 2−m) ⊆ Bρ(x, 2−m) for some B0 ∈
Bt0 . Furthermore, since Bt0 is discrete, there is l0 ∈ ω with l0 ≥ m such that

(†) Bd(x, 2−l0 ) ∩B 6= ∅ and B ∈Bt0 imply B = B0.
Finally, there exists k0 ∈ ω such that x ∈ A(l0+1)k0 .

Note first that x ∈ B0 ∩Ct0l0 ∩A(l0+1)k0 . Hence

x ∈ intBρ(x, 2−(l0+1)) ⊆ intBρ(B0 ∩Ct0l0 ∩A(l0+1)k0, 2
−(l0+1)).

We claim that

Bρ(B0 ∩Ct0l0 ∩A(l0+1)k0, 2
−(l0+1)) ⊆ Bρ(x, 2−h),

which will show that
⋃
k,l,tRklt is a basis for τρ. Let y ∈ Bρ(B0 ∩ Ct0l0 ∩

A(l0+1)k0, 2
−(l0+1)). Then y ∈ Bρ(B0, 2−(l0+1)). That is, ρ(z,y) < ρ(z,z) +

2−(l0+1) for some z ∈ B0. Hence ρ(z,y) < ρ(z,z) + 2−m. On the other hand,



The local triangle axiom 61

since B0 ⊆ Bρ(x, 2−m), we have ρ(x,z) < ρ(x,x) + 2−m. From the last two
inequalities we conclude that ρ(x,y) < ρ(x,x) + 2−h, again using Proposition
2.17. That is, we have shown that y ∈ Bρ(x, 2−h).

It remains to prove that ord(x,Rk0(l0+1)t0 ) = 1. Let x ∈ Bρ(B0∩Ct0(l0+1)∩
A(l0+2)k0, 2

−(l0+2)). Then there is y ∈ B0 ∩ Ct0(l0+1) ∩ A(l0+2)k0 such that
ρ(y,x) < ρ(y,y) + 2−(l0+2).

We will demonstrate that

ρ(x,y) < ρ(x,x) + 2−(l0+1).

If ρ(x,x) > ρ(y,y), then ρ(x,y) = ρ(y,x) < ρ(x,x) + 2−(l0+1), using symmetry
of ρ. Otherwise, since x,y ∈ A(l0+2)k0 , we have ρ(y,y) < ρ(x,x) + 2

−(l0+2) and
so ρ(x,y) = ρ(y,x) < ρ(y,y) + 2−(l0+2) < ρ(x,x) + 2−(l0+1).

Hence d(x,y) = 2ρ(x,y) − ρ(x,x) − ρ(y,y) < 2−l0 , which means that
y ∈ Bd(x, 2−l0 ) ∩ B. By (†), we thus have B = B0. We have shown that
ord(x,Rk0(l0+1)t0 ) = 1. Now, it is immediate that the collection

⋃
k,l,tRklt is

a quasi-development for τρ. �

5. The self-distance mapping in distance spaces

The self-distance map µρ : X → [0,∞)op (called also a weight function) as-
sociated with a partial semimetric ρ proves to be an important object to study.
We start with some basic properties of the mapping. Later, in Section 5, we
will see that in Domain Theory weight functions correspond to measurements
in the sense of Martin [13]. In Section 6.4 we use measurements to build partial
metrics on algebraic domains.

Note that the codomain of the self-distance map is the set of non-negative
real numbers with the opposite of the natural order.

In this section we consider the interplay between self-distance maps and the
specialisation order in partial semimetric spaces.

Proposition 5.1. The self-distance map associated with a partial semimetric
ρ is monotone and strictly monotone with respect to the specialisation orders
of its domain and codomain.

Proof. For any x,y ∈ X, x vτρ y is equivalent to ρ(x,y) = ρ(x,x) by Propo-
sition 2.11. But by (ssd), ρ(y,y) ≤ ρ(x,y) = ρ(x,x). That is, µρy ≤ µρx.
Hence the map µρ is monotone.

Here, strict monotonicity is the condition

∀x,y ∈ X. (x vρ y and µρx = µρy) imply x = y
and one can note that this condition is an equivalent formulation of the T0
axiom of the space (using Corollary 2.12). �

For the self-distance map µρ associated with a partial symmetric ρ define

(5.1) µρ(x,ε) := {y ∈ X | y vτρ x ∧ µρy < µρx + ε}.
We say that µρ(x,ε) is the set of elements of X which are ε-close to x.



62 P. Waszkiewicz

Lemma 5.2. Let 〈X,ρ〉 be a partial semimetric space. Then

∀ε > 0. ∀x ∈ P. µρ(x,ε) ⊆ Bρ(x,ε).

Proof. Suppose z ∈ µρ(x,ε). Since z vρ x, we have ρ(x,z) = µρz by semi-
metrizability. Therefore, ρ(x,z) = µρz < µρx + ε. That is, z ∈ Bρ(x,ε). �

Proposition 5.3. Let 〈X,ρ〉 be a partial semimetric space, x ∈ X and S any
subset of X. If S v x and µρx = inf{µρs | s ∈ S}, then x is the supremum of
S.

Proof. Let x ∈ U ∈ τρ and let u be any upper bound of S. Since U is open,
there exists ε > 0 such that x ∈ Bρ(x,ε) ⊆ U.

By assumption, there exists s ∈ S with µρs < µρx + ε. Since s v x,
s ∈ µρ(x,ε). By Lemma 5.2, s ∈ Bρ(x,ε) and so s ∈ U. But the latter set is
upper, and therefore u ∈ U. We have shown that for any U ∈ τρ, x ∈ U implies
u ∈ U and hence x vτρ u follows. This means that x =

⊔
S. �

5.1. Stability condition for partial semimetrics. It happens that there
exists a class of partial semimetric spaces, where the distance topology can be
recovered from self-distance maps. Continuous domains in their Scott topology
(see Section 6.2) are our major example of such spaces. Here, we develop the
basics. For more information, consult [13, 25, 24].

Definition 5.4. Let 〈X,τρ〉 be a partial symmetric space. We say that the
map ρ is stable if for all x,y ∈ X we have

ρ(x,y) := inf{µρz | z vτρ x,y}.

Lemma 5.5. Let 〈X,ρ〉 be a partial semimetric space. Then

∀ε > 0. ∃δ > 0. ∀x ∈ P. µρ(x,δ) ⊆ int(Bρ(x,ε)).

Proof. Let x ∈ X and ε > 0. Then by definition, there exists δ > 0 such that
x ∈ Bρ(x,δ) ⊆ int(Bρ(x,ε)). By Lemma 5.2, µρ(x,δ) ⊆ int(Bρ(x,ε)). �

Lemma 5.6. Let 〈X,ρ〉 be a stable partial semimetric space. Then for every
x ∈ X and ε > 0 we have

Bρ(x,ε) ⊆↑µρ(x,ε).

Proof. Let x ∈ X, ε > 0 and y ∈ Bρ(x,ε). Then ρ(x,y) < µρx+ε. By stability,
there exists z vρ x,y with µρz < ρ(x,y) + ε and hence z ∈ µρ(x,ε). Then,
y ∈↑µρ(x,ε) follows. �

Theorem 5.7. Let (X,ρ) be a stable partial semimetric space. Then

{↑µρ(x,ε) | ε > 0}

is a neighborhood base at x in τρ.

Proof. Let x ∈ X and ε > 0. Then Bρ(x,ε) = ↑µρ(x,ε) by Lemma 5.2 and
Lemma 5.6. �



The local triangle axiom 63

6. Distance for continuous domains

As we have mentioned in the introduction, a major motivation for studying
general distance spaces was the desire to understand the concept of distance
for continuous domains. In previous sections we outlined a general theory,
which, we believe, proves especially useful in Domain Theory. In this section
we demonstrate that for continuous domains the theory of partial semimetric
spaces coincides with the theory of measurements introduced by Martin [13]
and studied further in the author’s PhD thesis [25]. Furthermore, we introduce
so called Lebesgue measurements (Definition 6.5), which correspond to partial
semimetrics, which satisfy condition (L). Finally, we prove a characterization
of partial metrizability of algebraic domains using Lebesgue measurements.

First, let us recall some terminology of Domain Theory. See [1] for more
information. Let P be a poset. A subset A ⊆ P of P is directed if it is nonempty
and any pair of elements of A has an upper bound in A. If a directed set A has
a supremum, it is denoted

⊔↑A. A poset P in which every directed set has a
supremum is called a dcpo.

Let x and y be elements of a poset P. We say that x approximates (is way-
below ) y if for all directed subsets A of P, y v

⊔↑A implies x v a for some
a ∈ A. We denote this by x � y. If x � x then x is called a compact element.
The subset of compact elements of a poset P is denoted K(P). Now, ↓↓x is
the set of all approximants of x below it. ↑↑x is defined dually. We say that a
subset B of a dcpo P is a basis for P if for every element x of P, the set ↓↓x∩B
is directed with supremum x. A poset is called continuous if it has a basis. It
can be shown that a poset P is continuous iff ↓↓x is directed with supremum
x, for all x ∈ P. A poset is called a continuous domain if it is a continuous
dcpo. Note that K(P) ⊆ B for any basis B of P. If K(P) is itself a basis, the
domain P is called algebraic.

In a continuous domain, every basis is an example of a so called abstract
basis, which is a set B together with a transitive relation ≺ on B, such that

(INT) M ≺ x implies ∃y ∈ B. M ≺ y ≺ x

holds for all elements x and finite subsets M of B. For an abstract basis 〈B,≺〉
and an element x ∈ B set x∗ := {y ∈ B | y ≺ x}.

For x ∈ B we also define x∗ := {y ∈ B | x ≺ y}. The collection of all sets of
the form x∗ is a basis for a topology on B called the pseudoScott topology [11].

In this paper, an abstract basis such that the relation ≺ is reflexive is named
a reflexive abstract basis. For any algebraic domain P, the set K(P) is an
example of a reflexive abstract basis.

For an abstract basis 〈B,≺〉 let I(B) be the set of all ideals (directed, lower
subsets) ordered by inclusion. It is called the (rounded ) ideal completion of B.
For any algebraic domain P the rounded ideal completion of K(P) is isomorphic
to P, in symbols: I(K(P)) ∼= P.



64 P. Waszkiewicz

Upper sets inaccessible by directed suprema form a topology called the Scott
topology. The specialisation order of the Scott topology on a poset coincides
with the underlying order. The collection {↑↑x | x ∈ P} forms a basis for the
Scott topology on a continuous domain P. The Scott topology satisfies only
weak separation axioms: it is always T0 on a poset but T1 only if the order is
trivial. The Scott topology on a poset P will be denoted σ(P) (or σ for short).

6.1. Measurements. We say that a monotone mapping µ: P → [0,∞)op in-
duces the Scott topology on a poset P if

∀U ∈ σ(P). ∀x ∈ P. ∃ε > 0. µ(x,ε) ⊆ U,
where

µ(x,ε) := {y ∈ P | y v x and µy < µx + ε}.
We denote this by µ−→σ(P).

Definition 6.1. If P is a continuous poset, µ: P → [0,∞)op a Scott-continuous
map with µ −→ σ(P), then we will say that µ measures P or that µ is a
measurement on P.

Our definition of a measurement is a special case of the one given by Martin.
In the language of [13] our maps are measurements, which induce the Scott
topology everywhere.

6.2. Partial semimetrics versus measurements.

Definition 6.2 (Martin). Let P be a continuous poset with a measurement
µ: P → [0,∞)op. The map pµ : P ×P → [0,∞)op defined by

pµ(x,y) :=
⊔
{µz | z � x,y} = inf{µz | z � x,y}

is the partial semimetric associated with µ (cf. Proposition 6.3 below).

Note that the definition is well-formed if any two elements x,y of P are
bounded from below. This condition, however, may be omitted: whenever
x,y have no lower bound, we scale the measurement to µ∗ : P → [0, 1)op with
µ∗(x) := µx/(1 + µx) for any x ∈ P. Such map is again a measurement (cf.
Lemma 5.3.1 of [13], page 135). Now, we define p∗µ to be:

∀x,y ∈ P. p∗µ(x,y) =
{

inf{µ∗z | z � x,y} if ∃z ∈ P. z v x,y
1 otherwise.

Proposition 6.3. Let µ: P → [0,∞)op be a measurement on a continuous
poset P . Then:

(1) pµ is a Scott-continuous map from P ×P to [0,∞)op.
(2) pµ(x,x) = µx for all x ∈ P .
(3) Bpµ(x,ε) = ↑µ(x,ε) for all x ∈ P and ε > 0. That is, pµ is a stable

partial semimetric, which induces the Scott topology.
(4) For a sequence (xn) and any x ∈ P , xn → x in the Scott topology on

P iff lim pµ(xn,x) = µx.



The local triangle axiom 65

Proof. For the proof of statements (1)–(3) consult [13]. Lastly, (4) follows from
Proposition 2.10. �

We are now ready to discuss the coincidence of the theory of partial semi-
metric spaces with the theory of measurements in the framework of continuous
domains.

Theorem 6.4. For a continuous poset P the following are equivalent:
(1) P admits a stable partial semimetric compatible with the Scott topology;
(2) P admits a partial semimetric compatible with the Scott topology, which

is Scott-continuous as a map from P ×P to [0,∞)op;
(3) P admits a measurement.

Proof. The implication (1)⇒(2) is trivial. For (2)⇒(3), the self-distance map-
ping of the partial semimetric has the defining measurement property µ → σ(P)
by Lemma 5.5. In addition, it is Scott-continuous since the partial semimetric
is. Lastly, (3)⇒(1) is a consequence of Proposition 6.3.(3). �

6.3. Lebesgue measurements. For the purpose of the next definition we
introduce the following notation.

(6.2) µ(A,ε) :=
⋃
{µ(x,ε) | x ∈ A},

where A is a subset of a continuous domain P, the map µ: P → [0,∞)op is
monotone and ε > 0.

Definition 6.5. Let P be a continuous domain. A Scott-continuous map
µ: P → [0,∞)op is a Lebesgue measurement on P if for all Scott-compact (we
may assume saturated) subsets K ⊆ P and for all Scott-open subsets U ⊆ P,

K ⊆ U ⇒ ∃ε > 0. µ(K,ε) ⊆ U.

One can immediately see from the definitions that Lebesgue measurements
are measurements.

Proposition 6.6. Let P be a continuous domain equipped with a measurement
µ: P → [0,∞)op. The following are equivalent:

(1) µ is a Lebesgue measurement.
(2) If ↑x ⊆ U for some x in P and a Scott-open subset U of P , then there

exists ε > 0 such that µ(↑x,ε) ⊆ U.

Proof. For the nontrivial direction, suppose that K ⊆ U, where K is a Scott-
compact saturated subset of P and U is Scott-open in P. Then for all k ∈ K
choose an element l � k with l ∈ U. The collection {↑↑l | l � k} is an open
cover of K. Hence a finite subcollection ↑↑l1,↑↑l2, . . . ,↑↑ln covers K already. By
assumption, for every li, i = 1, . . . ,n there is εi > 0 with µ(↑li,εi) ⊆ U. Set
ε := min{εi | i = 1, 2, . . . ,n}. Then K ⊆ µ(K,ε) ⊆ U. �

Next, we demonstrate that whenever µ is a Lebesgue measurement on a
continuous domain, its induced partial semimetric pµ satisfies condition (L).



66 P. Waszkiewicz

Proposition 6.7. Let P be a continuous domain measured by µ. The following
are equivalent:

(1) µ is a Lebesgue measurement;
(2) For all x ∈ P and for all sequences (xn), (yn) of P ,

pµ(x,xn) → µx and pµ(xn,yn) → µxn imply pµ(x,yn) → µx.

Proof. Let x ∈ P and take a Scott-open set U with x ∈ U. The assumption
pµ(x,xn) → µx is equivalent to saying that (xn) →σ x by Proposition 6.3.
That is,

∃N1 ∈ ω. ∀n ≥ N1. xn ∈ U.
By definition, pµ(xn,yn) = inf{µz | z � xn,yn}, for each n ∈ ω. Therefore,
there exists a sequence (zn) with zn � xn,yn and lim µzn = lim pµ(xn,yn), for
all n ∈ ω. Since by assumption, lim pµ(xn,yn) = lim µxn, we have lim µzn =
lim µxn. This means

∀ε > 0. ∃N2 ∈ ω. ∀n ≥ N2. µzn −µxn < ε.

Note that the set K := {xn | n ≥ N1}∪{x} is a compact subset of U, and
since µ is a Lebesgue measurement, the condition specialises to:

∃λ > 0. ∀xn ∈ K. ∀z ∈ P. [z v xn and µz < µxn + λ] ⇒ z ∈ U.

Hence zn ∈ U for all n ≥ max{N1,N2(λ)}. Since zn � yn for all n ∈ ω, (yn)
is cofinally in U. That is, (yn) →σ x, or equivalently, pµ(x,yn) → µx.

For the converse, observe that by Proposition 2.19 for any Scott-open set
U ⊆ P and x ∈ U we have qpµ(↑x,P \U) > 0. This is however equivalent to
saying that there exists ε > 0 such that

Bpµ(↑x,ε) :=
⋃
xvy

Bpµ(y,ε) ⊆ U.

But the map pµ is stable and hence Bpµ(y,ε) = ↑µ(y,ε) by the proof of Theo-
rem 5.7. We conclude that µ(↑x,ε) ⊆ U. Therefore, the map µ is a Lebesgue
measurement by Proposition 6.6. �

The last result can be easily extended and stated in a form analogous to
Theorem 6.4.

Theorem 6.8. For a continuous domain P the following are equivalent:
(1) P admits a stable partial semimetric with (L) for the Scott topology;
(2) P admits a partial semimetric with (L) for the Scott topology, which is

Scott-continuous as a mapping from P ×P to [0,∞)op;
(3) P admits a Lebesgue measurement.

Proof. It is enough to show (2)⇒(3). Let p: P × P → [0,∞) be a partial
semimetric for the Scott topology, which satisfies (L). Then for any Scott-
compact subset K and for any Scott-open subset U of P with K ⊆ U, we have
qρ(K,P \ U) > 0, by Proposition 2.19. It is equivalent to say that for some
ε > 0 we have K ⊆ Bp(K,ε) ⊆ U. Denote the self-distance map for p by µp.



The local triangle axiom 67

Now, by Lemma 5.2, we have µp(K,ε) ⊆ Bp(K,ε) ⊆ U and so the mapping µp
is a Lebesgue measurement on P. �

6.4. Partial metrization of algebraic domains. In this section we apply
the knowledge about Lebesgue measurements to obtain a complete character-
ization of partial metrizability of the Scott topology on an algebraic domain.
We start from a similar result obtained by Künzi and Vajner in Proposition 3
p.73 of [10].

Theorem 6.9 (Künzi and Vajner). A poset X admits a partial metric for its
Alexandrov topology iff there is a function | · |: X → [0,∞) such that

(∗) ∀x ∈ X. ∃ε > 0. ∀y ∈↑x. ∀z ∈↓y \↑x. |z|− |y| ≥ ε.

For the following crucial lemma, recall the notation from the beginning of
Section 6.

Lemma 6.10. Let (B,≺) be a reflexive abstract basis equipped with a map-
ping µ: B → [0,∞)op, which satisfies (∗). Then there exists a partial metric
compatible with the Scott topology on the rounded ideal completion I(B) of B.

Proof. Note that by the discussion in [10] p.74 we can always assume that µ is
bounded by 1. Therefore, applying Theorem 6.9 we conclude that there exists a
partial metric p: B×B → [0,∞) which is bounded by 3 and such that µp = µ.
The partial metric p satisfies

(+) ∀x ∈ X. ∃εx > 0. ↑x = {y | p(x,y) = p(x,x)} = {y | p(x,y) <
p(x,x) + εx},

Now, extend the function p to I(B) in the following way: for each I,J ∈I(B)
define p̂: I(B) ×I(B) → [0,∞) by

p̂(I,J) := inf{p(x,y) | x ∈ I,y ∈ J}.
Since p is continuous as a map from B equipped with the pseudoScott topology
to [0,∞)op in its Scott topology, the mapping p̂: I(B) ×I(B) → [0,∞)op is
Scott continuous. Note that for every x,y ∈ B we have p̂(x∗,y∗) = p(x,y).
This means that p induces the subspace Scott topology on the image of B in
I(B) under the canonical embedding.
Step 1: First we will prove that the map so defined satisfies all the partial
metric axioms except (T0). For the sharp triangle inequality, note that for all
x ∈ I, y ∈ J and z ∈ K, where I,J,K ∈ I(B), we have p(x,z) ≤ p(x,y) +
p(y,z) −p(y,y) and this inequality extends to infima. Similarly we prove the
(ssd) axiom. Symmetry is trivial.

Step 2: If I ⊆ J in I(B), then {p(x,y) | x ∈ I,y ∈ I}⊆ {p(x,y) | x ∈ I, y ∈
J}. Therefore, p̂(I,I) ≥ p̂(I,J). Hence p̂(I,I) = p̂(I,J) by (ssd).
Step 3: To prove that p̂ induces the Scott topology on I(B) we will show
that ↑↑x∗ = Bp̂(x∗,ε) for some x ∈ B and ε > 0. Suppose x∗ � K for some
K ∈ I(B). Then x∗ ⊆ K and so p̂(x∗,K) = p̂(x∗,x∗) by Step 2. Hence,
↑↑x∗ ⊆ Bp̂(x∗,ε) for any ε > 0. Conversely, if for some L ∈I(B) we have that



68 P. Waszkiewicz

p̂(x∗,L) < p̂(x∗,x∗) + εx, then p̂(x∗,w∗) < p̂(x∗,x∗) + εx for some w ∈ L. This
is however equivalent to p(x,w) < p(x,x) + εx and so w ∈ Bp(x,εx) = ↑x by
(+). That is, x ≺ w. This means that x ∈ L and so x∗ � L. We have shown
that Bp̂(x∗,εx) ⊆↑↑x∗.

Step 4: By Step 3, the order induced by p̂ agrees with subset inclusion, which
is the specialisation order for the Scott-topology on I(B). Hence if p̂(I,J) =
p̂(I,I) = p̂(J,J) for some I,J ∈I(B), then I ⊆ J and J ⊆ I. Therefore, I = J
and so p̂ is a partial metric on I(B). �

Lemma 6.11. Let P be an algebraic domain with a monotone map µ: P →
[0,∞)op. Then the local triangle axiom for µ implies (∗) for the restriction of
µ to K(P).

Proof. Take any x ∈ K(P). The set ↑x is Scott open and Scott-compact in P.
By the local triangle axiom,

qµ(↑x,P \↑x) > 0.

Therefore, there exists an ε > 0 such that

qµ(↑x,P \↑x) > ε.

This implies in particular that for every y ∈ ↑x and z ∈ ↓y \ ↑x we have
pµ(y,z) ≥ µy + ε. Hence µz = pµ(y,z) ≥ µy + ε. This shows (∗). �

Therefore, we have obtained a complete characterization of partial metriz-
ability on algebraic domains:

Theorem 6.12. Let P be an algebraic dcpo. The following are equivalent:
(1) P admits a Lebesgue measurement µ: P → [0,∞)op.
(2) P admits a partial metric compatible with the Scott topology on P .

Proof. By Lemma 6.11 and 6.10, every Lebesgue measurement on P extends
to a partial metric for the Scott topology on I(P) ∼= P. Conversely, by Corol-
lary 2.18, the self-distance map for a partial metric for the Scott topology sat-
isfies (L) and is Scott-continuous by a result from [9]. Hence it is a Lebesgue
measurement on P by Theorem 6.8. �

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Received December 2001

Revised January 2003



70 P. Waszkiewicz

P. Waszkiewicz

School of Computer Science,
The University of Birmingham,
Edgbaston,
Birmingham B15 2TT,
UK.

E-mail address : P.Waszkiewicz@cs.bham.ac.uk


	The local triangle axiom in topology and domain theory. By P. Waszkiewicz