@ Appl. Gen. Topol. 22, no. 2 (2021), 461-481doi:10.4995/agt.2021.15610
© AGT, UPV, 2021

Quantale-valued Cauchy tower spaces and

completeness

Gunther Jäger
a
and T. M. G. Ahsanullah

b

a
School of Mechanical Engineering, University of Applied Sciences Stralsund, 18435 Stralsund,

Germany (gunther.jaeger@hochschule-stralsund.de)
b
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi

Arabia (tmga1@ksu.edu.sa)

Communicated by J. Rodŕıguez-López

Abstract

Generalizing the concept of a probabilistic Cauchy space, we intro-
duce quantale-valued Cauchy tower spaces. These spaces encompass
quantale-valued metric spaces, quantale-valued uniform (convergence)
tower spaces and quantale-valued convergence tower groups. For spe-
cial choices of the quantale, classical and probabilistic metric spaces
are covered and probabilistic and approach Cauchy spaces arise. We
also study completeness and completion in this setting and establish
a connection to the Cauchy completeness of a quantale-valued metric
space.

2010 MSC: 54E15; 54A20; 54A40; 54E35; 54E70.

Keywords: Cauchy space; quantale-valued metric space; quantale-valued
uniform convergence tower space; completeness; completion;
Cauchy completeness.

1. Introduction

Cauchy spaces, axiomatized in [17], are a natural setting for studying com-
pleteness and completion [32]. In the probabilistic case such spaces are intro-
duced as certain “towers indexed by [0, 1]” by Richardson and Kent [33] and
by Nusser [27]. The connection to probabilistic metric spaces [34], however,
was not clarified there. In this paper, we generalize the approaches of [33, 27]

Received 13 May 2021 – Accepted 06 August 2021

http://dx.doi.org/10.4995/agt.2021.15610


G. Jäger and T. M. G. Ahsanullah

by allowing the index set to be a quantale. In this way, a quantale-valued
metric space, in particular a probabilistic metric space, possesses a “natural”
Cauchy structure, which then can in turn be used to study completeness and
completion. Further examples include quantale-valued uniform convergence
tower spaces and quantale-valued uniform spaces as well as quantale-valued
convergence groups. We study the basic categorical properties of the category
of quantale-valued Cauchy tower spaces and characterize those spaces that are
quantale-valued metrical. Finally we discuss completeness and completion and
we establish a connection with the Cauchy completeness [6] of a quantale-valued
metric space.

2. Preliminaries

Let L be a complete lattice with distinct top element ⊤ and bottom element
⊥. In L we can define the well-below relation α ✁ β if for all subsets D ⊆ L
such that β ≤

∨

D there is δ ∈ D such that α ≤ δ. Then α ≤ β whenever
α ✁ β and, for a subset B ⊆ L, we have α ✁

∨

β∈B β iff α ✁ β for some β ∈ B.
Sometimes we consider also a weaker relation, the way-below relation, α ≪ β
if for all directed subsets D ⊆ L such that β ≤

∨

D there is δ ∈ D such that
α ≤ δ. The properties of this relation are similar to the properties of the well-
below relation, replacing arbitrary subsets by directed subsets. But we also
have α ∨ β ≪ γ if α, β ≪ γ, [9].

A complete lattice is completely distributive, if and only if we have α =
∨

{β : β ✁ α} for any α ∈ L [31] and it is continuous if and only if we have
α =

∨

{β : β ≪ α} for any α ∈ L, [9]. Clearly α ✁ β implies α ≪ β and
hence every completely distributive lattice is also continuous. For more results
on lattices we refer to [9].

Lemma 2.1. Let L be a continuous lattice. Then

(1)
∨

δ≪α(δ ∗ δ) = α ∗ α.
(2) If ǫ ≪ α ∗ α, then there is δ ≪ α such that ǫ ≪ δ ∗ δ.

Proof. (1) We have
∨

δ≪α(δ ∗ δ) ≤ α ∗ α =
∨

δ≪α δ ∗
∨

γ≪α γ =
∨

δ,γ≪α δ ∗ γ ≤
∨

δ∨γ≪α(δ ∨ γ) ∗ (δ ∨ γ) ≤
∨

η≪α(η ∗ η).

(2) follows directly from (1) as the set {δ ∈ L : δ ≪ α} is directed. �

The triple L = (L, ≤, ∗), where (L, ≤) is a complete lattice, is called a
commutative and integral quantale if (L, ∗) is a commutative semigroup with
the top element of L as the unit, and ∗ is distributive over arbitrary joins, i.e.
if we have (

∨

i∈J αi) ∗ β =
∨

i∈J(αi ∗ β) for all αi, β ∈ L, i ∈ J. In a quantale
we can define an implication operator, α → β =

∨

{γ ∈ L : α ∗ γ ≤ β}, which
can be characterized by γ ≤ α → β ⇐⇒ γ ∗ α ≤ β. A quantale is called
divisible [11] if for β ≤ α there exists γ ∈ L such that β = α ∗ γ. In a divisible
quantale we have

∨

δ✁α(δ ∗ δ) = α ∗ α, see [30].
Prominent examples of quantales are e.g. the unit interval [0, 1] with a left-

continuous t-norm [34] or Lawvere’s quantale, the interval [0, ∞] with the op-
posite order and addition α ∗ β = α + β (extended by α + ∞ = ∞ + α = ∞),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 462



Quantale-valued Cauchy tower spaces and completeness

see e.g. [7]. A further noteworthy example is the quantale of distance distri-
bution functions. A distance distribution function ϕ : [0, ∞] −→ [0, 1], satisfies
ϕ(x) = supy<x ϕ(y) for all x ∈ [0, ∞]. The set of all distance distribution func-
tions is denoted by ∆+. With the pointwise order, the set ∆+ then becomes
a completely distributive lattice [7] with top-element ε0. A quantale operation
on ∆+, ∗ : ∆+ × ∆+ −→ ∆+, is also called a sup-continuous triangle function
[34].

We consider in the sequel only commutative and integral quantales L =
(L, ≤, ∗) with underlying complete lattices that are completely distributive.

For a set X, we denote its power set by P(X) and the set of all filters F, G, ...
on X by F(X). The set F(X) is ordered by set inclusion and maximal elements
of F(X) in this order are called ultrafilters. The set of all ultrafilters on X is
denoted by U(X). In particular, for each x ∈ X, the point filter [x] = {A ⊆
X : x ∈ A} is an ultrafilter. If F ∈ F(X) and f : X −→ Y is a mapping, then
we define f(F) ∈ F(Y ) by f(F) = {G ⊆ Y : f(F) ⊆ G for some F ∈ F}. For
filters Φ, Ψ ∈ F(X × X) we define Φ−1 to be the filter generated by the filter
base {F −1 : F ∈ Φ} where F −1 = {(x, y) ∈ X × X : (y, x) ∈ F} and Φ ◦ Ψ
to be the filter generated by the filter base {F ◦ G : F ∈ Φ, G ∈ Ψ}, whenever
F ◦ G 6= ∅ for all F ∈ Φ, G ∈ Ψ, where F ◦ G = {(x, y) ∈ X × X : (x, s) ∈
F, (s, y) ∈ G for some s ∈ X}.

For details and notation from category theory we refer to [1] and [29].

Definition 2.2 ([14]). Let L = (L, ≤, ∗) be a quantale. A pair (X, q = (qα)α∈L)
is called an L-convergence tower space [14], where (qα : F(X) −→ P(X))α∈L is
a family of mappings satisfying:

(LCTS1) x ∈ qα([x]), ∀x ∈ X and ∀α ∈ L;
(LCTS2) ∀F, G ∈ F(X), with F ≤ G, and α ∈ L implies qα(F) ⊆ qα(G);
(LCTS3) ∀α, β ∈ L with α ≤ β implies qβ(F) ⊆ qα(F), ∀F ∈ F(X);
(LCTS4) x ∈ q⊥(F), ∀x ∈ X, F ∈ F(X).

If, moreover, (X, q) satisfies

(LCTS5) qα(F) ∩ qα(G) ≤ qα(F ∧ G), ∀α ∈ L, for F, G ∈ F(X),

then the pair (X, q) is called an L-limit tower space. If (X, q) satisfies x ∈
q∨A(F) whenever x ∈ qα(F) ∀α ∈ A, it is called left-continuous. A mapping
f : (X, q) −→

(

X′, q′
)

between L-convergence tower spaces is called continuous
if, for all x ∈ X, and for all F ∈ F(X), f(x) ∈ q′α(f(F)) whenever x ∈ qα(F).
The category of all L-convergence tower spaces and continuous mappings is
denoted by L-CTS.

If L = {0, 1}, then L-convergence tower spaces can be identified with clas-
sical convergence spaces, [5, 29]. If L = ([0, ∞], ≥ +) is Lawvere’s quantale,
then an L-limit tower space is a limit tower space [4] and a left-continuous
L-limit tower space is an approach limit spaces in the sense of Lowen [20]. For
L = ([0, 1], ≤, ∗), we obtain probabilistic convergence spaces in the sense of
Richardson and Kent [33] and if L = (∆+ ≤, ∗), then an L-convergence tower
space is a probabilistic convergence space in the definition of [12].

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 463



G. Jäger and T. M. G. Ahsanullah

3. L-Cauchy tower spaces

Definition 3.1. Let L = (L, ≤, ∗) be a quantale. A pair (X, C) = (X, (Cα)α∈L)
is called an L-Cauchy tower space, where Cα ⊆ F(X) for all α ∈ L, if

(LChyTS1) [x] ∈ Cα for all x ∈ X, α ∈ L;
(LChyTS2) G ≥ F ∈ Cα implies G ∈ Cα;
(LChyTS3) α ≤ β, F ∈ Cβ implies F ∈ Cα;
(LChyTS4) C⊥ = F(X).
(LChyTS5) F ∈ Cα, G ∈ Cβ, F ∨ G exists, implies F ∧ G ∈ Cα∗β.

An L-Cauchy tower space is called left-continuous if F ∈ Cα for all α ∈ A ⊆
L implies F ∈ C∨ A. We call a mapping f : (X, C) −→ (X

′, C′) Cauchy-
continuous if F ∈ Cα implies f(F) ∈ C

′
α. The category with the L-Cauchy tower

spaces as objects and Cauchy-continuous mappings as morphisms is denoted
by L-ChyTS.

For L = ([0, 1], ≤, ∗) with a continous t-norm ∗, an L-Cauchy tower space is
a probabilistic Cauchy space in the definition of Nusser [26] and if ∗ = min,
then we obtain the probabilistic Cauchy spaces of Kent and Richardson [18].
In case of Lawever’s quantale L, an L-Cauchy tower space is a Cauchy tower
space in the definition of [25], which can be identified in the left-continuous
case with an approach Cauchy space [22].

We note that for (X, C) ∈ |L-ChyTS|, the “top level” (X, C⊤) is a classical
Cauchy space [17].

Proposition 3.2. The category L-ChyTS is topological. If the quantale opera-
tion is the minimum, i.e. if ∗ = ∧, then it is also a Cartesian closed category.

Proof. The topologicalness can be shown in a straight-forward way. We only
describe the initial constructions. For (fj : X −→ (Xj, Cj))j∈J we define for
F ∈ F(X), F ∈ Cα ⇐⇒ fj(F) ∈ C

j
α for all j ∈ J. Then (X, C) is an L-Cauchy

tower space and a mapping g : (Y, D) −→ (X, C) is Cauchy-continuous if and

only if fj ◦ g : (Y, D) −→ (Xj, Cj) is Cauchy-continuous for all j ∈ J.
To show the Cartesian closedness, we can follow the proof in [18]. We only

state the function space structures. We define for two spaces (X, C), (X′, C′) ∈
|L-ChyTS| the set of all Cauchy-continuous mappings by Hom(X, X′) = {f :
(X, C) −→ (X′, C′) : f Cauchy-continuous} and define for H ∈ F(Hom(X, X′)),

H ∈ Ccα ⇐⇒ ∀β ≤ α∀F ∈ Cβ : ev(H × F) ∈ C
′
β.

Here, ev : Hom(X, X′) × X −→ X′, ev(f, x) = f(x), is the evaluation map-
ping. Then (Hom(X, X′), Cc) is an L-Cauchy tower space and the evaluation
mapping is Cauchy-continuous. Furthermore, for a Cauchy continuous mapping
f : (X, C)×(X′, C′) −→ (X′′, C′′) we define f∗ : (X, C) 7−→ (Hom(X, X′), Cc)
by f∗(x) = fx with fx(x

′) = f(x, x′) for x′ ∈ X′. Then f∗ is Cauchy-
continuous. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 464



Quantale-valued Cauchy tower spaces and completeness

We define, for an L-Cauchy tower space (X, C), the underlying L-convergence

tower space (X, qC) by

x ∈ qCα (F) ⇐⇒ F ∧ [x] ∈ Cα.

If ∗ = ∧, then the axiom (LCTS5) is satisfied: For F ∧ [x] ∈ Cα, G ∧ [x] ∈ Cβ,
then (F ∧ [x]) ∨ (G ∧ [x]) exists and hence (F ∧ G) ∧ [x] = (F ∧ [x]) ∧ (G ∧ [x]) ∈
Cα∧α = Cα.

Definition 3.3. An L-Cauchy tower space (X, C) is called a T1-space if [x] ∧
[y] ∈ C⊤ implies x = y. It is called a T2-space if F ∧ [x], F ∧ [y] ∈ C⊤ implies
x = y.

As the concepts of T1-space and T2-space only involve the Cauchy space
(X, C⊤) we immediately have that an L-Cauchy tower space (X, C) is a T1-
space if and only if it is a T2-space.

4. Example: L-metric spaces

An L-metric space, see e.g. [7, 19], is a pair (X, d) of a set X and a mapping
d : X × X −→ L which satisfies the axioms

(LM1) d(x, x) = ⊤ for all x ∈ X;
(LM2) d(x, y) ∗ d(y, z) ≤ d(x, z) for all x, y, z ∈ X.

We call an L-metric space symmetric if it satisfies

(LMs) d(x, y) = d(y, x) for all x, y ∈ X.

A mapping between two L-metric spaces, f : (X, d) −→ (X′, d′) is called an
L-metric morphism if d(x1, x2) ≤ d

′(f(x1), f(x2)) for all x1, x2 ∈ X. We de-
note the category of L-metric spaces with L-metric morphisms by L-MET, the
subcategory with symmetric L-metric spaces is denoted by L-sMET

In case L = ({0, 1}, ≤, ∧), an L-metric space is a preordered set. If L =
([0, ∞], ≥, +) is Lawvere’s quantale, an L-metric space is a quasi-metric space.
If L = (∆+, ≤, ∗), an L-metric space is a probabilistic quasi-metric space, see
[7].

In [14] we defined, for an L-metric space (X, d), the L-convergence tower qd

by

x ∈ qdα(F) ⇐⇒
∨

F ∈F

∧

y∈F

d(x, y) ≥ α.

Similarly, we define an L-Cauchy tower, Cd, by

F ∈ Cdα ⇐⇒
∨

F ∈F

∧

x,y∈F

d(x, y) ≥ α.

We note that F ∈ Cd⊤ if and only if for all ǫ ✁ ⊤ there is F ∈ F such that F
has a “diameter”

∧

x,y∈F d(x, y) ≥ ǫ. For Lawvere’s quantale this is exactly
the definition of a Cauchy filter in a metric space. We note that because F is
a filter,

∨

F ∈F

∧

x,y∈F d(x, y) is a directed join. Hence we can replace ǫ ✁ ⊤ by
ǫ ≪ ⊤ here.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 465



G. Jäger and T. M. G. Ahsanullah

Proposition 4.1. Let (X, d) ∈ |L-MET|. Then (X, Cd) ∈ |L-ChyTS|.

Proof. (LChyTS1) From
∨

F ∈[x]

∧

u,v∈F d(u, v) ≥ d(x, x) ≥ α it follows [x] ∈

Cdα.
(LChyTS2), (LChyTS3) and (LChyTS4) are obvious.
(LChyTS5) Let F ∈ Cdα, G ∈ C

d
β and let F ∨ G exist. Let α

′
✁ α and β′ ✁ β.

Then there are F ∈ F such that for all x, y ∈ F we have d(x, y) ≥ α′ and G ∈ G
such that for all u, v ∈ G we have d(u, v) ≥ β′. As F∨G exists, there is z ∈ F ∩G
and hence we have, for all x ∈ F and all v ∈ G, d(x, z) ≥ α′, d(z, v) ≥ β′ and
d(z, x) ≥ α′ and d(v, z) ≥ β′. Hence, for all x ∈ F, v ∈ G we conclude
d(x, v) ≥ d(x, z)∗d(z, v) ≥ α′ ∗β′ and for all x ∈ G, v ∈ F we conclude likewise
d(x, v) ≥ d(x, z) ∗ d(z, v) ≥ β′ ∗ α′. If both x, v ∈ F , then d(x, v) ≥ α′ ≥ α′ ∗ β′

and if both x, v ∈ G then d(x, v) ≥ β′ ≥ α′ ∗ β′. Hence for all x, v ∈ F ∪ G
we have d(x, v) ≥ α′ ∗ β′ and as the sets F ∪ G with F ∈ F, G ∈ G form
a basis of F ∧ G we conclude

∨

H∈F∧G

∧

x,v∈H d(x, v) ≥ α
′ ∗ β′. From the

complete distributivity finally we obtain
∨

H∈F∧G

∧

x,v∈H d(x, v) ≥ α ∗ β and

hence F ∧ G ∈ Cdα∗β. �

It is not difficult to show that for a symmetric L-metric space (X, d), the

space (X, Cd) is a T1-space if and only if (X, d) is separated, i.e. if d(x, y) = ⊤
implies x = y.

Proposition 4.2. Let f : (X, d) −→ (X′, d′) be an L-MET-morphism. Then

f : (X, Cd) −→ (X′, Cd
′

) is an L-ChyTS-morphism.

Proof. Let F ∈ Cdα. Then
∨

H∈f(F)

∧

u,v∈H d
′(u, v) ≥

∨

F ∈F

∧

x,y∈F d
′(f(x), f(y))

≥
∨

F ∈F

∧

x,y∈F d(x, y) ≥ α and hence f(F) ∈ C
d′

α . �

Hence we have a functor F : L-MET −→ L-ChyTS, F((X, d)) = (X, Cd),
F(f) = f. This functor, restricted on L-sMET, is injective on objects. Let
(X, d) 6= (X, d′). Then there are x, y ∈ X such that d(x, y) 6= d′(x, y), i.e.
without loss of generality d(x, y) 6≤ d′(x, y). It is not difficult to show that
[x]∧[y] ∈ Cdα ⇐⇒ d(x, y)∧d(y, x) = d(x, y) ≥ α. As we have [x]∧[y] ∈ C

d
d(x,y)

but [x] ∧ [y] /∈ Cd
′

d(x,y) we see that C
d 6= Cd

′

.

We define now for (X, C) ∈ |L-ChyTS| a mapping dC : X × X −→ L by

dC(x, y) =
∨

[x]∧[y]∈Cα

α.

Proposition 4.3. Let (X, C) ∈ |L-ChyTS|. Then (X, dC) ∈ |L-sMET|. If

(X, C) is a T1-space and is left-continuous, then (X, dC) is separated.

Proof. (LM1) We have dC(x, x) =
∨

[x]∈Cα
α = ⊤ by (LChyTS1).

(LM2) We have by the distributivity of the quantale operation over joins

dC(x, y) ∗ dC(y, z) =
∨

[x]∧[y]∈Cα,[y]∧[z]∈Cβ
α ∗ β. Clearly ([x] ∧ [y]) ∨ ([y] ∧ [z])

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 466



Quantale-valued Cauchy tower spaces and completeness

exists and by (LChyTS5) and (LChyTS2) we conclude dC(x, y) ∗ dC(y, z) ≤
∨

[x]∧[z]∈Cα∗β
α ∗ β ≤

∨

[x]∧[z]∈Cγ
γ = dC(x, z).

(LMs) is obvious.

For the separation, let ǫ ✁ ⊤ = dC(x, y). Then [x] ∧ [y] ∈ Cǫ. The left-
continuity yields [x] ∧ [y] ∈ C⊤ and hence x = y by (T1). �

Proposition 4.4. Let f : (X, C) −→ (X′, C′) be an L-ChyTS-morphism. Then

f : (X, dC) −→ (X′, dC
′

) is an L-MET-morphism.

Proof. We have, using f([x] ∧ [y]) = f([x]) ∧ f([y]) = [f(x)] ∧ [f(y)] that

dC(x, y) =
∨

[x]∧[y]∈Cα
α ≤

∨

[f(x)]∧[f(y)]∈C′α
α = dC

′

(f(x), f(y)). �

Proposition 4.5. Let (X, d) ∈ |L-sMET|. Then dC
d
= d.

Proof. We use again [x] ∧ [y] ∈ Cdα ⇐⇒ d(x, y) ≥ α and conclude d
Cd(x, y) =

∨

[x]∧[y]∈Cdα
α =

∨

d(x,y)≥α α = d(x, y). �

Hence the functor G which assigns to an L-Cauchy tower space (X, C)

the symmetric L-metric space (X, dC) and leaves morphisms unchanged, is
left-inverse for F, in the sense that we have G ◦ F((X, d) = (X, d) for all
(X, d) ∈ |L-sMET|. However, L-sMET is in general not a reflective subcategory
of L − ChyTS. The example in Remark 7.9 [21] can be used as a counterexample
in the case of Lawvere’s quantale.

We consider now the following axiom for (X, C) ∈ |L-ChyTS|.

(LChyM) F ∈ Cα ⇐⇒ ∀ǫ ✁ α∃Fǫ ∈ F∀x, y ∈ Fǫ : [x] ∧ [y] ∈ Cǫ.

Proposition 4.6. Let (X, d) ∈ |L-sMET|. Then (X, Cd) satisfies (LChyM).

Proof. Let first F ∈ Cdα and let ǫ ✁ α. Then there is Fǫ ∈ F such that for all
x, y ∈ Fǫ we have d(x, y) ≥ ǫ. The latter is equivalent to [x] ∧ [y] ∈ C

d
ǫ .

Let now for all ǫ ✁ α exist Fǫ ∈ F such that for all x, y ∈ Fǫ we have
[x] ∧ [y] ∈ Cdǫ . Then

∨

F ∈F

∧

u,v∈F

d(u, v) ≥
∧

u,v∈Fǫ

d(u, v) ≥
∧

[u]∧[v]∈Cdǫ

d(u, v) =
∧

d(u,v)≥ǫ

d(u, v) ≥ ǫ.

The complete distributivity yields
∨

F ∈F

∧

u,v∈F d(u, v) ≥ α, i.e. F ∈ C
d
α. �

Proposition 4.7. Let (X, C) ∈ |L-ChyTS|. If (X, C) satisfies (LChyM), then

Cd
C

α = Cα for all α ∈ L.

Proof. Let first F ∈ Cd
C

α . Then
∨

F ∈F

∧

x,y∈F

∨

[x]∧[y]∈Cβ
β ≥ α. Let ǫ ✁ α.

Then there is F ∈ F such that for all x, y ∈ F there is β ≥ ǫ with [x] ∧ [y] ∈
Cβ ⊆ Cǫ. The axiom (LChyM) then implies F ∈ Cα.

Conversely, let F ∈ Cα. The axiom (LChyM) implies that for all ǫ ✁ α
there is Fǫ ∈ F such that for all x, y ∈ Fǫ we have [x] ∧ [y] ∈ Cǫ. Therefore
∨

F ∈F

∧

x,y∈F

∨

[x]∧[y]∈Cβ
β ≥

∧

x,y∈Fǫ

∨

[x]∧[y]∈Cβ
β ≥ ǫ and again the complete

distributivity yields F ∈ Cd
C

α . �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 467



G. Jäger and T. M. G. Ahsanullah

From this we conclude the following result.

Theorem 4.8. The subcategory of L-ChyTS with objects the spaces that satisfy
the axiom (LChyM) is isomorphic to the category L-sMET.

Now we shall look into alternative characterizations of the L-Cauchy tower

of an L-metric space in terms of the L-convergence tower (X, qd).

Proposition 4.9. Let (X, d) ∈ |L-MET|. Then the following are equivalent.
(i) For all ǫ ✁ α there is x ∈ X such that Bd(x, ǫ) ∈ F;
(ii)

∨

x∈X

∨

x∈qd
β
(F) β ≥ α.

Proof. (i)⇒(ii): Let ǫ ✁ α. Then there is x ∈ X and F ∈ F such that
F ⊆ Bd(x, ǫ). Hence for all y ∈ F we have d(x, y) ≫ ǫ and we conclude
∨

F ∈F

∧

y∈F d(x, y) ≥ ǫ. This is equivalent to x ∈ q
d
ǫ (F) and hence we conclude

∨

x∈qd
β
(F) β ≥ ǫ. Therefore

∨

x∈X

∨

x∈qd
β
(F) β ≥ ǫ and the complete distributivity

yields
∨

x∈X

∨

x∈qd
β
(F) β ≥ α.

(ii)⇒(i): Let ǫ ✁ α. Then there is x ∈ X and β ✄ ǫ such that x ∈ qdβ(F).
Hence we have

∨

F ∈F

∧

y∈F d(x, y) ≥ β ✄ ǫ and there is F ∈ F such that for all

y ∈ F we have d(x, y) ✄ ǫ. As α ✁ β implies α ≪ β, we conclude F ⊆ Bd(x, ǫ),
i.e. Bd(x, ǫ) ∈ F. �

Proposition 4.10. Let (X, d) ∈ |L-MET| and consider the following state-
ments:
(i) F ∈ Cdα;
(ii)

∨

x∈X

∨

x∈qd
β
(F) β ≥ α;

(iii) F ∈ Cdα∗α.
Then (i) implies (ii). If L is divisible and (X, d) is symmetric, (ii) implies (iii).

Proof. Let first F ∈ Cdα. For F ∈ F we fix xF ∈ F . Then α ≤
∨

F ∈F

∧

x,y∈F d(x, y) ≤
∨

F ∈F

∧

y∈F d(xF , y). This shows xF ∈ q
d
α(F) for all

xF ∈ F and we conclude
∨

x∈X

∨

x∈qd
β
(F) β ≥ α.

Let now L be divisible and (X, d) be symmetric and let
∨

x∈X

∨

x∈qd
β
(F) β ≥

α ✄ ǫ. Then there is xǫ ∈ X and β ✄ ǫ such that x ∈ q
d
β(F). This means that

∨

F ∈F

∧

y∈F d(xǫ, y) ≥ β ✄ ǫ and hence there is Fǫ ∈ F such that for all y ∈ Fǫ
we have d(xǫ, y) ≥ ǫ. For u, v ∈ Fǫ we have by symmetry and the triangle
inequality d(u, v) ≥ d(u, xǫ)∗d(xǫ, v) ≥ ǫ∗ǫ. Hence

∨

F ∈F

∧

u,v∈F d(u, v) ≥ ǫ∗ǫ

and the complete distributivity yields F ∈ Cdα∗α. �

Remark 4.11. (1) For Lawvere’s quantale L = ([0, ∞], ≥, +) and the idem-
potent ⊤-element ⊤ = 0, the above characterization yields, using
x ∈ qdβ(F) ⇐⇒ infF ∈F supy∈F d(x, y) ≤ β,

F ∈ Cd0 ⇐⇒ inf
x∈X

inf
F ∈F

sup
y∈F

d(x, y) = 0.

This is the definition of a Cauchy filter in the metric space viewed as
an approach space [20].

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 468



Quantale-valued Cauchy tower spaces and completeness

(2) A sequence (xn) in a probabilistic metric space [34] is called a strong
Cauchy sequence if for all t > 0 there is n0 ∈ IN such that for all
n, m ≥ n0 we have d(xn, xm)(t) > 1 − t. This definition employs the
strong uniformity, i.e. the system of entourages generated by the sets
U(t) = {(x, y) ∈ X × X : d(x, y)(t) > 1 − t}, t > 0. Equivalently, as is
pointed out in [34], we can also use the system of entourages generated
by the sets U(t, ǫ) = {(x, y) ∈ X × X : d(x, y)(t) > 1 − ǫ}, t, ǫ > 0. As
in [16], using the quantale of distance distribution functions (∆+, ≤, ∗),
we can see that the sets Bd(ϕ) = {(x, y) ∈ X × X : d(x, y) ✄ ϕ},
ϕ ✁ ǫ0 generate the same uniformity. From this we can deduce that a
sequence (xn) is a strong Cauchy sequence in the probabilistic metric
space (X, d) if and only if the generated filter is in Cdǫ0.

Note that for a divisible quantale L, a symmetric L-metric space (X, d) and
an idempotent element α ∈ L, in Proposition 4.10, (i) and (ii) are equivalent. In
particular, for a frame L, we can characterize the L-Cauchy tower of a symmetric
L-metric space by (ii) or by the equivalent statement (i) in Proposition 4.9. The
following example shows that we cannot omit the idempotency here.

Example 4.12. Let X = IR and L = ([0, ∞], ≥, +) be Lawvere’s quantale and
d the usual metric on IR. We consider, for α > 0, the sequence xn =

α
2
+ 1

n
if n

is even and xn = −
α
2
− 1

n
if n is odd and denote F the filter generated by this

sequence. Then

∨

F ∈F

∧

x,y∈F

d(x, y) = inf
k∈IN

sup
m,n≥k

∣

∣

∣

∣

α

2
+

1

n
− (−

α

2
−

1

m
)

∣

∣

∣

∣

= inf
k∈IN

∣

∣

∣

∣

α +
2

k

∣

∣

∣

∣

= α

and hence we have F ∈ Cdα.

However, as
∨

x∈X

∨

F ∈F

∧

y∈F

d(x, y) = inf
x∈IR

inf
k∈IN

sup
m≥k

(d(x,
α

2
+

1

m
) ∨ d(x, −

α

2
−

1

m
)) = inf

x∈IR
(
∣

∣

∣
x −

α

2

∣

∣

∣
∨
∣

∣

∣
x +

α

2

∣

∣

∣
) =

α

2
, we have

∨

x∈X

∨

x∈qd
β
(F) β =

α
2
but F /∈

Cdα/2.

Proposition 4.13. Let (X, d) ∈ |L-MET|. Then qC
d

α (F) ⊆ q
d
α(F) and if (X, d)

is symmetric, we have qdα(F) ⊆ q
Cd
α∗α(F).

Proof. Let first x ∈ qC
d

α (F). Then F∧ [x] ∈ C
d
α. Let ǫ✁ α. Then there is F ∈ F

such that for all u, v ∈ F ∪ {x} we have d(u, v) ≥ ǫ. We conclude for u = x
that d(x, v) ≥ ǫ for all v ∈ F and hence

∨

F ∈F

∧

y∈F d(x, y) ≥ ǫ. The complete

distributivity implies x ∈ qdα(F).
Let now (X, d) be symmetric and x ∈ qdα(F). Let ǫ ≪ α. Noting that the

set D = {
∧

y∈F d(x, y) : F ∈ F} is directed, there is F ∈ F such that for all

y ∈ F we have d(x, y) ≥ ǫ. Let u, v ∈ F ∪ {x}. We distinguish four cases.
Case 1: u 6= x, v 6= x. Then d(x, u), d(x, v) ≥ ǫ and hence d(u, v) ≥ d(u, x) ∗

d(x, v) ≥ ǫ ∗ ǫ.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 469



G. Jäger and T. M. G. Ahsanullah

Case 2: u = x, v 6= x. Then d(u, v) = d(x, v) ≥ ǫ ≥ ǫ ∗ ǫ.
Case 3: u 6= x, v = x. Similar to case 2.
Case 4: u = v = x. Then d(u, v) = d(x, x) = ⊤ ≥ ǫ ∗ ǫ.
Hence we have

∨

F ∈F

∧

u,v∈F ∪{x} d(u, v) ≥ ǫ∗ǫ and the continuity of L yields

F ∧ [x] ∈ Cdα∗α, i.e. x ∈ q
Cd
α∗α(F). �

Again, for idempotent α ∈ L, i.e. if α∗α = α, we have the equality qC
d

α (F) =
qdα(F). In particular in a frame L, this equality is guaranteed for all α ∈ L.

Example 4.14. We use Example 4.12 and show that α
4
∈ qC

d

α (F). To this end,
we consider the sequence (yn) = (x1,

α
4
, x2,

α
4
, x3,

α
4
, ...), the generated filter of

which is G = F ∧ [α
4
]. We then have

∨

G∈G

∧

x,y∈G d(x, y) =

inf
k∈IN supm,n≥k |yn − ym| = α and hence we have F ∧ [

α
4
] ∈ Cdα, i.e.

α
4

∈

qC
d

α (F) = q
Cd

α/2+α/2
(F). However, we see in a similar way that

inf
k∈IN supn≥k

∣

∣

α
4
− xn

∣

∣ = 3
4
α 6≤ α

2
, so that α

4
/∈ qd

α/2
(F). So we have qC

d

α/2+α/2
(F)

6⊆ qdα(F).

5. Example: L-uniform convergence tower spaces and L-uniform
tower spaces

Definition 5.1 ([15]). A pair
(

X, Λ = (Λα)α∈L
)

, is called an L-uniform con-
vergence tower space, if Λα ⊆ F(X × X), α ∈ L, satisfy the following:

(LUCTS1) [(x, x)] ∈ Λα for all x ∈ X, α ∈ L;
(LUCTS2) Φ ∈ Λα whenever Φ ≤ Ψ and Ψ ∈ Λα;
(LUCTS3) Φ, Ψ ∈ Λα implies Φ ∧ Ψ ∈ Λα;
(LUCTS4) Λβ ⊆ Λα whenever α ≤ β;
(LUCTS5) Φ−1 ∈ Λα whenever Φ ∈ Λα;
(LUCTS6) Φ ◦ Ψ ∈ Λα∗β whenever Φ ∈ Λα, Ψ ∈ Λβ and Φ ◦ Ψ exists;
(LUCTS7) Λ⊥ = F(X × X).

A mapping f :
(

X, Λ
)

−→
(

X′, Λ′
)

is called uniformly continuous if, for all
Φ ∈ F(X × X), (f × f)(Φ) ∈ Λ′α, whenever Φ ∈ Λα. The category of L-uniform
convergence tower spaces and uniformly continuous mappings is denoted by
L-UCTS.

If L = ([0, 1], ≤, ∗), then we obtain Nusser’s probabilistic uniform conver-
gence spaces [26, 27]. For L = (∆+, ≤, ∗) we obtain the probabilistic uniform
convergence spaces in [2].

For (X, Λ) ∈ |L-UCTS|, F ∈ F(X), x ∈ X and α ∈ L we define

x ∈ qΛα(F) ⇐⇒ F × [x] ∈ Λα.

It is not difficult to show that (X, qΛ = (qΛα)α∈L) ∈ |L-CTS| is an L-limit tower
space.

For (X, Λ) ∈ |L-UCTS| and α ∈ L, a filter F ∈ F(X) is called an α-Cauchy

filter, written as F ∈ CΛα if and only if F × F ∈ Λα.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 470



Quantale-valued Cauchy tower spaces and completeness

Proposition 5.2. Let (X, Λ) ∈ |L-UCTS|. Then (X, CΛ) ∈ |L-ChyTS|.

Proof. (LChyTS1) Since for all x ∈ X and α ∈ L, [x] × [x] ∈ Λα, we have

[x] ∈ CΛα .

(LChyTS2) Let G ≥ F with F ∈ CΛα . Then F×F ∈ Λα and hence, by (LUCTS2),

G × G ∈ Λα, i.e. G ∈ C
Λ
α .

(LChyTS3) Let α ≤ β and F ∈ CΛβ . Then F × F ∈ Λβ, then by (LUCTS4),

F × F ∈ Λα which in turn yields F ∈ C
Λ
α .

(LChyTS4) Follows at once from the definition.

(LChyTS5) Let α, β ∈ L, and let F ∈ CΛα and G ∈ C
Λ
β such that F ∨ G exists.

Then F × F ∈ Λα and G × G ∈ Λβ. Then F × G = (F × G) ◦ (F × G) ∈ Λα∗β.
Also, G × F ∈ Λα∗β. Since

(F ∧ G) × (F ∧ G) = (F × F) ∧ (F × G) ∧ (G × F) ∧ (G × G) ∈ Λα∗β,

this implies F ∧ G ∈ CΛα∗β. �

Proposition 5.3. Let (X, Λ) ∈ |L-UCTS|. Then qC
Λ

α (F) ⊆ q
Λ
α(F) ⊆ q

CΛ
α∗α(F).

Proof. Let x ∈ qC
Λ

α (F), then F∧[x] ∈ C
Λ
α . This implies that (F∧[x])×(F∧[x]) ∈

Λα. As (F ∧ [x]) × (F ∧ [x]) ≤ F × [x], (LUCTS2) implies x ∈ q
Λ
α(F).

If x ∈ qΛα(F), then F × [x] ∈ Λα and with (LUCTS5) then also [x] × F =
(F×[x])−1 ∈ Λα. From (LUCTS6) we obtain F×F = (F×[x])◦([x]×F) ∈ Λα∗α.
This yields with (LUCTS3) (F ∧ [x]) × (F ∧ [x]) = (F × F) ∧ (F × [x]) ∧ ([x] ×

F) ∧ ([x] × [x]) ∈ Λα∗α. Thus, F ∧ [x] ∈ C
Λ
α∗α which means x ∈ q

CΛ
α∗α(F). �

Again, for idempotent α ∈ L, we have equality, qC
Λ

α (F) = q
Λ
α(F). In partic-

ular this is the case if L is a frame.

Definition 5.4 ([15]). Let L = (L, ≤, ∗) be a quantale. A pair
(

X, U
)

with

U = (Uα)α∈L a family of filters on X × X is called an L-uniform tower space if
for all α ∈ L the following holds:

(LUTS1) Uα ≤ [∆] with [∆] =
∧

x∈X[(x, x)];

(LUTS2) Uα ≤ (Uα)
−1;

(LUTS3) Uα∗β ≤ Uα ◦ Uβ;
(LUTS4) Uα ≤ Uβ whenever α ≤ β;
(LUTS5) U⊥ =

∧

F(X × X);
(LUTS6) U∨ A ≤

∨

α∈A Uα whenever ∅ 6= A ⊆ L.

A mapping f :
(

X, U
)

−→
(

X′, U′
)

is called uniformly continuous if U′α ≤
(f × f)(Uα) for all α ∈ L. The category with objects all L-uniform tower
spaces and uniformly continuous mappings as morphisms is denoted by L-UTS.

If L = ([0, 1], ≤, ∗) with a t-norm ∗, then we obtain Florescu’s probabilistic
uniform spaces [8], for L = (∆+, ≤, ∗) we obtain the probabilistic uniform
spaces in [2]. For Lawvere’s quantale an L-uniform tower space is an approach
uniform space [23].

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 471



G. Jäger and T. M. G. Ahsanullah

For
(

X, U = (Uα)α∈L
)

∈ |L − UTS| and α ∈ L, a filter F ∈ F(X) is called an

α-Cauchy filter written as F ∈ CUα if and only if F × F ≥ Uα.

Lemma 5.5. Let (X, U) ∈ |L-UTS|. Then (X, CU) ∈ |L-ChyTS|.

Proof. (LChyTS1) Clearly by (LUTS1), Uα ≤ [x]×[x], implies [x] ∈ C
U
α , for all

x ∈ X and α ∈ L. (LChyTS2) Let F, G ∈ F(X) with F ≤ G, and F ∈ CUα . Then

F × F ≥ Uα but then G × G ≥ Uα, which in turn implies G ∈ C
U
α . (LChyTS3)

Let α ≤ β and F ∈ CUβ , implying F × F ≥ Uβ. But then Uβ ≥ Uα by (LUTS4)

for α ≤ β. So, F × F ≥ Uα which gives F ∈ C
U
α . The condition (LChyTS4)

is trivially true. Finally, we check the condition (LChyTS5): let F ∈ CUα and

G ∈ CUβ such that F ∨ G exists. Then F × F ≥ Uα and G × G ≥ Uβ. Then
(F× F)◦ (G × G) ≥ Uα ◦ Uβ which in view of the condition (LUTS3) yields that
F×G = (F×F)◦(G×G) ≥ Uα∗β. This means that F×G ≥ Uα∗β. Furthermore,
note that we also have: G×F ≥ Uα∗β, F×F ≥ Uα∗β, and G×G ≥ Uα∗β; this is
due to the condition (LUTS5) and the fact that α ∗ β ≤ α, β. Hence we arrive
at the following:

(F ∧ G) × (F ∧ G) = (F × F) ∧ G × G) ∧ (F × G) ∧ (G × F) ≥ Uα∗β,

that is, (F ∧ G) × (F ∧ G) ≥ Uα∗β which in turn implies that F ∧ G ∈ C
U
α∗β. �

Given an L-metric space (X, d) we define Λdα ⊆ F(X × X) for α ∈ L, by [15]

Φ ∈ Λdα ⇐⇒
∨

F ∈Φ

∧

(x,y)∈F d(x, y) ≥ α.

Theorem 5.6. The L-Cauchy tower space (X, Cd) of an L-metric space (X, d)

is the same as the L-Cauchy tower space (X, CΛ
d
) induced by the L-uniform

convergence tower space (X, Λd) of an L-metric space. That is, Cd = CΛ
d
.

Proof. This follows from
∨

H∈F×F

∧

(x,y)∈H d(x, y) =
∨

F ∈F

∧

x,y∈F d(x, y) for

F ∈ F(X), as the sets F × F with F ∈ F are a basis of F × F. �

For an L-metric space (X, d) we define Uǫ = {(x, y) ∈ X ×X : d(x, y) ≫ ǫ}.
For ǫ ≪ α, these sets form the basis of a filter Udα and we have U

d
α =

∧

Φ∈Λdα
Φ,

see [15]. We call (X, Ud) the L-uniform tower space induced by the L-metric
space (X, d).

Proposition 5.7. Let (X, d) ∈ |L-MET|. Then CU
d

α = C
d
α for all α ∈ L.

Proof. Let F ∈ CU
d

α . Then F×F ∈ U
d
α and for all ǫ ≪ α there is F ∈ F such that

F × F ⊆ Uǫ. Hence for all ǫ ≪ α there is F ∈ F such that
∧

x,y∈F d(x, y) ≥ ǫ.

This implies
∧

x,y∈F d(x, y) ≥ α from which we conclude F ∈ C
d
α.

Conversely, we note that the set {
∧

x,y∈F d(x, y) : F ∈ F} is directed. If

F ∈ Cdα, then
∨

F ∈F

∧

x,y∈F d(x, y) ≥ α. Hence, for ǫ ≪ α there is F ∈ F such

that for all x, y ∈ F we have d(x, y) ≫ ǫ. This means F × F ⊆ Uǫ and we have
F × F ≥ Udα. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 472



Quantale-valued Cauchy tower spaces and completeness

6. Example: L-limit tower groups

Let (X, ·) be a group with identity element e ∈ X and let F, G, H ∈ F(X). We
define F⊙G as the filter with filter basis {F ·G : F ∈ F, G ∈ G} and F−1 as the
filter with filter basis {F −1 : F ∈ F}. Then, for F, G, H ∈ F(X) and x, y ∈ X,
the following properties are easily verified: F ⊙ F−1 ≤ [e]; [x] ⊙ [x]−1 = [e];
[x−1] = [x]−1; [x · y] = [x] · [y]; (F ⊙ G) ⊙ H = F ⊙ (G ⊙ H); (F−1)−1 = F;
(F ⊙ G)−1 = G−1 ⊙ F−1; [e] ⊙ F = F ⊙ [e] = F; (F ∧ G)−1 = F−1 ∧ G−1 and
(F ∧ G) ⊙ H = (F ⊙ H) ∧ (G ⊙ H).

Definition 6.1. Let L = (L, ≤, ∗) be a quantale, and (X, ·) be a group with
identity e. Then a triple (X, ·, q = (qα)α∈L) is called an L-convergence tower
group (respectively, an L-limit tower group) if the following conditions are ful-
filled:

(LCTG) If (X, q) is an L-convergence tower space (resp. an L-limit tower space)
(LCTGM) x ∈ qα(F), y ∈ qβ(G) implies xy ∈ qα∗β (F ⊙ G), for all F, G ∈ F(X),

for all α, β ∈ L, and x, y ∈ X;
(LCTGI) x ∈ qα(F) implies x

−1 ∈ qα(F
−1), for all F ∈ F(X), x ∈ X and α ∈ L.

The category of all L-convergence tower groups and continuous group homo-
morphisms is denoted by L-CTGrp (respectively, the category of all L-limit tower
groups and continuous group homomorphisms is denoted by L-LIMGrp.)

If L = {0, 1}, then we obtain classical convergence groups [28]. If L =
([0, 1], ≤, ∗) with a continuous t-norm ∗, then we get probabilistic convergence
group under a t-norm in the sense of [13]. If L = ([0, ∞], ≥, +), then a left-
continuous L-convergence tower group is an approach group [24]. If L = (∆+, ≤
, ∗), we obtain a probabilistic convergence group in the definition of [3].

For (X, ·, q) ∈ |L − LIMTGrp|, a filter F ∈ F(X) is called an α-Cauchy filter,
written F ∈ Cqα, if and only if e ∈ qα

(

F
−1 ⊙ F

)

.

Proposition 6.2. Let (X, ·, q) ∈ |L-LIMGrp|. Then
(

X, Cq
)

∈ |L-ChyTS|.

Proof. (LChyTS1) Since e ∈ qα([x]
−1 ⊙ [x]) implies [x] ∈ Cqα.

(LChyTS2) Let F ≤ G with F ∈ Cqα. This implies e ∈ qα
(

F
−1 ⊙ F

)

. But then

e ∈ qα
(

G
−1 ⊙ G

)

which gives G ∈ Cqα.

(LChyTS3) Let α, β ∈ L with α ≤ β and F ∈ C
q
β. Then e ∈ qβ

(

F
−1 ⊙ F

)

implies e ∈ qα
(

F
−1 ⊙ F

)

. Hence F ∈ Cqα.
(LChyTS4) Obvious.
(LChyTS5) Let F ∈ Cα and G ∈ C

q
β such that F ∨ G exists. It is not difficult

to prove that F ⊙ G−1 ≤ [e]. As e ∈ qα
(

F
−1 ⊙ F

)

and e ∈ qβ
(

G
−1 ⊙ G

)

, we

obtain with condition (LCTGM) e = ee ∈ qα∗β
(

F
−1 ⊙ F ⊙ G−1 ⊙ G

)

which

implies that e ∈ qα∗β
(

F
−1 ⊙ G

)

. Similarly, we can get e ∈ qα∗β
(

G
−1 ⊙ F

)

.
Since

(F ∧ G)−1 ⊙ (F ∧ G) = (F−1 ∧ G−1) ⊙ (F ∧ G)

= (F−1 ⊙ F) ∧ (F−1 ⊙ G) ∧ (G−1 ⊙ F) ∧ (G−1 ⊙ G),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 473



G. Jäger and T. M. G. Ahsanullah

we conclude from condition (LCTS5) e ∈ qα∗β
(

(F ∧ G)−1 ⊙ (F ∧ G)
)

. Hence
F ∧ G ∈ C

q
α∗β. �

Definition 6.3 ([14]). A triple (X, ·, d) is called an L-metric group if d is
invariant, i.e., d(x, y) = d(xz, yz) = d(zx, zy) for all x, y, z ∈ X. A group
homomorphism f : (X, ·) −→ (X′, ·′) between the L-metric groups (X, ·, d) and
(X′, ·′, d′) is called an L-METGrp-morphism if it is an L-MET-morphism between
(X, d) and (X′d′). The category of L-metric groups is denoted by L-METGrp.

Lemma 6.4. Let (X, ·, d) be a symmetric L-metric group. Then (X, ·, qd) is
an L-convergence tower group.

Proof. We need to show the conditions (LCTGM) and (LCTGI). For (LCTGM),
let x ∈ qdα(F) and y ∈ q

d
β(G). For ǫ ✁ α and δ ✁ β, there is F ∈ F and G ∈ G

such that for all u ∈ F and for all v ∈ G we have d(x, u) ≥ ǫ and d(y, v) ≥ δ.
Then d(e, x−1u) ≥ ǫ and d(yv−1, e) ≥ δ and hence d(yv−1, x−1u) ≥ ǫ ∗ δ. This
implies d(xy, uv) ≥ ǫ ∗ δ for all u ∈ F, v ∈ G and hence d(xy, h) ≥ ǫ ∗ δ for all
h ∈ F ⊙ G. Therefore

∨

H∈F⊙G

∧

h∈H d(xy, h) ≥ ǫ ∗ δ and from the complete

distributivity we obtain xy ∈ qdα∗β(F ⊙ G).

(LCTGI) follows with the symmetry from d(x, y) = d(xy−1, e) = d(y−1, x−1).
�

Proposition 6.5. Let (X, d) ba a symmetric L-metric group. Then Cdα = C
qd
α

for all α ∈ L.

Proof. F ∈ Cq
d

α is equivalent to
∨

F ∈F

∧

x,y∈F d(e, x
−1y) ≥ α, which is, by

the invariance of the L-metric, equivalent to
∨

F ∈F

∧

x,y∈F d(x, y) ≥ α, i.e.

equivalent to F ∈ Cdα. �

7. Completeness and completion

Following [33, 27] we call (X, C) ∈ |L-ChyTS| complete if for all α ∈ L,
F ∈ Cα implies the existence of x ∈ X such that F ∧ [x] ∈ Cα.

With this definition, the “point of convergence” x = x(F, α) not only de-
pends on the filter F but may also depend on the “level” α ∈ L. In the
left-continuous case, we can omit this extra dependency.

Proposition 7.1. Let (X, C) ∈ |L-ChyTS| be left-continuous. Then (X, C) is
complete if for all F ∈ F(X) there is x = x(F) such that for all α ∈ L, F ∈ Cα
implies F ∧ [x] ∈ Cα.

Proof. Let F ∈ F(X) and define A = {α ∈ L : F ∈ Cα} and δ =
∨

A. By
left-continuity F ∈ Cδ and hence there is x = x(F, δ) such that F ∧ [x] ∈ Cδ. If
F ∈ Cα, then α ≤ δ and hence F ∧ [x] ∈ Cδ ⊆ Cα and we can choose x for each
α. �

The following examples show that the left-continuity is essential.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 474



Quantale-valued Cauchy tower spaces and completeness

Example 7.2. Let L = {⊥, α, β, ⊤} with α, β incomparable, ∗ = ∧ and let X
be an infinite set. There is an ultrafilter U ∈ U(X) with U 6≥ [x] for all x ∈ X.
We fix x0, y0 ∈ X with x0 6= y0 and define an L-Cauchy tower as follows. We
let C⊥ = F(X), F ∈ Cα iff F ≥ U ∧ [x0] or if F = [x] for some x ∈ X. Similarly,
F ∈ Cβ iff F ≥ U∧ [y0] or F = [x] for some x ∈ X, and finally F ∈ C⊤ if F = [x]
for some x ∈ X. We only show (LChyTS5) as the other axioms are easy. Let
F ∈ Cγ, G ∈ Cδ and F ∨ G exist. We distinguish three cases.

Case 1: γ = α, δ = β. Then α ∧ β = ⊥ and hence F ∧ G ∈ Cα∧β.
Case 2: F, G ∈ Cα. If F = [x] and G = [y] then x = y because F ∨ G exists

and hence F ∧ G = [x] ∈ Cα. If F = [x] and G = U ∧ [x0], then [x] ∨ (U ∧ [x0])
exists. If x 6= x0 we define F = X \ {x}. Then either F ∈ U or its complement
F c ∈ U and also F ∈ [x0]. If F ∈ U then F ∈ U ∧ [x0] and hence also
F ∈ [x] ∨ (U ∧ [x0]), in contradiction to F

c ∈ [x] and F ∩ F c = ∅. If F /∈ U
then F c ∈ U and hence U = [x], again a contradiction. We conclude x = x0,
i.e. F = [x0] and G = U ∧ [x0] and therefore F ∧ G = G ∈ Cα = Cα∧α.

Case 3: F, G ∈ Cβ. This is similar.
We note that (X, C) is complete but since we have U ∈ Cα, U ∈ Cβ and

U /∈ Cα∨β = C⊤, it is not left-continuous.
Clearly, U ∈ Cα and U ∧ [x0] ∈ Cα. Likewise, U ∈ Cβ and U ∧ [y0] ∈ Cβ.

As we have seen above U ∧ [x] ∈ Cα implies x = x0 and hence in particular
U ∧ [y0] /∈ Cα. This shows that the point of convergence for U is different for
α and β.

Example 7.3. We consider X = [0, ∞) and Lawvere’s quantale L = ([0, ∞], ≥
, +). Again we choose an ultrafilter U ∈ U(X) with U 6= [x] for all x ∈ [0, ∞).
We define C∞ = F(X) and for α < ∞ we define F ∈ Cα if F = [x] for some
x ∈ X or if F ≥ U ∧ [x] for some 0 < x < α. Then ([0, ∞), C) is an L-Cauchy
tower space. We again only show (LChyTS5). Let F ∈ Cα, G ∈ Cβ and let F∨G
exist. The case F = [x] and G = [y] implies x = y and then F∧G = [x] ∈ Cα∧β.
If F = [x] and G ≥ U ∧ [y] with 0 < y < β implies again x = y and we obtain
F∧G = G ∈ Cβ ⊆ Cα+β. It remains the case G ≥ U∧[x], G ≥ U∧[y]. If x = y,
trivially F ∧ G ∈ Cα+β. If x 6= y, then we choose two sets F1, F2 ⊆ [0, 1] with
non-empty and finite complement and F1 ∩ F2 = ∅ such that x ∈ F1, y ∈ F2.
Then F2 ∈ U ∧ [x], as F2 ∈ [x] and if F2 /∈ U, then the complement F

c
2 ∈ U

and as F c2 is a finite set, then U = [z] for z ∈ F
c
2 . Similarly, F1 ∈ U ∧ [y]. As

H = (U ∧ [x]) ∨ (U ∧ [y]) exists, we get the contradiction ∅ = F1 ∩ F2 ∈ H.
As U ∈ Cα for all 0 < α but U /∈ C0 we see that ([0, ∞), C) is not left-

continuous. Clearly, the space is complete. However, the point of convergence
varies with the level α: We have U ∈ Cα for all 0 < α < ∞. We fix α ∈ (0, ∞).
Then there is x with 0 < x < α such that U ∧ [x] ∈ Cα. For β with 0 < β < x,
however, U ∧ [x] /∈ Cβ, because U ∧ [x] 6≥ U ∧ [y] for x 6= y. Hence we cannot
choose for each level the same point of convergence for U.

We note that for (X, d) ∈ |L-MET|, the space (X, Cd) is left-continuous.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 475



G. Jäger and T. M. G. Ahsanullah

Proposition 7.4. Let (Xj, Cj) be complete L-Cauchy tower spaces for all j ∈
J. Then the product space (

∏

j∈J Xj, π-C) is also complete.

Proof. Let F ∈ (π-C)α. Then, for all j ∈ J, prj(F) ∈ C
j
α. By completeness

there is, for each j ∈ J, a point xj ∈ Xj such that prj(F) ∧ [xj] ∈ C
j
α. Hence

with x = (xj)j∈J , then for all j ∈ J, prj(F ∧ [x]) = prj(F) ∧ [prj(x)] =
prj(F) ∧ [xj] ∈ C

j
α which implies F ∧ [x] ∈ (π-C)α. �

Remark 7.5 (Completeness in L-UTS). We define for an L-uniform convergence

tower space (X, U) the underlying L-convergence tower qU by x ∈ qUα (F) ⇐⇒
F×[x] ≥ Uα. Noting that (F∧[x])×(F∧[x]) = (F×F)∧(F×[x])∧(F×[x])

−1 ∧

[(x, x)] and F×F = (F×[x])◦([x]×F) we immediately obtain x ∈ qC
U

α (F) ⇐⇒

x ∈ qUα∗α(F). For the definition of completeness however, we can use either

L-convergence tower, i.e. if we define (X, U) complete if (X, CU) is complete,
then this is equivalent to demanding that F × F ≥ Uα implies F × [x] ≥ Uα for
some x ∈ X.

Remark 7.6 (Completeness in L-UCTS.). Similarly, if we define in L-UCTS that

a space (X, Λ) is complete if (X, CΛ) is complete, then this is equivalent to
F × F ∈ Λα implies F × [x] ∈ Λα for some x ∈ X.

Let (X, C) be a non-complete L-Cauchy tower space. We call a pair
((X′, C′), κ) with a complete L-Cauchy tower space (X′, C′) and an initial and
injective mapping κ : (X, C) −→ (X′, C′) such that κ(X) is dense in (X′, C′),
a completion of (X, C). Here, a set A ⊆ X is called dense in (X, C) if for all
x ∈ X there is F ∈ F(X) such that A ∈ F and F ∧ [x] ∈ C⊤ and a mapping
κ : (X, C) −→ (X′, C′) is initial if F ∈ Cα if and only if κ(F) ∈ C

′
α.

In the sequel, we describe a completion construction which goes back to [18]
and [27]. The proofs in [27] can simply be adapted, replacing the quantale
L = ([0, 1], ≤, ∗) with a continuous t-norm ∗ on [0, 1] by an arbitrary quantale.
For this reason, they are not presented.

We consider a non-complete L-Cauchy tower space (X, C) and define NC =
{F ∈ F(X) : F ∈ C⊤, F ∧ [x] /∈ C⊤∀x ∈ X}. Furthermore, we consider the
following equivalence relation on C⊤: F ∼ G ⇐⇒ F ∧ G ∈ C⊤ and we denote
the equivalence class of F ∈ C⊤ by 〈F〉 = {G ∈ C⊤ : F ∼ G}. We define
X∗ = {〈[x]〉 : x ∈ X} ∪ {〈F〉 : F ∈ NC} and denote the inclusion mapping

ιX = ι : X −→ X
∗, x 7−→ ι(x) = 〈[x]〉. We note that if (X, C) is a T1-space,

then ι is an injection. In fact, if ι(x) = ι(y), then [x] ∧ [y] ∈ C⊤ and by (T1)
then x = y.

Let Φ ∈ F(X∗) and let α 6= ⊥. We define Φ ∈ C∗α if there is F ∈ Cα such
that Φ ≥ ι(F) or if there is F ∈ Cα and there are F1, ..., Fn ∈ NC such that
F ∨ Fi exists for all i = 1, ..., n and Φ ≥ ι(F) ∧

∧n
i=1[〈Fi〉]. Furthermore, we put

C∗⊥ = F(X).
We will consider the completion axiom (LCA): for all F ∈ Cα with F ∧ [x] /∈

Cα for all x ∈ X, there is V ∈ NC such that F ∧ V ∈ Cα.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 476



Quantale-valued Cauchy tower spaces and completeness

Then ((X∗, C∗), ι) is a completion of (X, C) if (LCA) is true. We can say
more.

Proposition 7.7 ([27, 18]). Let (X, C) ∈ |L-ChyTS|. Then (X, C) has a com-
pletion if and only if it satisfies (LCA).

For the completion ((X∗, C∗), ι), the following universal property is true.

Theorem 7.8 ([27, 18]). Let (X, C), (Y, D) ∈ |L-ChyTS| and let f : X −→ Y
be Cauchy-continuous. Then there exists a Cauchy-continuous mapping f∗ :
(X∗, C∗) −→ (Y ∗, D∗) such that the following diagram commutes:

(X, C)
f

−→ (Y, D)
ιX ↓ ↓ ιY

(X∗, C∗)
f∗

−→ (Y ∗, D∗)

Corollary 7.9 ([27]). Let (X, C), (Y, D) ∈ |L-ChyTS| and let (X, C) satisfy
(LCA) and let (Y, D) be a complete T1-space. If f : X −→ Y is Cauchy-
continuous then there is a unique Cauchy-continuous extension f∗ : (X∗, C∗) −→
(Y, D) such that f∗ ◦ ι = f.

It is at present not clear if for an L-metric space (X, d) the completion

(X, (Cd)∗) is again L-metrical, i.e. if there is an L-metric e in X∗ such that
(Cd)∗α = C

e
α for all α ∈ L. We can, however, show the one half of the axiom

(LChyM).

Proposition 7.10. Let (X, d) ∈ |L-MET|. If Φ ∈ (Cd)∗α then for all ǫ ✁ α
there is φǫ ∈ Φ such that for all x

∗, y∗ ∈ φǫ we have [x
∗] ∧ [y∗] ∈ (Cd)∗ǫ.

Proof. Let Φ ∈ (Cd)∗α. Then there are F ∈ C
d
α and F1, ..., Fn ∈ NC such that

F ∨ Fk exists for k = 1, ..., n, n ≥ 0, and Φ ≥ ι(F)
∧n

k=1[〈Fk〉]. Let ǫ ✁ α.
By the axiom (LChyM) there is Fǫ ∈ F such that for all x, y ∈ Fǫ we have
[x] ∧ [x] ∈ Cdǫ . We define φǫ = ι(Fǫ) ∪ {〈F1〉, ..., 〈Fn〉} ∈ Φ. Let x

∗, y∗ ∈ φǫ. We
distinguish three cases.

Case 1: x∗, y∗ ∈ ι(Fǫ). Then x
∗ = ι(x), y∗ = ι(y) for x, y ∈ Fǫ and as

[x] ∧ [y] ∈ Cdǫ we conclude [x
∗] ∧ [y∗] = ι([x] ∧ [y]) ∈ (Cd)∗ǫ.

Case 2: x∗ = 〈Fk〉, y
∗ = 〈Fl〉. Then trivially [〈Fk〉] ∧ [〈Fl〉] ≥ ι(F) ∧

∧n
k=1[〈Fk〉] and hence [x

∗] ∧ [y∗] ∈ (Cd)∗ǫ .
Case 3: x∗ = 〈[x]〉 ∈ ι(Fǫ), y

∗ = 〈Fk〉. We have [Fǫ] = {G ⊆ X : Fǫ ⊆
G} ≤ F and [Fǫ] ≤ [x] as x ∈ Fǫ. As F ∨ Fk exists therefore also F ∨ [Fǫ]
exists. Moreover, we have

∨

G∈[Fǫ]

∧

u,v∈G d(u, v) ≥
∧

u,v∈Fǫ
d(u, v) ≥ ǫ as

d(u, v) ≥ ǫ is equivalent to [u] ∧ [v] ∈ Cdǫ . This shows [Fǫ] ∈ C
d
ǫ . We conclude

[x∗] ∧ [y∗] = ι([x]) ∧ [〈Fk〉] ≥ ι([Fǫ]) ∧ [〈Fk〉], i.e. [x
∗] ∧ [y∗] ∈ (Cd)∗ǫ. �

8. The L-metric case: Cauchy completeness

We follow concepts and notations introduced in [6], see also [10]. Let (X, d) ∈
|L-MET|. A mapping Φ : X −→ L is an order ideal if d(y, x) ∗ Φ(x) ≤ Φ(y) for
all x, y ∈ X. It is called an order filter if Φ(x) ∗ d(x, y) ≤ Φ(y) for all x, y ∈ X.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 477



G. Jäger and T. M. G. Ahsanullah

Clearly, d(y, x) ≤ Φ(x) → Φ(y) and using the L-metric dL : L × L −→ L
defined by dL(α, β) = α → β for α, β ∈ L, we see that an order ideal is
an L-metric morphism from (X, dop) to (L, dL) and similarly, an L-order filter
is an L-metric morphism from (X, d) to (L, dL). The following lemmas give
important examples.

Lemma 8.1 ([6]). Let (X, d) be an L-metric space and a ∈ X. Then ↓ (a)
defined by ↓ (a)(x) = d(x, a) is an order ideal and ↑ (a) defined by ↑ (a)(x) =
d(a, x) is an order filter.

Lemma 8.2. Let (X, d) be an L-metric space and let F be a filter on X. Then
Φ : X −→ L, defined by Φ(x) =

∨

F ∈F

∧

y∈F d(x, y) is an order ideal and

Ψ : X −→ L defined by Ψ(x) =
∨

F ∈F

∧

y∈F d(y, x) is an order filter.

Proof. We only show the first case. We have for x, y ∈ X

d(y, x) ∗ Φ(x) = d(y, x) ∗
∨

F ∈F

∧

z∈F

d(x, z) =
∨

F ∈F

d(y, x) ∗
∧

z∈F

d(x, z)

≤
∨

F ∈F

∧

z∈F

d(y, x) ∗ d(x, z) ≤
∨

F ∈F

∧

z∈F

d(y, z) = Φ(y).

�

Definition 8.3 ([6]). Let (X, d) be an L-metric space and let Φ : X −→ L be
an order ideal and Ψ : X −→ L be an order filter. The pair (Φ, Ψ) is called a
cut on X

(1) ⊤ =
∨

x∈X Φ(x) ∗ Ψ(x);
(2) Φ(x) ∗ Ψ(y) ≤ d(x, y) for all x, y ∈ X.

We note that for a given a ∈ X, the pair (↓ (a), ↑ (a)) is a cut on X.

Proposition 8.4. Let (X, d) be an L-metric space and let F ∈ Cd⊤. Then
(Φ, Ψ) as defined in Lemma 8.2 is a cut on X.

Proof. Let ǫ ≪ ⊤ and choose δ ≪ ⊤ such that ǫ ≪ δ ∗ δ. Then δ ≪ ⊤ =
∨

F ∈F

∧

x,y∈F d(x, y) and hence there is F ∈ F such that for all x, y ∈ F we

have d(x, y) ≫ δ and d(y, x) ≫ δ. We conclude for all x ∈ F

ǫ ≪ δ ∗ δ ≤
∧

y∈F

d(x, y) ∗
∧

y∈F

d(y, x)

≤
∨

F ∈F

∧

y∈F

d(x, y) ∗
∨

F ∈F

∧

y∈F

d(y, x) = Φ(x) ∗ Ψ(x).

Hence ǫ ≪
∨

x∈X Φ(x) ∗ Ψ(x) and taking the join for all ǫ ≪ ⊤ we obtain
⊤ =

∨

x∈X Φ(x) ∗ Ψ(x).
Moreover, we have

Φ(x) ∗ Ψ(y) =
∨

F ∈F

∧

z∈F

d(x, z) ∗
∨

G∈F

∧

z∈G

d(z, y)

≤
∨

F,G∈F

∧

z∈F ∩G

d(x, z) ∗ d(z, y) ≤ d(x, y).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 478



Quantale-valued Cauchy tower spaces and completeness

Hence (Φ, Ψ) is a cut on X. �

We call a symmetric L-metric space (X, d) complete if for F ∈ Cd⊤ there is

a ∈ X such that a ∈ qd⊤(F). From Proposition 4.3 we know that q
d
⊤(F) = q

Cd
⊤ (F)

and hence the completeness of (X, d) is the completeness of the Cauchy space

(X, Cd⊤). Clearly, the requirement that (X, C
d) is complete is stronger.

We shall now establish a relation to the concept of Cauchy completeness of
an L-metric space as defined by Lawvere [19], see also [6].

Definition 8.5. Let (X, d) be an L-metric space. Then (X, d) is called Cauchy
complete if for all cuts (Φ, Ψ) there is a ∈ X that represents the cut (Φ, Ψ) in
the sense that Φ =↓ (a) and Ψ =↑ (a).

Theorem 8.6. Let (X, d) be a symmetric L-metric space. Then (X, d) is
complete if and only if (X, d) is Cauchy complete.

Proof. Let (X, d) be Cauchy complete and let F ∈ Cd⊤. We consider the order
ideal Φ and the order filter Ψ defined in Lemma 8.2. From Proposition 8.4
we see that (Φ, Ψ) is a cut on X. By assumption, there is a ∈ X such that
Φ(x) = d(x, a) for all x ∈ X and Ψ(x) = d(a, x) for all x ∈ X. Then ⊤ =
Ψ(a) ≤

∨

F ∈F

∧

z∈F d(z, a). which means that a ∈ q
d
⊤(F) and (X, d) is complete.

Conversely, let (X, d) be complete and let (Φ, Ψ) be a cut. For ǫ ≪ ⊤ we
define Fǫ = {x ∈ : Φ(x)∗Ψ(x) ≫ ǫ}. From ⊤ =

∨

x∈X Φ(x)∗Ψ(x) we conclude
that Fǫ 6= ∅ and hence {Fǫ : ǫ ≪ ⊤} is a basis for a filter F. We show that
F ∈ Cd⊤. Let ǫ ≪ ⊤ and choose δ ≪ ⊤ such that ǫ ≪ δ ∗ δ. For all x, y ∈ Fδ
we then have

δ ∗ δ ≤ Φ(x) ∗ Ψ(x) ∗ Φ(y) ∗ Ψ(y) ≤ Φ(x) ∗ Ψ(y) ≤ d(x, y),

and hence

ǫ ≪ δ ∗ δ ≤
∨

F ∈F

∧

x,y∈F

d(x, y).

Taking the join over all ǫ ≪ ⊤ yields F ∈ Cd⊤. Hence there is a ∈ X such that
a ∈ qd⊤(F).

We note that this means

⊤ =
∨

ǫ≪⊤

∧

z∈Fǫ

d(z, a)

and, Ψ : (X, d) −→ (L, dL) being an L − MET-morphism, this yields

⊤ =
∨

ǫ≪⊤

∧

z∈Fǫ

(Ψ(z) → Ψ(a)) =
∨

ǫ≪α

((
∨

z∈Fǫ

Ψ(z)) → Ψ(a)).

For δ ≪ ⊤ there is ǫδ ≪ ⊤ such that

δ ∗
∨

z∈Fǫδ

Ψ(z) ≤ Ψ(a).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 479



G. Jäger and T. M. G. Ahsanullah

As ǫδ ≤ ǫδ ∨ δ ≪ ⊤ we have Fǫδ∨δ ⊆ Fǫδ and hence

δ ∗
∨

z∈Fǫδ∨δ

Ψ(z) ≤ Ψ(a).

For z ∈ Fǫδ∨δ we know Ψ(z) ≥ Φ(z)∗Ψ(z) ≥ ǫδ ∨δ ≥ δ and we conclude δ ∗δ ≤
Ψ(a). Taking the join over all δ ≪ ⊤ we obtain ⊤ = Ψ(a). Similarly we can
show ⊤ = Φ(a). Φ being an order ideal implies d(x, a) = d(x, a) ∗ Φ(a) ≤ Φ(x)
for all x ∈ X and from Φ(x) = Φ(x) ∗ Ψ(a) ≤ d(x, a) we obtain Φ(x) = d(x, a)
for all x ∈ X. Hence Φ =↓ (a). Similarly we can show Ψ =↑ (a) and hence
a ∈ X represents the cut (Φ, Ψ) and (X, d) is Cauchy complete. �

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