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@ Appl. Gen. Topol. 19, no. 1 (2018), 173-187doi:10.4995/agt.2018.7997
c© AGT, UPV, 2018

Some aspects of Isbell-convex quasi-metric

spaces

Olivier Olela Otafudu

School of Mathematics, University of the Witwatersrand Johannesburg 2050, South Africa

(olivier.olelaotafudu@wits.ac.za)

Communicated by H.-P. A. Künzi

Abstract

We introduce and investigate the concept of geodesic bicombing in

T0-quasi-metric spaces. We prove that any Isbell-convex T0-quasi-

metric space admits a geodesic bicombing which satisfies the equiv-

ariance property. Additionally, we show that many results on geodesic

bicombing hold in quasi-metric settings, provided that nonsymmetry

in quasi-metric spaces holds.

2010 MSC: 54E50; 30L05.

Keywords: Isbell-convexity; geodesic bicombing; injectivity.

1. Introduction

Let (X,d) be a metric space. A map σ : X × X × [0,1] → X is said to be a
geodesic bicombing if for every (x,y) ∈ X × X,

(1.1) σ(x,y,0) = x, σ(x,y,1) = y

and

(1.2) d(σ(x,y,t),σ(x,y,t′) = |t − t′|d(x,y)

whenever t,t′ ∈ [0,1] (see [4, 10]). Furthermore, a geodesic bicombing σ on a
metric space (X,d) is conical if

d(σ(x,y,t),σ(x′,y′, t)) ≤ (1 − t)d(x,x′) + td(y,y′)

whenever t,t′ ∈ [0,1] and x,y,x′,y′ ∈ X.

Received 11 September 2017 – Accepted 14 November 2017

http://dx.doi.org/10.4995/agt.2018.7997


O. Olela Otafudu

In [4] Descombes and Lang discussed and compared three different convexity
notions (convex and consistent, convex and conical) for geodesic bicombings.
They proved that Busemann spaces, and in particular CAT(0) spaces admit
geodesic bicomings which are convex and consistent. Additional to this, they
proved that every Gromov hyperbolic group acts geometrically on a proper
finite-dimensional metric space with convex and consistent geodesic bicombing.

Isbell [6] and Dress [5] developed independently the concept of injective hull
in the category of metric spaces with nonexpansive maps as morphisms. Isbell
proved that the injective hull (unique up to isometry) of a metric space is
hyperconvex by appealing to Zorn’s lemma. Later on Lang [10] presented a
new proof which is more constructive of the same result. Furthermore, Lang
proved that every metric space which is injective, admits a conical geodesic
bicombing.

The injective hull construction for metric spaces has been generalized in the
category of T0-quasi-metric spaces with nonexpansive maps as morphisms (see
[7]). Furthermore, an explicit description of the algebraic and vector lattice
operations on the Isbell-convex hull of an asymmetrically normed linear vector
space is proved in [3]. Naturally this led to the speculation that the Isbell-
convex hull of an Isbell-convex T0-quasi-metric space admits a conical geodesic
bicombing. The aim of this article is to give a careful and complete proof of
the aforementioned speculation.

We also discuss the continuity of a geodesic bicombing on a T0-quasi-metric
space. Furthermore, we prove that for a conical geodesic bicombing σ on a
T0-quasi-metric space (X,q), if a set A is bounded σ-convex on the set P0(X)
of nonempty subsets of (X,q), then its double closure clτ(q)A∩clτ(q−1)A is also
bounded σ-convex on P0(X). Let us mention that a conical geodesic bicombing
on a T0-quasi-metric space enjoys some property with the Takahashi convex
structure on the same T0-quasi-metric space. For details on Takahashi convex
structures on a T0-quasi-metric space, we refer the reader to [9].

2. Preliminaries

We start by recalling some useful concepts that we are going to use in the
sequel.

Definition 2.1. Let X be a nonempty set and q : X × X → [0,∞) be a map.
Then q is a quasi-pseudometric on X if

(a) q(x,x) = 0 whenever x ∈ X, and
(b) q(x,z) ≤ q(x,y) + q(y,z) whenever x,y,z ∈ X.

If q is a quasi-pseudometric on a set X, then the pair (X,q) is called a a quasi-
pseudometric space. Moreover, we say that q is a T0-quasi-metric provided that
it satisfies the additional condition that for any x,y ∈ X, q(x,y) = 0 = q(y,x)
implies that x = y. The set X together with a T0-quasi-metric on X is called
a quasi-metric space.

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Some aspects of Isbell-convex quasi-metric spaces

Remark 2.2. Note that if q is a quasi-metric on X, then q−1 : X × X →
[0,∞) defined by q−1(x,y) = q(y,x) whenever x,y ∈ X is also a quasi-
pseudometric on X, called the conjugate quasi-pseudometric of q. As usual,
a quasi-pseudometric d on X such that q = q−1 is called a pseudometric on
X. Furthermore, the map qs = max{q,q−1} is a pseudometric on X. If q is a
T0-quasi-metric on X, then q

s is a metric on X.

Let (X,q) be a quasi-pseudometric space and for each x ∈ X and r ∈ [0,∞),
let Cq(x,r) = {y ∈ X : q(x,y) ≤ r} be the τ(q

−1)-closed ball of centre x and
radius r. Furthermore, the open ball with centre x and radius r is represented
by Bq(x,r) = {y ∈ X : q(x,y) < r}.

Example 2.3. Let (X,q) be a T0-quasi-metric space. For any x,y ∈ X with
x 6= y and q(x,y) + q(y,x) 6= 0, the function uq(x,y),q(y,x) : R × R → R defined
by

uq(x,y),q(y,x)(λ,λ
′

) =

{

(λ − λ
′

)q(x,y) if λ ≥ λ
′

(λ
′

− λ)q(y,x) if λ < λ
′

is a T0-quasi-metric.

Take any T0-quasi-metric space (X,q). If q(x,y) = 1 and q(y,x) = 0 when-
ever x,y ∈ X, then the T0-quasi-metric uq(x,y),q(y,x) is the standard T0-quasi-
metric u on R, where u(x,y) = max{0,x − y} = x−̇y whenever x,y ∈ R.

Consider a T0-quasi-metric space (X,q). Let P0(X) be the set of all nonempty
subsets of X. We recall that for any given P ∈ P0(X), q(P,x) = inf{q(p,x) :
p ∈ P} and q(x,P) = inf{q(x,p) : p ∈ P} for all x ∈ X.

For any P,Q ∈ P0(X), the so-called Hausdorff (-Bourbaki) quasi-pseudometric
qH on P0(X) is defined by

qH(P,Q) = sup
x∈Q

q(P,x) ∨ sup
x∈P

q(x,Q).

It is well-known that qH is an extended quasi-pseudometric (qH may attain
the value ∞, then the triangle inequality is interpreted in the obvious way).
Moreover, qH is a T0-quasi-metric if we restrict the set P0(X) to the nonempty
subsets of P of X which satisfy P = clτ(q)P ∩ clτ(q−1)P (see [2, 8]).

Definition 2.4. ([9, Definition 7]) Let (X,q) be a T0-quasi-metric space. For
any subset P of X, we call clτ(q)P ∩clτ(q−1)P the double closure of P . Moreover
if P = clτ(q)P ∩ clτ(q−1)P , we say that P is doubly closed.

Definition 2.5. ([7, Definition 2]) A quasi-pseudometric space (X,q) is called
Isbell-convex (or q-hyperconvex) provided that for any family (xi)i∈I of points
in X and families (ri)i∈I and (si)i∈Iof nonnegative real numbers satisfying
q(xi,xj) ≤ ri + sj whenever i,j ∈ I, the following condition hold:

⋂

i∈I

[Cq(xi,ri) ∩ Cq−1(xi,si)] 6= ∅.

For more details about the theory of Isbell-convex T0-quasi-metric spaces,
we refer the reader to [3, 7, 11].

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O. Olela Otafudu

3. Geodesic bicombing

We start this section with the following observation.

Remark 3.1. We observed that the condition (1.2) is unsuitable for a T0-quasi-
metric space (X,d) that is not a metric: Indeed if d is a T0-quasi-metric space
with properties (1.1) and (1.2), then it satisfies

|0 − 1|d(x,y) = d(σ(x,y,0),σ(x,y,1)) = d(σ(y,x,1),σ(y,x,0)) = |1 − 0|d(y,x)

whenever x,y ∈ X and thus d would be a metric. Therefore, for a T0-quasi-
metric space (X,d) we propose the condition (1.2) differently in Definition 3.2.

Let I = [0,1] be the set of real unit interval. Using a different terminology,
it was essentially observed by Remark 3.1 that in a T0-quasi-metric space the
concept of geodesic bicombing can be modified in the following way:

Definition 3.2. Let (X,q) be a T0-quasi-metric space. A geodesic bicombing
σ on (X,q) is a map σ : X × X × I → X such that for each (x,y) ∈ X × X,
σ(x,y,0) = x, σ(x,y,1) = y and

(3.1) q(σ(x,y,λ),σ(x,y,λ′)) = (λ′ − λ)q(x,y) if λ′ ≥ λ

and

(3.2) q(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)q(y,x)) if λ′ < λ

whenever λ,λ′ ∈ I.

Definition 3.3. Let σ be a geodesic bicombing on a T0-quasi-metric space
(X,q). We say that σ satisfies the equivariance property if

σ(x,y,λ) = σ(y,x,1 − λ) whenever x,y ∈ X and λ ∈ [0,1].

Lemma 3.4. If σ is a geodesic bicombing on a T0-quasi-metric space (X,q),
then σ is a geodesic bicombing on the conjugate T0-quasi-metric space (X,q

−1).
Furthermore, σ is a geodesic bicombing on the metric space (X,qs)

Proof. Suppose that σ is a geodesic bicombing on (X,q). Let x,y ∈ X and
λ,λ′ ∈ I. Obviously, σ(x,y,0) = x, σ(x,y,1) = y.

If λ′ ≥ λ, then

q−1(σ(x,y,λ),σ(x,y,λ′)) = q(σ(x,y,λ′),σ(x,y,λ)) = (λ′ − λ)q−1(x,y).

If λ > λ′, we have

q−1(σ(x,y,λ),σ(x,y,λ′) = q(σ(x,y,λ′),σ(x,y,λ)) = (λ − λ)q−1(y,x).

So σ is a geodesic bicombing on (X,q−1).
If λ′ ≥ λ for λ,λ′ ∈ I, then we have

qs(σ(x,y,λ),σ(x,y,λ′)) = max{q(σ(x,y,λ),σ(x,y,λ′)),q−1(σ(x,y,λ),σ(x,y,λ′))}

= max{(λ′−λ)q(x,y),(λ′−λ)q−1(x,y)} = (λ′−λ)qs(x,y).

Similarly if λ > λ′,

qs(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)qs(x,y).

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Some aspects of Isbell-convex quasi-metric spaces

Hence qs(σ(x,y,λ),σ(x,y,λ′)) = |λ − λ′|qs(x,y) whenever λ,λ′ ∈ I. �

Example 3.5. If we equip (R,u) with σ(x,y,λ) = (1 − λ)x + λy whenever
x,y ∈ R and λ ∈ I, then σ is a geodesic bicombing on (R,u) and σ is called
the standard geodesic bicombing on (R,u).

Indeed, if λ′ ≥ λ, we have

u(σ(x,y,λ),σ(x,y,λ′)) = u((1 − λ)x + λy,(1 − λ′)x + λ′y)

= max{0,(1 − λ)x + λy − [(1 − λ′)x + λ′y]}.

So

u(σ(x,y,λ),σ(x,y,λ′)) = max{0,(λ′ − λ)x − (λ′ − λ)y} = (λ′ − λ)u(x,y).

By similar arguments if λ > λ′, then

u(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)u(y,x).

Lemma 3.6. Let σ be a geodesic bicombing on a T0-quasi-metric space (X,q).
Then the map σ−1 defined by σ−1(x,y,λ) = σ(y,x,1 − λ) whenever x,y ∈ X
and λ ∈ I, is a geodesic bicombing on (X,q). The geodesic bicombing σ−1 is
called reversible geodesic bicombing of σ (see [4, p.2]).

Proof. Let x,y ∈ X and λ,λ′ ∈ I. We have σ−1(x,y,0) = σ(y,x,1) = x and
σ−1(x,y,1) = σ(y,x,0) = y.

If λ′ ≥ λ, then (1 − λ) > (1 − λ′). It follows that

q(σ−1(x,y,λ),σ−1(x,y,λ′)) = q(σ(y,x,1 −λ),σ(y,x,1 −λ′)) = (λ′ −λ)q(y,x).

If λ > λ′, then one sees that

q(σ−1(x,y,λ),σ−1(x,y,λ′)) = (λ − λ′)q(x,y).

�

Example 3.7. Let C be a convex subset of a real linear space X equipped with
the asymmetric norm ‖.|. Then σ(x,y,λ) = (1 − λ)x + λy whenever x,y ∈ C
and λ ∈ I, is a geodesic bicombing on (C,d), where d(x,y) = ‖x− y| whenever
x,y ∈ C.

Lemma 3.8. Let (X,q) be a T0-quasi-metric space with a geodesic bicombing
σ. Then we have σ(x,x,λ) = x whenever x ∈ X and λ ∈ I.

Proof. Let x ∈ X and λ ∈ I. Then

q(x,σ(x,x,λ)) ≤ q(x,σ(x,x,0))+q(σ(x,x,0),σ(x,x,λ)) = q(x,x)+λq(x,x) = 0.

Thus q(x,σ(x,x,λ)) = 0. Furthermore,

q(q(σ(x,x,λ),x) ≤ q(σ(x,x,λ),σ(x,x,0))+q(σ(x,x,0),x) = λq(x,x)+q(x,x) = 0.

Hence q(σ(x,x,λ),x) = 0 = q(x,σ(x,x,λ)). We have σ(x,x,λ) = x by T0-
property of (X,q). �

Note that a geodesic bicombing need not to be unique. To obtain the fol-
lowing embedding we assume that the geodesic bicombing is unique.

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O. Olela Otafudu

Lemma 3.9 (compare [9, Proposition 4]). Suppose that σ is the unique ge-
odesic bicombing on the T0-quasi-metric space (X,q). If for every x,y ∈ X
with x 6= y, the map ψ : (I,uq(x,y),q(y,x)) → (X,q) defined by ψ(λ) = σ(x,y,λ)
whenever λ ∈ I is an isometric embedding. (Here uq(x,y),q(y,x) is the restriction
of the T0-quasi-metric uq(x,y),q(y,x) given in Example 2.3 to I).

Proof. For λ,λ′ ∈ I, we show that

q(σ(x,y,λ),σ(x,y,λ′)) = uq(x,y),q(y,x)(λ,λ
′).

If λ′ ≥ λ, then q(σ(x,y,λ),σ(x,y,λ′)) = (λ′−λ)q(x,y) = uq(x,y),q(y,x)(λ,λ
′).

If λ > λ′, then q(σ(x,y,λ),σ(x,y,λ′)) = (λ−λ′)q(y,x) = uq(x,y),q(y,x)(λ,λ
′).
�

Lemma 3.10. Let σ be a geodesic bicombing on a T0-quasi-metric space (X,q).
Then whenever x,y,u ∈ X and λ ∈ I, σ satisfies the following inequalities:

(3.3) q(u,σ(x,y,λ)) ≤ q(u,x) + λq(x,y)

(3.4) q(u,σ(x,y,λ)) ≤ q(u,y) + (1 − λ)q(u,x)

(3.5) q(σ(x,y,λ),u) ≤ λq(y,x) + q(x,u)

(3.6) q(σ(x,y,λ),u) ≤ (1 − λ)q(x,y) + q(u,y).

Proof. We prove only (3.3) and (3.6), then (3.4) and (3.5) follow analogously.
Let x,y,x′,y′ ∈ X and λ ∈ I.

Then q(u,σ(x,y,λ)) ≤ q(u,σ(x,y,0)) + q(σ(x,y,0),σ(x,y,λ)) by triangle
inequality. Since σ(x,y,0) = x and from the equality (3.1) (λ ≥ 0), it follows
that

q(u,σ(x,y,λ)) ≤ q(u,x) + λq(x,y).

Furthermore, q(σ(x,y,λ),u) ≤ q(σ(x,y,λ),σ(x,y,1)) + q(σ(x,y,1),u), then

q(σ(x,y,λ),u) ≤ (1 − λ)q(x,y) + q(y,u)

by σ(x,y,1) = y and from the equality (3.1) (1 ≥ λ). �

Proposition 3.11 (compare [9, Proposition 2]). Let σ be a geodesic bicombing
on a T0-quasi-metric space (X,q). Then for each x ∈ X and λ ∈ I, σ is
continuous at (x,x,λ), where X carries the topology τ(q) (or τ(q−1)).

Proof. Consider the convergent sequence ((xn,yn,λn)) in X × X × I. Suppose
that ((xn,yn,λn)) converges to (x,x,λ) with respect to the topology induced
by q on X. The topology on I does not really matter. We have to prove that
the sequence (σ(xn,yn,λn)) converges to (σ(x,x,λ)). From Lemma 3.8, we

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Some aspects of Isbell-convex quasi-metric spaces

know that σ(x,x,λ) = x whenever λ ∈ I. By inequality (3.4) of Lemma 3.10,
whenever n ∈ N we have

q(x,σ(xn,yn,λn)) ≤ q(x,yn) + (1 − λn)q(x,xn).

Therefore, the sequence (σ(xn,yn,λn)) converges to (σ(x,x,λ)). One obtains
the similar result by using the topology induced by q−1 on X and inequality
(3.3). �

4. Conical geodesic bicombing

Definition 4.1. Let σ be a geodesic bicombing on a T0-quasi-metric space
(X,q). Then σ is said to be conical if

(4.1) q(σ(x,y,λ),σ(x′,y′,λ)) ≤ (1 − λ)q(x,x′) + λq(y,y′)

whenever x,y,x′,y′ ∈ X and λ ∈ I. Furthermore, the geodesic bicombing
σ is called convex if the function λ 7→ q(σ(x,y,λ),σ(x′,y′,λ)) is convex on I
whenever x,y,x′,y′ ∈ X and λ ∈ I.

The following ideas are not new and were inspired from [9].
Let σ be a conical geodesic bicombing on a T0-quasi-metric space (X,q).

A subset C of X is called σ-convex provided that σ(c,c′,λ) ∈ C whenever
c,c′ ∈ C and λ ∈ I. Observe that X is σ-convex subset of itself. Moreover,
each σ-convex subset C of X carries a natural conical geodesic bicombing,
which is the restriction of σ to C × C × I.

Proposition 4.2. Let σ be a conical geodesic bicombing on a T0-quasi-metric
space (X,q). Then whenever x ∈ X and r > 0, the closed balls Cq(x,r) and
Cq−1(x,s) and the open balls Bq(x,r) and Bq−1(x,s) are σ-convex subsets of
X.

Proof. We only prove that Cq(x,r) is σ-convex, the proofs related to the balls
Cq−1(x,s), Bq(x,r) and Bq−1(x,s) follow analogously. Suppose that x ∈ X,
r > 0 and λ ∈ I. Let y,z ∈ Cq(x,r). Then

q(x,σ(y,z,λ)) = q(σ(x,x,λ),σ(y,z,λ))

by Lemma 3.8. Furthermore,

q(x,σ(y,z,λ)) ≤ (1 − λ)q(x,y) + λq(x,z) ≤ (1 − λ)r + λr = r

since σ is conical, q(x,y) ≤ r and q(x,z) ≤ r. Thus σ(y,z,λ) ∈ Cq(x,r). �

Obviously, one can prove that the intersection of any family of σ-convex
subsets of (X,q) is σ-convex too.

Let CB0(X) be the subcollection of bounded σ-convex elements of P0(X).
In this case qH is a quasi-pseudometric since qH(A,B) < ∞. For more details
about how qH(A,B) < ∞, we refer the reader to [9, p.13].

Let σ be a conical geodesic bicombing on a T0-quasi-metric space (X,q). For
any A,B ∈ CB0(X) and λ ∈ I, set σ(A,B,λ) := {σ(a,b,λ) : a ∈ A,b ∈ B}.

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O. Olela Otafudu

Then we have that σ(A,B,λ) is nonempty and bounded. Let a,a′ ∈ A and
b,b′ ∈ B. Then we have that

q(σ(a,b,λ),σ(a′,b′,λ)) ≤ (1−λ)q(a,a′)+λq(b,b′) ≤ (1−λ)diam(A)+λdiam(B)

since σ is conical. One can add some conditions on σ in order the set σ(A,B,λ)
preserves σ-convexity.

We leave the proof of the following lemma to the reader.

Lemma 4.3 (compare [9, Lemma 3]). Let σ be a conical geodesic bicombing
on a T0-quasi-metric space (X,q). If A ∈ CB0(X), then its double closure
clτ(q)A ∩ clτ(q−1)A is contained in CB0(X).

5. Injective spaces

We start by recalling the construction of Isbell-convex hull (injective hull)
of a T0-quasi-metric space (X,q) and we refer the reader to [7] for more details.

Let FP(X,q) be the set of all pairs of functions f = (f1,f2), where fi :
X → [0,∞)(i = 1,2). A function pair (f1,f2) is said to be ample on (X,q) if
q(x,y) ≤ f2(x) + f1(y) whenever x,y ∈ X. The set of all function pairs which
are ample on (X,q) will be denoted by A(X,q). For a ∈ X, it is easy to see
that the function pair fa(x) = (q(a,x),q(x,a)) is element of A(X,q). For each
(f1,f2),(g1,g2) ∈ A(X,q), the map D defined by

D((f1,f2),(g1,g2)) = sup
x∈X

(f1(x)−̇g1(x)) ∨ sup
x∈X

(g2(x)−̇f2(x))

is an extended T0-quasi-metric on A(X,q).
Let (f1,f2) ∈ A(X,q). We say that (f1,f2) is extremal (or minimal) on

(X,q) if take (g1,g2) ∈ A(X,q) such that g1(x) ≤ f1(x) and g2(x) ≤ f2(x) for
x ∈ X, then f1(x) = g1(x) and f2(x) = g2(x).

The Isbell-convex hull or the injective hull of (X,q) is the set E(X,q) of all
extremal function pairs on (X,d).

It is well-known that if a function pair (f1,f2) is extremal, then f1 : (X,q
−1) →

(R,u) and f2 : (X,q) → (R,u) are nonexpansive map, that is

f1(x) − f1(y) ≤ q(y,x)

and

f2(x) − f2(y) ≤ q(x,y)

whenever x,y ∈ X. Furthermore,

f1(x) = sup
y∈X

(q(y,x)−̇f2(y))

and

f2(x) = sup
y∈X

(q(x,y)−̇f1(y))

whenever x ∈ X. For a ∈ X, it is easy to see that

fa(x) = (q(a,x),q(x,a)) ∈ E(X,q).

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Some aspects of Isbell-convex quasi-metric spaces

Let FPNexp(X,R) be the set of all function pairs whose the first component
is nonexpansive on (X,q−1) and the second component is nonexpansive on
(X,q). Observe that if (f1,f2) ∈ E(X,q), then (f1,f2) ∈ FPNexp(X,R).

We now consider the set A1(X,q) := A(X,q) ∩ FPNexp(X,R).

Lemma 5.1. Let (X,q) be a T0-quasi-metric space. If we equip FPNexp(X,R)
with the restriction of the extended T0-quasi-metric D, then D((f1,f2),(g1,g2)) <
∞. (Moreover D is a T0-quasi-metric).

Proof. Let (f1,f2) ∈ FPNexp(X,R). Then q(x,y) ≤ f2(x) + f1(y) whenever
x,y ∈ X and so

sup
x∈X

(q(x,y)−̇f2(x)) ≤ f1(y).

Moreover, we have

sup
x∈X

(f1(x)−̇q(y,x)) ≤ f1(y)

whenever x,y ∈ X, since f1 is nonexpansive on (X,q
−1).

Thus D((f1,f2),((fy)1,(fy)2)) ≤ f1(y) whenever y ∈ X. By similar ar-
guments one shows that D(((fy)1,(fy)2),(f1,f2)) ≤ f2(y) whenever y ∈ X.
Therefore, for y ∈ X we have D((f1,f2),(g1,g2)) ≤ f1(y) + g2(y) < ∞ when-
ever (f1,f2),(g1,g2) ∈ FPNexp(X,R). �

The following useful result is due to [1]. Its proof is based on Zorn’s Lemma
but a different proof of Proposition 5.2 can be given without appealing to Zorn’s
Lemma.

Proposition 5.2. Let (X,q) be a T0-quasi-metric space. There exists a retrac-
tion map p : A(X,q) → E(X,q), i.e., a map that satisfies the conditions

(a) D(p((f1,f2)),p((g1,g2))) ≤ D((f1,f2),(g1,g2)) whenever (f1,f2),(g1,g2) ∈
A(X,q).

(b) p((f1,f2)) ≤ (f1,f2) whenever (f1,f2) ∈ A(X,q).(In particular p((f1,f2)) =
(f1,f2) whenever (f1,f2) ∈ E(X,q)).

Remark 5.3. From Proposition 5.2, it follows that if (f1,f2),(g1,g2) ∈ A(X,q),
then D(p((f1,f2)),p((g1,g2))) can be ∞. But if (f1,f2),(g1,g2) ∈ FPNexp(X,R),
then D(p((f1,f2)),p((g1,g2))) is finite. Therefore, the restriction of the map
p : A(X,q) → E(X,q) to FPNexp(X,R) is a nonexpansive retraction.

Proposition 5.4. Let (X,q) be a T0-quasi-metric space. Then the T0-quasi-
metric spaces (FPNexp(X,R),D) and (E(X,q),D) are injective.

Proof. Let ∅ 6= A ⊆ B ⊆ X. Consider a map F : A → FPNexp(X,R)
defined by F(a) = fa = (q(a,.),q(.,a)) whenever a ∈ A. Obviously Fa is a
function pair which is ample, where q(a,.) is nonexpansive on (X,q−1) and
q(.,a) is nonexpansive on (X,q).

Let b ∈ B, we set fb = ((fb)1,(fb)2) where,

(fb)1(x) := inf
a∈A

{(fa)1(x) + q(b,a)}

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O. Olela Otafudu

and

(fb)2(x) := inf
a∈A

{(fa)2(x) + q(a,b)}

whenever x ∈ X.
We have to show that fb ∈ FPNexp(X,R).
For each x,y ∈ X, we have

(fb)1(x) − (fb)1(y) = inf
a∈A

{(fa)1(x) + q(b,a)} − inf
a′∈A

{(fa′)1(y) + q(b,a
′)}

≤ (fa)1(x) + q(b,a) − (fa)1(y) − q(b,a) with a = a
′

≤ q(a,y) + q(y,x) − q(a,y)

= q(y,x).

Similarly,

(fb)2(x) − (fb)2(y) ≤ q(x,y)

whenever x,y ∈ X. So (fb)1 and (fb)2 are nonexpansive.
To show that the function pair fb is ample, let x,y ∈ X. Then

(fb)2(x) + (fb)1(y) ≥ inf
a,a′∈A

{(fa)2(x) + q(a,b) + (fa′)1(x) + q(b,a
′)}

≥ inf
a,a′∈A

{(fa)2(x) + (fa′)1(x) + q(a,a
′)}

≥ inf
a∈A

{(fa)2(x) + q(a,y)} = inf
a∈A

{q(x,a) + q(a,y)}

≥ q(x,y).

Therefore, fb ∈ FPNexp(X,R).
Let b,b′ ∈ B and x ∈ X. We show that D(fb,fb′) ≤ q(b,b

′).
Indeed,

(fb′)2(x) − q(b,b
′) = inf

a∈A
{(fa)2(x) + q(a,b

′)} − q(b,b′)

= inf
a∈A

{(fa)2(x) + q(a,b
′) − q(b,b′)}

≤ inf
a∈A

{(fa)2(x) + q(a,b)}

≤ (fb)2(x).

Hence

(5.1) sup
x∈X

(

(fb′)2(x)−̇(fb)2(x)

)

≤ q(b,b′)

whenever b,b′ ∈ B and x ∈ X.
Similarly, we have

(5.2) sup
x∈X

(

(fb)1(x)−̇(fb′)1(x)

)

≤ q(b,b′)

for b,b′ ∈ B and x ∈ X. Combining (5.1) and (5.2) we have

D(fb,fb′) ≤ q(b,b
′)

for b,b′ ∈ B. We now show that fb = fb whenever b ∈ A.

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Some aspects of Isbell-convex quasi-metric spaces

If b ∈ A, then

(5.3) (fb)1(x) ≤ q(b,x) = (fb)1(x) whenever x ∈ X

since (fb)1(x) ≤ q(b,x) + q(b,b) and b ∈ A.
Moreover, (fb)1(x) = q(b,x) ≤ q(b,a) + q(a,x) = (fa)1(x) + q(b,a) for all

x ∈ X. Thus

(5.4) (fb)1(x) ≤ inf
a∈A

{(fa)1(x) + q(b,a)} = (fb)1(x)

for all x ∈ X and b ∈ B.
Hence, (fb)1(x) = (fb)1(x) for all x ∈ X and b ∈ A from (5.3) and (5.4).
Analogously, one shows that (fb)2(x) = (fb)2(x) whenever x ∈ X and b ∈ A.

Therefore the map F : B → FPNexp(X,R) defined by F(b) = fb whenever
b ∈ B, extends F . So (FPNexp(X,R),D) is injective. The injectivity of
(E(X,q),D) follows from Remark 5.3 since (E(X,q),D) is nonexpansive retract
of (FPNexp(X,R),D). ✷

The following observations are not new since they have been discussed in
[10] from the metric point of view.

Remark 5.5. Let (X,q) be a T0-quasi-metric space.

(a) If L : (E(X,q),D) → (E(X,q),D) is a nonexpansive map that fixes
eX(X) pointwise, then L is an identity on E(X,q).

Indeed, if f = (f1,f2) ∈ E(X,q) such that L(f) = (g1,g2) for some
g = (g1,g2) ∈ E(X,q), then,

g1(x) = D(g,fx) = D(L(f),L(fx))

≤ D(f,fx) = f1(x) for x ∈ X.

Similarly, g2(x) ≤ f2(x) whenever x ∈ X. By minimality of (f1,f2),
we have g1(x) = f1(x) and g2(x) = f2(x) whenever x ∈ X. Hence
f = (f1,f2) = (g1,g2) = g. Therefore, L(f) = f whenever f ∈ E(X,q).

(b) Since (E(X,q),D) is injective and eX is quasi-essential by [11, Remark
16], (E(X,q),eX) is an injective hull of (X,q).

(c) If (Y,i) is another injective hull of (X,q), then there exists an iso-
metric embedding of (X,q), and then there exists a unique isometry
I : (E(X,q),D) → (Y,i) such that I ◦ eX = i by [11, Proposition 10].

In [1], Agyingi et al. proved that if (Y,qY ) is a T0-quasi-metric space
and X is a subspace of (Y,qY ), then there exists an isometric embedding
τ : (E(X,q),D) → (Y,qY ) such that τ(f)|X = f whenever f ∈ E(X,q).

Proposition 5.6. Let (X,q) be a T0-quasi-metric space. If L : (X,q) →
(X,q) is an isometry, then there exists a unique isometry L̄ : (E(X,q),D) →
(E(X,q),D) such that L̄◦eX = eX ◦L. Furthermore, L̄(f) = (f1◦L

−1,f2◦L
−1)

whenever f ∈ E(X,q).

Proof. Suppose L : (X,q) → (X,q) is an isometry. Then eX ◦ L : (X,q) →
(E(X,q),D) is quasi-essential and since (E(X,q),D) is injective, it follows that
(E(X,q),eX ◦ L) is injective hull of (X,q) by Remark 5.5 (b).

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O. Olela Otafudu

Moreover, by Remark 5.5 (c), there exists a unique isometry such that L̄ ◦
eX = eX ◦ L.

If f = (f1,f2) ∈ E(X,q) and x ∈ X, then

(L̄(f))1(x) = D(L̄(f),fx) = D(L̄(f), L̄(L̄
−1(fx))) = D(f,L̄

−1(fx)).

Since eX ◦ L
−1 = L̄−1 ◦ eX,

fL−1(x) = (eX ◦ L
−1)(x) = (L̄−1 ◦ eX)(x) = L̄

−1(fx),

whenever x ∈ X.
Hence

(L̄(f))1(x) = D(f,L̄
−1(fx)) = D(f,fL−1(x)) = f1(L

−1(x)) = (f1 ◦ L
−1)(x).

By similar arguments we have

(L̄(f))2(x) = (f1 ◦ L
−1)(x)

whenever x ∈ X. �

Proposition 5.7. Let (X,q) be a T0-quasi-metric space. If L : (X,q) →
(X,q) is an isometry, then the function pair ψ(f) = L̄(f) is ample whenever
f = (f1,f2) ∈ A(X,q). Furthermore, we have L̄(p(f)) = p(L̄(f)) whenever
f = (f1,f2) ∈ A(X,q), where p is the map in Proposition 5.2 and L̄ is the
unique isometry map in Proposition 5.6.

Proof. Let f = (f1,f2) ∈ A(X,q). Then for any x,y ∈ X, we have

(L̄(f))2(x) + (L̄(f))1(y) = (f2 ◦ L
−1)(x) + (f1 ◦ L

−1)(y)

= f2(L
−1(x)) + f1(L

−1(y))

≥ q(L−1(x),L−1(y))

= q(x,y).

Let y ∈ X. Consider

f∗1 (y) = sup
x′∈X

{q(x′,y)−̇f2(x
′)},

f∗2 (y) = sup
x′∈X

{q(y,x′)−̇f1(x
′)}

and the operator q(f) = (1
2
(f1 + f

∗

1 ),
1
2
(f2 + f

∗

2 )) defined in the proof (given in
[1]) of Proposition 5.2. Then

(f∗1 ◦ L
−1)(y) = f∗1 (L

−1(y)) = sup
x′∈X

{q(x′,L−1(y))−̇f2(x
′)}

= sup
L−1(L(x′))∈X

{q(L−1(L(x′)),L−1(y))−̇f2(L
−1(L(x′)))}

= f1(L
−1)∗(y) = (f1 ◦ L

−1)∗(y).

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Some aspects of Isbell-convex quasi-metric spaces

Thus, we have that (f∗1 ◦ L
−1)(y) = (f1 ◦ L

−1)∗(y) whenever y ∈ X. Hence,
whenever x ∈ X we have

(q(f)1 ◦ L
−1)(x) =

(

1

2
(f1 + f

∗

1 ) ◦ L
−1

)

(x) =
1

2

(

f1(L
−1) + f∗1 (L

−1)

)

(x)

=
1

2

(

f1 ◦ L
−1 + f∗1 ◦ L

−1

)

(x)

= q(f ◦ L−1)1(x).

Similarly, we can show that (q(f)2 ◦ L
−1)(x) = q(f ◦ L−1)2(x) whenever

x ∈ X. Therefore,

L̄(p(f))1 = p(f)1 ◦ L
−1 = p(f ◦ L−1)1 = p(L̄(f))1

and

L̄(p(f))2 = p(f)2 ◦ L
−1 = p(f ◦ L−1)2 = p(L̄(f))2.

�

Proposition 5.8. Every Isbell-convex T0-quasi-metric space admits a conical
geodesic bicombing which satisfies the equivariance property.

Proof. Suppose that (X,q) is an Isbell-convex T0-quasi-metric space. Let x,y ∈
X and λ ∈ [0,1], we define a function pair ϕλxy = (ϕ

λ
xy,1,ϕ

λ
xy,2) by

ϕ
λ
xy,1(z) = (1 − λ)(fx)1(z) + λ(fy)1(z)

and

ϕλxy,2(z) = (1 − λ)(fx)2(z) + λ(fy)2(z)

whenever z ∈ X. We will prove that ϕλxy ∈ A1(X,q).

We first show that ϕλxy is ample. Let z,z
′ ∈ X, then

ϕλxy,2(z) + ϕ
λ
xy,1(z

′) = (1 − λ)q(z,x) + λq(z,y) + (1 − λ)q(x,z′) + λq(y,z′)

= (1 − λ)[q(z,x) + q(x,z′)] + λ[q(z,y) + q(y,z′)]

≥ (1 − λ)q(z,z′) + λq(z,z′)

= q(z,z′).

We now show that ϕλxy,2 is a nonexpansive map on (X,q) and the proof of the

fact that ϕλxy,1 is a nonexpansive map on (X,q
−1) follow by similar arguments.

Let z,z′ ∈ X, then

ϕλxy,2(z) − ϕ
λ
xy,2(z

′) = [(1 − λ)q(z,x) + λq(z,y)] − [(1 − λ)q(z′,x) + λq(z′,y)]

= (1 − λ)[q(z,x) − q(z′,x)] + λ[q(z,y) − q(z′,y)]

≤ q(z,z′).

Thus ϕλxy ∈ A1(X,q).
Since (X,q) an Isbell-convex T0-quasi-metric space, (X,q) is injective. Then

the map eX : (X,q) → E(X,q) defined by eX(x) = fx whenever x ∈ X, is an
isometry. We consider the retraction map p : A(X,q) → E(X,q) in Proposition
5.2. For any x,y ∈ X and λ ∈ [0,1], we set σ(x,y,λ) := (e−1

X
◦ p) ◦ ϕλxy.

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O. Olela Otafudu

Now we have to show that σ is a conical geodesic bicombing on X.
On sees that σ is well defined. Observe that if λ = 0, then ϕλxy = ((fx)1,(fx)2).

Moreover, if λ = 1, then ϕλxy = ((fy)1,(fy)2). It follows that

σ(x,y,0) = (e−1
X

◦ p) ◦ (((fx)1,(fx)2)) = e
−1
X

(eX(x)) = x

and

σ(x,y,1) = (e−1
X

◦ p) ◦ (((fy)1,(fy)2)) = e
−1
X

(eX(y)) = y,

since ((fx)1,(fx)2),((fy)1,(fy)2) ∈ E(X,q). Let x,y ∈ X and λ,λ
′ ∈ [0,1].

Then

q(σ(x,y,λ),σ(x,y,λ′)) = D(eX(σ(x,y,λ)),eX (σ(x,y,λ
′)))

= D[eX(e
−1
X

(p(ϕλxy))),eX(e
−1
X

(p(ϕλ
′

xy)))]

= D(p(ϕλxy),p(ϕ
λ
′

xy)).

Hence

q(σ(x,y,λ),σ(x,y,λ′)) ≤ D(ϕλxy,ϕ
λ
′

xy)

since p is a retraction.
Furthermore,

D(ϕλxy,ϕ
λ
′

xy) = sup
z∈X

[(1 − λ)q(x,z) + λq(y,z)−̇(1 − λ′)q(x,z) + λ′q(y,z)].

If λ′ ≥ λ, then by triangle inequality we have

D(ϕλxy,ϕ
λ
′

xy) ≤ (λ
′ − λ)q(x,y).

If λ′ < λ, then by triangle inequality we have

D(ϕλxy,ϕ
λ
′

xy) ≤ (λ − λ
′)q(y,x).

It follows that if λ′ ≥ λ, then

(5.5) q(σ(x,y,λ),σ(x,y,λ′)) ≤ (λ′ − λ)q(x,y)

and if λ′ < λ, then

(5.6) q(σ(x,y,λ),σ(x,y,λ′)) ≤ (λ − λ′)q(y,x).

Observe that for any x,y ∈ X and 0 ≤ λ ≤ λ′ ≤ 1, since σ(x,y,0) = x and
σ(x,y,1) = y we have the following equality from the inequality (5.5)

q(σ(x,y,λ),σ(x,y,λ′)) = (λ′ − λ)q(x,y).

Similarly, we obtain from inequality (5.6)

q(σ(x,y,λ),σ(x,y,λ′)) = (λ − λ′)q(y,x).

Therefore, σ is a geodesic bicombing on X. It remains to show that σ
satisfies the property (4.1) to be conical. Let x,y,x′,y′ ∈ X and λ ∈ [0,1].
Then

D(ϕλxy,ϕ
λ
x′y′) = sup

z∈X

[(1 − λ)q(x,z) + λq(y,z)−̇(1 − λ)q(x′,z) + λq(y′,z)]

≤ (1 − λ)q(x,x′) + λq(y,y′).

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Some aspects of Isbell-convex quasi-metric spaces

Hence

q(σ(x,y,λ),σ(x′,y′,λ) ≤ D(ϕλxy,ϕ
λ
x′y′) ≤ (1 − λ)q(x,x

′) + λq(y,y′).

Thus σ is a conical geodesic bicombing on X.
The equivariance follows from the observations below: for z ∈ X, we have

ϕ
1−λ
yx,1(z) = λ(fy)1(z) + (1 − λ)(fx)1(z) = ϕ

λ
xy,1(z)

and
ϕ
1−λ
yx,2(z) = λ(fy)2(z) + (1 − λ)(fx)2(z) = ϕ

λ
xy,2(z)

whenever x,y ∈ X and λ ∈ [0,1]. �

Acknowledgements. The author would like to thank the South African Na-
tional Research Foundation (NRF) and the Faculty of Science Research Com-
mittee (FRC) of University of the Witwatersrand for partial financial support.

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c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 187