@ Appl. Gen. Topol. 16, no. 2(2015), 167-181doi:10.4995/agt.2015.3323
c© AGT, UPV, 2015

Lebesgue quasi-uniformity on textures

Selma Özçağ

Department of Mathematics, Hacettepe University, Ankara, Turkey. (sozcag@hacettepe.edu.tr)

Abstract

This paper considers the Lebesgue property on quasi di-uniform tex-

tures. It is well known that the quasi-uniform space with a compact

topology has the Lebesgue property. This result is extended to direla-

tional quasi-uniformities and dual dicovering quasi-uniformities. Addi-

tionally we discuss the completeness of Lebesgue di-uniformities.

2010 MSC: 54E15; 54A05; 06D10; 03E20.

Keywords: Texture; di-uniformity; quasi-uniformity; Lebesgue quasi-
uniformity.

1. Introduction

A texturing on a set S is a point-separating, complete, completely distribu-
tive lattice S of subsets of S with respect to inclusion, which contains S and ∅,
and for which arbitrary meet

∧

coincides with intersection
⋂

and finite joins
∨

with unions
⋃

. The pair (S, S) is called a texture.

This definition was first introduced by L. M. Brown to represent Hutton
algebras and lattices of L fuzzy sets in a point based setting [4]. However the
development of the theory has proceeded largely independently and the work
on di-uniformities has shown that it has much closer links with topological
ideas than might be expected. Di-uniformity on a texture was first defined
in [13] by giving descriptions in terms of direlations, dicovers and dimetrics
and the concepts of completeness and total boundedness were introduced in
[14]. The effect of a complementation and the relation with quasi-uniformity
and uniformity were discussed in [15]. In this context the work [15] pointed
out that di-uniformities provide a more unified setting for the study of quasi-
uniformity and uniformity than does the classical approach.

Received 19 October 2014 – Accepted 4 April 2015

http://dx.doi.org/10.4995/agt.2015.3323


S. Özçağ

As is well known a quasi-uniformity is obtained by omitting the symmetry
condition in the definition of a uniformity. We recall the notion of direlational
uniform texture space as follows.

Definition 1.1 ([13]). Let (S, S) be a texture and U a family of direlations on
(S, S). If U satisfies the conditions,

(1) (i, I) ⊑ (d, D) for all (d, D) ∈ U. That is, U ⊆ RDR.
(2) (d, D) ∈ U, (e, E) ∈ DR and (d, D) ⊑ (e, E) implies (e, E) ∈ U.
(3) (d, D), (e, E) ∈ U implies (d, D) ⊓ (e, E) ∈ U.
(4) Given (d, D) ∈ U there exists (e, E) ∈ U satisfying (e, E) ◦ (e, E) ⊑

(d, D).
(5) Given (d, D) ∈ U there exists (c, C) ∈ U satisfying (c, C)← ⊑ (d, D).

then U is called a direlational uniformity on (S, S), and (S, S, U) is known as a
direlational uniform texture.

This definition is formally same as the usual definition of diagonal uniformity.
It should be noted, that the symmetry condition (5) which guarantees a base
of symmetric direlations for the direlational uniformity is quite different from
the notion of symmetry for relations. In [15] an important result was obtained
that a direlational uniformity on the discrete texture (X, P(X)) corresponds
not to uniformity but to quasi uniformity. When the symmetry condition (5)
is removed we obtain a direlational quasi-uniform texture space (S, S, Uq) [17].

Another representation for di-uniformities is in terms of dicovers. We recall
from [2] that by a difamily we mean a set C = {(Aj, Bj) | j ∈ J} of elements of
S×S and C is called a dicover of (S, S) if

⋂

j∈J1
Bj ⊆

∨

j∈J2
Aj for all partitions

(J1, J2) of J. A dicover corresponds to a dual cover in the sense of [1] and this
notion is related to the notion of pairs of covers with a common index used
by Gantner and Steinlage [8] to characterize quasi uniformities. As in the
classical case dicovers generate symmetric direlations and are not appropriate
to characterize quasi di-uniformities. Hence in [17] the authors used a new
notion called dual dicover to introduce dual dicovering quasi-uniformity. Below
we recall these definitions.

Dual dicover([17]) A dual difamily Cd =
{((

C
1,1
j , C

1,2
j

)

,
(

C
2,1
j , C

2,2
j

))

| j ∈

J
}

of elements of (S × S) × (S × S) is called a dual dicover of (S, S) if
{(

C
1,1
j ∩

C
2,1
j , C

1,2
j ∪ C

2,2
j

)

| j ∈ J
}

is a dicover of (S, S).

Definition 1.2 ([17]). Let (S, S) be a texture. If υq is a family of dual dicovers
satisfying the conditions

(1) Given Cd ∈ υ
q there exists an anchored dual dicover Dd ∈ υ

q with
Dd ≺ Cd,

(2) Cd ∈ υ
q, Cd ≺ Dd implies Dd ∈ υ

q,
(3) Cd, Dd ∈ υ

q implies Cd ∧ Dd ∈ υ
q,

(4) Given Cd ∈ υ
q there exists Dd ∈ υ

q with Dd ≺(⋆) Cd.

we say υq is a dual dicovering quasi-uniformity on (S, S).

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Lebesgue quasi-uniformity on textures

In [17] besides these definitions there is another approach by using quasi-
pseudometrics. Since this work will be based on the direlational and dual
dicovering representations we will omit it.

This paper is a continuation of the work [16] where Lebesgue and co-Lebesgue
di-uniformities were first introduced and the relationship between Lebesgue
quasi-uniformity on X and the corresponding Lebesgue di-uniformity on dis-
crete texture (X, P(X)) was investigated. Moreover, the notions of Lebesgue
quasi di-uniformity and dual dicovering Lebesgue quasi-uniformity were in-
troduced and discussed some of their properties. In this work our source of
inspiration is [11] where the notion of pair Lebesgue quasi-uniformity was first
introduced by J. Marin and S. Romaguera and we confine our attention to dual
dicovering bi-Lebesgue quasi di-uniformities. The aim of this work is to con-
tinue to develop the notion of Lebesgue property on quasi di-uniform textures
and investigate dicompleteness of Lebesgue di-uniform textures.

After a brief introduction, in section 2 we introduce the notion of a bi-
Lebesgue quasi di-uniformity and show that on plain textures each quasi di-
uniformity with a dicompact topology is a bi-Lebesgue quasi di-uniformity.
We obtained the analogous result for the dual dicovering bi-Lebesgue quasi
di-uniform spaces which will be defined in Definition 2.11. In section 3 we
first consider the dual covering Lebesgue quasi-uniformy in the sense of Brown
[1] and discuss the completeness of Lebesgue di-uniformities on the discrete
texture.

General references on ditopological texture spaces include [1, 2, 3, 4, 5, 6]
and constant reference will be made to [13, 14, 15, 16, 17] for definitions and
results relating to di-uniformities. Our standart references for quasi uniformity
are [7, 8, 9]. For the conveince of the reader we recall some more special
definitions.

Let (S, S) be a texture. For s ∈ S the sets

Ps =
⋂

{A ∈ S | s ∈ A} and Qs =
∨

{A ∈ S | s /∈ A}

are called respectively, the p-sets and q-sets of (S, S). For A ∈ S the core A♭ of
A is given by A♭ = {s ∈ S | A * Qs}. The set A

♭ does not necessarily belong
to S.

In general, a texturing of S need not be closed under set complementation, but
sometimes we have a notion of complementation.

Complementation: [2] A mapping σ : S → S satisfying σ(σ(A)) = A, ∀ A ∈ S
and A ⊆ B =⇒ σ(B) ⊆ σ(A), ∀ A, B ∈ S is called a complementation on
(S, S) and (S, S, σ) is then said to be a complemented texture.

Examples :

1. For any set X, (X, P(X), πX), πX(Y ) = X \ Y for Y ⊆ X, is the com-
plemented discrete texture representing the usual set structure of X. Clearly,
Px = {x} and Qx = X \ {x} for all x ∈ X.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 169



S. Özçağ

2. For I = [0, 1] define I = {[0, t] | t ∈ [0, 1]} ∪ {[0, t) | t ∈ [0, 1]}, ι([0, t]) =
[0, 1 − t) and ι([0, t)) = [0, 1 − t], t ∈ [0, 1]. Then (I, I, ι) is a complemented
texture, which we will refer to as the unit interval texture. Here Pt = [0, t] and
Qt = [0, t) for all t ∈ I.

Ditopology: A dichotomous topology on (S, S) or ditopology for short, is a
pair (τ, κ) of subsets of S, where the set of open sets τ satisfies

1. S, ∅ ∈ τ,
2. G1, G2 ∈ τ =⇒ G1 ∩ G2 ∈ τ and
3. Gi ∈ τ, i ∈ I =⇒

∨

i
Gi ∈ τ,

and the set of closed sets κ satisfies
1. S, ∅ ∈ κ,
2. K1, K2 ∈ κ =⇒ K1 ∪ K2 ∈ κ and
3. Ki ∈ κ, i ∈ I =⇒

⋂

Ki ∈ κ.

For A ∈ S the sets [A] =
⋂

{K ∈ κ | A ⊆ K} and ]A[=
∨

{G ∈ τ | G ⊆ A} are
called the closure and interior of A.

A plain texture is one for which the texturing is closed under arbitrary
unions or equivalently join coincides with union in S. There is a considerable
simplification in plain textures. We have Ps * Qs for each s ∈ S. Hence, for
A ∈ S, s ∈ A, Ps ⊆ A and A * Qs are equivalent to each other.

One of the most useful notions in the theory of di-uniformities is that of
direlation.

Direlations: [5] Let (S, S), (T, T) be textures. We use P (s,t), Q(s,t) to denote

the p-sets and q-sets for the product texture (S × T, P(S) ⊗ T). Then:

(1) r ∈ P(S) ⊗ T is called a relation from (S, S) to (T, T) if it satisfies

R1 r * Q(s,t), Ps′ * Qs =⇒ r * Q(s′,t).

R2 r * Q(s,t) =⇒ ∃ s
′ ∈ S such that Ps * Qs′ and r * Q(s′,t).

(2) R ∈ P(S) ⊗ T is called a corelation from (S, S) to (T, T) if it satisfies

CR1 P (s,t) * R, Ps * Qs′ =⇒ P (s′,t) * R.

CR2 P (s,t) * R =⇒ ∃ s
′ ∈ S such that Ps′ * Qs and P (s′,t) * R.

A pair (r, R) consisting of a relation r and corelation R is called a direlation.

Now let (r, R) be a direlation from (S, S) to (T, T). The inverses of r and R
are given by

r← =
⋂

{Q(t,s) | r * Q(s,t)}, R
← =

∨

{P (t,s) | P (s,t) * R}.

where R← is a relation and r← a corelation. The direlation (r, R)← = (R←, r←)
from (T, T) to (S, S) is called the inverse of (r, R).

For A ⊆ S the A–section of a relation r and A–section of a corelation R is
defined by

r→A =
⋂

{Qt | ∀ s, r * Q(s,t) =⇒ A ⊆ Qs},

R→A =
∨

{Pt | ∀ s, P (s,t) * R =⇒ Ps ⊆ A}

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 170



Lebesgue quasi-uniformity on textures

Compositions of direlations: Let (S, S), (T, T), (U, U) be textures.

(1) If p is a relation on (S, S) to (T, T) and q a relation on (T, T) to (U, U)
then their composition is the relation q ◦ p on (S, S) to (U, U) defined
by

q ◦ p =
∨

{P (s,u) | ∃ t ∈ T with p * Q(s,t) and q * Q(t,u)}.

(2) If P is a co-relation on (S, S) to (T, T) and Q a co-relation on (T, T)
to (U, U) then their composition is the co-relation Q ◦ P on (S, S) to
(U, U) defined by

Q ◦ P =
⋂

{Q(s,u) | ∃ t ∈ T with P (s,t) * P and P (t,u) * Q}.

(3) With p, q; P , Q as above, the composition of the direlations (p, P),
(q, Q) is the direlation

(q, Q) ◦ (p, P) = (q ◦ p, Q ◦ P).

2. Lebesgue quasi di-uniform spaces

In this section we consider the Lebesgue property on quasi di-uniform tex-
tures. We introduce Lebesgue and co-Lebesgue direlational quasi-uniformities.
We also define bi-Lebesgue quasi di-uniformity, dual dicovering bi-Lebesgue
quasi di-uniformity and give an analog of the well known result that each quasi-
uniformity compatible with a compact space is a Lebesgue quasi-uniformity.

The definition of a direlational uniformity U on a texture (S, S) has been
introduced in Definition 1.1. We obtain a direlational quasi-uniformity on (S, S)
by removing the symmetry condition from the definition of the direlational
uniformity. Now we begin by recalling the following definition.

Definition 2.1 ([17, Definition 2.1]). Let (S, S) be a texture and Uq a family
of direlations on (S, S). If Uq satisfies the conditions

(1) (i, I) ⊑ (d, D) for all (d, D) ∈ Uq,
(2) (d, D) ∈ Uq, (e, E) ∈ DR and (d, D) ⊑ (e, E) implies (e, E) ∈ Uq,
(3) (d, D), (e, E) ∈ Uq implies (d, D) ⊓ (e, E) ∈ Uq,
(4) Given (d, D) ∈ Uq there exists (e, E) ∈ Uq satisfying (e, E) ◦ (e, E) ⊑

(d, D),

then Uq will be called a direlational quasi-uniformity on (S, S) and (S, S, Uq) a
direlational quasi-uniform texture space.

As in the classical case for the direlational quasi-uniformity Uq on (S, S)

(Uq)← = {(d, D)← : (d, D) ∈ Uq}

is also a direlational quasi-uniformity on (S, S) and (S, S, (Uq)←) is called the
conjugate of (S, S, Uq) (see, [17]).

A direlational quasi-uniformity Uq on (S, S) induces a uniform ditopology
(τUq , κUq ) as follows, in exactly the same way that a direlational uniformity
does [13, Lemma 4.3].

(i) G ∈ τUq ⇐⇒ (G * Qs =⇒ ∃ (d, D) ∈ U
q with d[s] ⊆ G),

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 171



S. Özçağ

(ii) K ∈ κUq ⇐⇒ (Ps * K =⇒ ∃ (d, D) ∈ U
q with K ⊆ D[s]).

Here d[s] = d→Ps and D[s] = D
→Qs.

When we speak of the ditopology of (S, S, Uq) we will always mean the
uniform ditopology.

In order to consider Lebesgue direlational quasi-uniformity it is necessary to
recall [6] the open cover and closed cocover for the textures.

Let (τ, κ) be a ditopology on the texture (S, S) and let A ∈ S. The family
{Gi | i ∈ I} is said to be an open cover [6] of A if Gi ∈ τ for all i ∈ I and
A ⊆

∨

i∈I
Gi. Dually we may speak of a closed cocover of A, namely a family

{Fi | i ∈ I} with Fi ∈ κ for all i ∈ I satisfying
⋂

i∈I
Fi ⊆ A.

For the cocovers we need a notion of dual refinement.

Definition 2.2 ([16]). Let K1, K2 be cocovers. Then K1 will be called a dual
refinement of K2, and write K1 ⊳ K2 if for a given K2 ∈ K2 there exists
K1 ∈ K1 such that K1 ⊆ K2.

Now we may give:

Definition 2.3. A direlational quasi-uniformity Uq on (S, S) is called

(1) Lebesgue direlational quasi-uniformity provided that for each cover C
of S which is open for the uniform ditopology there is a direlation
(r, R) ∈ Uq such that {r[s] | s ∈ S♭} is a refinement of C.

(2) Co-Lebesgue direlational quasi-uniformity provided that for each co-
cover K of ∅ which is closed for the uniform ditopology there is a dire-
lation (r, R) ∈ Uq such that K is a dual refinement of {R[s] | s ∈ S♭}.

In [16, Proposition 2.6] it is proved that each direlational uniformity com-
patible with a compact (cocompact) ditopological texture space is a Lebesgue
(co-Lebesgue) direlational uniformity on (S, S). We now have the analogous
result for the direlational quasi-uniformities.

Let us recall the following definition.

Definition 2.4 ([6]). Let (τ, κ) be a ditopology on the texture (S, S) and A ∈ S.

(1) A is called compact if whenever {Gi | i ∈ I} is an open cover of A
then there is a finite subset J of I with A ⊆

⋃

j∈J
Gj. In particular the

ditopological texture space (S, S, τ, κ) is called compact if S is compact.
(2) A is called cocompact if whenever {Fi | i ∈ I} is a closed cocover of A

then there is a finite subset J of I with
⋂

j∈J
Fj ⊆ A. In particular

the ditopological texture space (S, S, τ, κ) is called cocompact if ∅ is
cocompact.

We recall from [6] that a ditopological texture space (S, S, τ, κ) is stable
(costable) if every F ∈ κ\{S} (G ∈ τ\{∅}) is compact (cocompact). The
ditopological texture space is called dicompact if it is compact, cocompact,
stable and costable.

We may now give the promised result.

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Lebesgue quasi-uniformity on textures

Proposition 2.5. Let (S, S, τ, κ) be a ditopological texture space.

(1) If Uq is a direlational quasi-uniformity compatible with a compact di-
topology, then Uq is a Lebesgue direlational quasi-uniformity on (S, S).

(2) If Uq is a direlational quasi-uniformity compatible with a cocompact
ditopology, then Uq is a co-Lebesgue direlational quasi-uniformity on

(S, S).

Proof. The proof follows from the same lines as in the proof of [16, Proposi-
tion 2.6]. �

The notion of Lebesgue quasi di-uniformity was introduced in [16] and the
anchored property for the dicover was omitted in that definition. Nevertheless
we now begin by introducing a new notion called bi-Lebegue quasi di-uniform
space.

First let us recall the notion of an anchored dicover which plays an important
role in the development of dicovering uniformities and bi-Lebesgue quasi di-
uniformities.

Definition 2.6 ([13]). A family C ⊆ S × S is called an anchored dicover if it
satisfies:

(1) P ≺ C, and
(2) Given A C B there exists s ∈ S satisfying

(a) A * Qu =⇒ ∃ A
′ C B′ with A′ * Qu and Ps * B

′, and
(b) Pv * B =⇒ ∃ A

′′ C B′′ with Pv * B
′′ and A′′ * Qs.

A dicover C = {(Aj, Bj) | j ∈ J} is finite if the set {Aj | j ∈ J} is finite
and cofinite if the set {Bj | j ∈ J} is finite. If C is defined on a ditopological
texture space (S, S, τ, κ), it is said to be open coclosed if Aj ∈ τ and Bj ∈ κ
and closed co-open if Aj ∈ κ and Bj ∈ τ for all j ∈ J. C is a refinement of D if
given j ∈ J we have L D M so that Aj ⊆ L and M ⊆ Bj. In this case we write
C ≺ D.

If (d, D) is a reflexive direlation on (S, S) then γ(d, D) = {(d[s], D[s]) | s ∈
S♭} is an anchored dicover of (S, S).

Definition 2.7. Let Uq be a quasi di-uniformity on a ditopological texture
space (S, S, τ, κ). Then Uq is a bi-Lebesque quasi di-uniformity provided that for
each open coclosed anchored dicover C of (S, S) there is a direlation (r, R) ∈ Uq

such that the dicover γ(r, R) = {(r[s], R[s]) | s ∈ S♭} refines C and (S, S, Uq) is
called a bi-Lebesgue quasi di-uniform texture space.

The following important theorem, which is proved in [3] gives a characteri-
zation of dicompactness in textures.

Theorem 2.8. The following are equivalent for (S, S, τ, κ).

(1) (S, S, τ, κ) is dicompact.
(2) Every closed co-open difamily with the finite exclusion property is bound.
(3) Every open coclosed dicover has a finite and cofinite subdicover.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 173



S. Özçağ

Now we may give the following theorem in the case of plain textures.

Theorem 2.9. Let (S, S, Uq) be a plain direlational quasi-uniform texture space
such that (τUq , κUq ) is dicompact. Then (S, S, U

q) is a bi-Lebesgue quasi di-
uniform space.

Proof. Let C = {(Aj, Bj) | j ∈ J} be an open coclosed anchored dicover of

(S, S). For each s ∈ S♭ and j(s) ∈ J we have Aj(s)C Bj(s) with Aj(s) ∈ τUq

and Bj(s) ∈ κUq . Since (S, S) is plain and C is an anchored dicover, for s
′ ∈ S♭

we have Aj(s) * Qs′ and there exists (es, Es) ∈ U
q with es[s

′] ⊆ Aj(s). Now
take Bj(s) ∈ κUq and Ps′ * Bj(s) then we have Bj(s) ⊆ Es[s

′]. Since Uq is
a direlational quasi-uniformity there exists (ds, Ds) ∈ U

q satisfying (ds, Ds) ◦
(ds, Ds) ⊑ (es, Es). Hence {( ] ds[s

′] [ , [ Ds[s
′] ] ) | s′ ∈ S♭} is an open coclosed

anchored dicover of (S, S) with Ps′ ⊆ ] ds[s
′] [ and [ Ds[s

′] ] ⊆ Qs′.

Since (S, S) is dicompact, the open coclosed dicover {( ] ds[s
′] [ , [ Ds[s

′] ] ) |
s′ ∈ S♭} has a finite cofinite subdicover {( ] dsk [s

′

k] [ , [ Dsk[s
′

k] ] ) | s
′ ∈ S♭} for

k = 1, ..., n by Theorem 2.8.

If we set (d, D) =
dn

k=1(dsk , Dsk), then (d, D) ∈ U
q. Since (S, S) is plain,

we have dsk [s
′

k] * Qs′ and Ps′ * Dsk[s
′

k] for s
′ ∈ S♭ and 1 ≤ k ≤ n. We shall

show that γ(d, D) = {(d[s′], D[s′]) | s′ ∈ S♭} ≺ C. For the given s′ ∈ S♭ there
is k ∈ {1, 2, 3, ..., n} such that esk[s

′

k] ⊆ Aj(sk) and Bj(sk) ⊆ Esk [s
′

k]. We need
to prove d[s′] ⊆ esk [s

′

k] ⊆ Aj(sk) and Bj(sk) ⊆ Esk [s
′

k] ⊆ D[s
′].

Now let us prove Bj(sk) ⊆ Esk [s
′

k] ⊆ D[s
′]. First suppose that Esk [s

′

k] *
D[s′]. Then there exists z ∈ S♭ with Esk [s

′

k] * Qz and Pz * D[s
′]. Because of

D =
⊔n

k=1 Dsk we have Dsk ⊆ D so Dsk [s
′] ⊆ D[s′] and we have Pz * Dsk [s

′].
By the definition of composition of co-relations [13, Definition 1.7] we have
Esk ⊆ Dsk ◦ Dsk ⊆ Q(s′

k
,z).

From Ps′ * Dsk[s
′

k] = D
→

sk
(Qs′

k
) we obtain P (s′

k
,s′) * Dsk and due to

Pz * Dsk[s
′] we have P (s′,z) * Dsk by [13, Lemma 1.5].

On the other hand E→sk (Qs′k) = Esk [s
′

k] * Qz gives z
′ ∈ S with Pz′ * Qz

and for t ∈ S♭

(2.1) P (t,z′) * Esk =⇒ Pt ⊆ Qs′k .

by [13, Definition 1.3.(2)]. From Esk ⊆ Q(s′
k
,z) and Pz′ * Qz we have P (s′k,z′) *

Esk and since Esk is a co-relation we have s
′′

k ∈ S with Ps′′k * Qs′k and P (s′′k ,z′) *
Esk by CR2. Now we may apply the implication (1) with t = s

′′

k to give the
contradiction Ps′′

k
⊆ Qs′

k
.

(2) The proof of d[s′] ⊆ esk[s
′

k] ⊆ Aj(sk) is dual to the above and is omitted.
�

We will use the term quasi di-uniformity [17] to refer to direlational quasi-
uniformities and dual dicovering quasi-uniformities in general.

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Lebesgue quasi-uniformity on textures

Let us recall from [17, Definition 3.8] that a dual difamily Cd is called an
anchored dual dicover if (i) Pd = {((Ps, Qs), (Ps, Qs))} | s ∈ S

♭} ≺ Cd and (ii)
Cd ≺ C

∆
d . If (r, R) is a reflexive direlation on (S, S) then the family γ

q(r, R) =

{(γ(r, R), γ(r, R)←) | s ∈ S♭} is an anchored dual dicover where γ(r, R)← =

{(R←[s], r←[s]) | s ∈ S♭}. Moreover a dual dicover Cd satisfying
(

C
1,1
j , C

1,2
j

)

∈

(τUq , κUq ) and
(

C
2,1
j , C

2,2
j

)

∈ (τ(Uq)←, κ(Uq)←) is called open co-closed.

We recall the definition of a refinement for the dual dicovers.

Definition 2.10 ([17]). Let Cd = {((C
1,1
j , C

1,2
j ), (C

2,1
j , C

2,2
j )) | j ∈ J} and Dd

be dual dicovers. Then Cd is a refinement of Dd, written Cd ≺ Dd, if given
j ∈ J we have ((D1,1, D1,2), (D2,1, D2,2)) ∈ Dd so that

(C
1,1
j , C

1,2
j ) ⊑ (D

1,1, D1,2) and (C
2,1
j , C

2,2
j ) ⊑ (D

2,1, D2,2)

⇐⇒ C
1,1
j ⊆ D

1,1 ; D1,2 ⊆ C
1,2
j and C

2,1
j ⊆ D

2,1 ; D2,2 ⊆ C
2,2
j

Now let us make the following definition:

Definition 2.11. Let (S, S, Uq) be a quasi di-uniform space. Uq is called a dual
dicovering bi-Lebesgue quasi di-uniformity if for each open coclosed anchored
dual dicover Cd of (S, S, U

q) there is a direlation (r, R) ∈ Uq such that γq(r, R)
refines Cd.

We also recall from [17, Definition 2.7] that the direlational uniformity with
subbase Uq ∪(Uq)← is called the direlational uniformity associated with Uq and
is denoted by U∗ = Uq ∨ (Uq)←.

J. Marin and S. Romaguera [11] obtained a result states that if (X, U) is
a quasi uniform space such that (X, τ(U∗)) is compact then (X, U) is a pair
Lebesgue quasi-uniform space.

We end this section by obtaining a similar result to the classical case.

Theorem 2.12. Let (S, S, Uq) be a plain quasi di-uniform texture space such
that (τU∗, κU∗) is dicompact. Then (S, S, U

q) is a dual dicovering bi-Lebesque
quasi di-uniform space.

Proof. Let Cd =
{((

C
1,1
j , C

1,2
j

)

,
(

C
2,1
j , C

2,2
j

))

| j ∈ J
}

be an open coclosed

anchored dual dicover. For each s ∈ S♭ and j(s) ∈ J we have

(C
1,1
j(s)

, C
1,2
j(s)

) Cd (C
2,1
j(s)

, C
2,2
j(s)

) with (C
1,1
j(s)

, C
1,2
j(s)

) ∈ (τUq × κUq ) and

(C
2,1
j(s)

, C
2,2
j(s)

) ∈ (τ(Uq)← × κ(Uq)←)

Since (S, S) is plain and Cd is an anchored dual dicover we have C
1,1
j(s)

* Qs′

for s′ ∈ S♭ and there exists (ds, Ds) ∈ U
q with ds[s

′] ⊆ C
1,1
j(s)

. In this case there

exist (rs, Rs) ∈ U
q with (rs, Rs)

2 ⊑ (ds, Ds) and rs
2[s′] ⊆ ds[s

′] ⊆ C
1,1
j(s)

since

Uq is a direlational quasi-uniformity.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 175



S. Özçağ

Now take C
2,1
j(s)

∈ τ(Uq)← . Since Cd is an anchored dual dicover and (S, S) is

plain, we have C
2,1
j(s)

* Qs′ for s
′ ∈ S♭. Hence, there exist (ds, Ds)

← ∈ (Uq)
←

and (rs, Rs)
← ∈ (Uq)

←

such that (Rs
←)2[s′] ⊆ Ds

←[s′] ⊆ C
2,1
j(s)

.

Dually for C
1,2
j(s)

∈ κUq and C
2,2
j(s)

∈ κ(Uq)← with Ps′ * C
1,2
j(s)

and Ps′ * C
2,2
j(s)

we have C
1,2
j(s)

⊆ Ds[s
′] ⊆ Rs

2[s′] and C
2,2
j(s)

⊆ ds
←[s′] ⊆ (rs

←)2[s′]. Thus

{((rs[s
′], Rs[s

′]), (Rs
←[s′], rs

←[s′])) | s′ ∈ S♭} is an anchored dual dicover by
[17, Proposition 3.10]. Then
{(

]

rs[s
′] ∩ Rs

←[s′]
[

), (
[

Rs[s
′] ∪ rs

←[s′]
]

) | s′ ∈ S♭} is an open coclosed an-

chored dicover of S satisfying Ps′ ⊆
]

rs[s
′]∩Rs

←[s′]
[

and
[

Rs[s
′]∪rs

←[s′]
]

⊆
Qs′

Since (S, S) is dicompact the open coclosed dicover
{(

]

rs[s
′] ∩ Rs

←[s′]
[

), (
[

Rs[s
′] ∪ rs

←[s′]
]

) | s′ ∈ S♭} has a finite cofinite subdi-

cover {(
]

rsk [s
′

k] ∩ Rsk
←[s′k]

[

), (
[

Rsk [s
′

k] ∪ rsk
←[s′k]

]

) | s′ ∈ S♭} for k = 1, ..., n
by Theorem 2.8.

Now we set (r, R) =
dn

k=1(rsk , Rsk ) and note that (r, R) ∈ U
q. Since (S, S)

is plain we have rsk [s
′

k] ∩ R
←

sk
[s′k] * Qs′ and Ps′ * Rsk [s

′

k] ∪ r
←

sk
[s′k] for s

′ ∈ S♭

and 1 ≤ k ≤ n.

We will complete the proof by showing γq(r, R) ≺ Cd. For the given s
′ ∈ S♭

there is k ∈ {1, 2, 3, ..., n} such that dsk [s
′

k] ⊆ C
1,1
j(sk)

, D←sk [s
′

k] ⊆ C
2,1
j(sk)

, C
1,2
j(sk)

⊆

Dsk[s
′

k] and C
2,2
j(sk)

⊆ d←sk [s
′

k]. Now let us prove that r[s
′] ⊆ dsk [s

′

k] ⊆ C
1,1
j(sk)

.

First suppose that r[s′] * dsk [s
′

k]. Then there exists z ∈ S
♭ with r[s′] * Qz

and Pz * dsk [s
′

k]. Since rsk [s
′

k]
⋂

R←sk[s
′

k] * Qs′ we have rsk [s
′

k] * Qs′ and

R←sk[s
′

k] * Qs′. Then we have rsk * Q
′

(s′
k
,s) and also by r =

⋂n
k=1 rsk for each

k = 1, ..., n we have r ⊆ rsk , whence r[s
′] ⊆ rsk [s

′] and we have rsk [s
′] * Qz

which gives rsk * Q(s′,z). Hence we obtain P (s′k,z) ⊆ r
2
sk

⊆ dsk . On the other

hand Pz * dsk [s
′

k] = d
→

sk
Ps′

k
gives Pz * Qz′ for z

′ ∈ S♭ and

(2.2) dsk * Q(v,z′) =⇒ Ps′k ⊆ Qv, for v ∈ S
♭

From P (s′
k
,z) ⊆ dsk and Pz * Qz′ we have dsk * Q(s′

k
,z′), and since r is a

relation we have s′′k ∈ S
♭ with Ps′

k
* Qs′′

k
such that dsk * Q(s′′

k
,z′) by R2.

Applying the implication (2) with v = s′′k we deduce Ps′k ⊆ Qs
′′

k
, which is a

contradiction. This verifies r[s′] ⊆ dsk [s
′

k] ⊆ C
1,1
j(s)

.

Now it is easy to prove that R←[s′] ⊆ D←sk [s
′

k] ⊆ C
2,1
j(sk)

. The other two

inclusions can be shown dually and hence the proof is omitted. This completes
the proof of γq(r, R) ≺ Cd, thus (S, S, U

q) is a dual dicovering bi-Lebesque quasi
di-uniform space. �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 176



Lebesgue quasi-uniformity on textures

3. Completeness of Lebesgue di-uniform spaces

We conclude this paper by discussing the completeness of Lebesgue di-
uniformities. The subjects of completeness and total boundedness in di-uniform
spaces are discussed in [14].

In the classical theory it is known from [7, 11] that every Lebesgue unifor-
mity is complete and every Lebesgue quasi-uniformity is convergence complete.
At the beginning of this work, we expect to find an analogue result for the
Lebesgue di-uniformities. However there are considerable difficulties such as
the convergence of the Cauchy di-filter to obtain a similar result for the general
textures. Since there is a close relationship between quasi-uniformities and di-
uniformities on discrete textures so we turn our attention to the completeness
of Lebesgue di-uniformities which correspond to Lebesgue quasi-uniformity on
the discrete textures.

Quasi-uniform spaces can be defined in various equivalent ways; by rela-
tions that satisfy all the axioms of a uniformity except symmetry; by quasi-
pseudometrics and by (pair, dual) covers. Gartner and Steinlage [8] presented a
description of quasi-uniformities in terms of pairs of covers and Marin and Ro-
maguera [11] used a similar notion called open pairs, that is {(Gα, Hα) | α ∈ A}
such that Gα is τQ-open and Hα is τQ−1-open and for each x ∈ X there is α ∈ A
with x ∈ Gα ∩Hα where τQ is the topology generated by Q and τQ−1 that gen-
erated by Q−1.

Brown [1] independently developed a theory of quasi-uniformities by using
a new concept of dual cover and showed the equivalence with the notion of
open pairs mentioned in [12]. In this context the notion of Q-completeness
was considered for the completeness of quasi-uniform spaces. Now we find it
convenient to use the representation in terms of dual covers in this section.

Throughout this section we are interested in the concept of completeness
namely Q-completeness for quasi-uniformities which is based on the use of dual
covers. Since dual covers are not well known concepts, we now recall from [1, 15]
some definitions and properties. However the dual covering quasi-uniformity
and the equivalence with the diagonal quasi-uniformity were studied widely in
[15].

Let X be a set. A family U = {(Aj, Bj) | j ∈ J} of subsets of X is called a
dual cover of X if

⋃

{(Aj
⋂

Bj) | j ∈ J} = X. If U and V are dual covers of
X we say U refines V and write U ≺ V if whenever AUB there exists CV D
satisfying A ⊆ C and B ⊆ D. Given a binary point relation d ∈ X we may
associate with d the dual family called dual cover

γ∗(d) = {(d[x], d−1[x]) | x ∈ X}

where, as usual d[x] = {y ∈ X | (x, y) ∈ d} and d−1[x] = {y ∈ X | (y, x) ∈ d}.
Now let us turn our attention to the completeness of quasi-uniformity and

Lebesgue quasi-uniformity. In the literature, several authors defined various
kinds of completeness on quasi-uniform spaces. We will mention particulary
two of these definitions. According to [7] a quasi uniform space (X, Q) is called

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 177



S. Özçağ

bicomplete if (X, Q∗) is a complete uniform space where Q∗ = Q ∨ Q−1. Since
Lebesgue uniformity is complete, Marin and Romaguera [11] obtained a result
which states that each pair Lebesgue quasi-uniformity is bicomplete.

On the other hand a quasi-uniform space (X, Q) is convergence complete
[7] provided that each Cauchy filter is τQ-convergent and since every Lebesgue
quasi-uniformity is convergence complete, Marin and Romaguera also proved
that each pair Lebesgue quasi-uniformity is convergence complete.

Starting from this point, we focus our attention on another type of complete-
ness namely Q-completeness in the Brown’s sense (see, [1]). We recall that a
bifilter B on the set X is defined as a product of two filters Bu and Bv on X
denoted by B = Bu × Bv. Any bifilter B is ı-regular if F ∩ G 6= ∅ whenever
(F, G) ∈ B.

If (X, u, v) is a bitopological space and x ∈ X then B(x) = {(H(x), K(x)) |
H(x) is a u-nhd. and K(x) is a v-nhd of x} is an ı-regular bifilter which we will
call the nhd. bifilter of x. The bifilter B converges to x if B(x) ⊆ B. If Q is
a dual covering quasi-uniformity compatible with (X, u, v) then the bifilter B
will be called Q-Cauchy if U ∩ B 6= ∅ for all U ∈ Q.

Definition 3.1 (see [1]). A quasi-uniform space (X, Q) is called Q-complete
if every ı-regular Q-Cauchy bifilter is convergent in the bitopological space
(X, τQ, τQ−1).

Now we shall work with dual covers in the sense of Brown [1] instead of
pair open cover and because of the equivalence of these two concepts we expect
to have similar results as given in the paper of Marin and Romaguera [11].
We first give the definition of a notion dual covering Lebesgue quasi-uniformity
which was defined by Marin and Romaguera under the name of pair Lebesgue
quasi-uniformity.

Definition 3.2. Let Q be a quasi-uniformity on X. We say that Q is a dual
covering Lebesgue quasi-uniformity if for each open dual cover U = {(Aj, Bj) |
j ∈ J} of (X, Q) there is d ∈ Q such that the dual cover {(d[x], d−1[x]) | x ∈ X}
refines {(Aj, Bj) | j ∈ J} (i.e. for each x ∈ X there is j ∈ J such that d[x] ⊆ Aj
and d−1[x] ⊆ Bj). Thus, we say that (X, Q) is a dual covering Lebesgue quasi-
uniform space.

Here we recall from [16] that a quasi-uniformity Q on a set X is a Lebesgue
quasi-uniformity provided that for each τQ-open cover G of X there is d ∈ Q
such that the cover {d[x] | x ∈ X} refines G.

Proposition 3.3. Let (X, Q) be a dual covering Lebesgue quasi-uniform space.
Then Q and Q−1 are Lebesgue quasi-uniformities.

Proof. Let Q be a dual covering Lebesgue quasi-uniformity on X. We shall
show that Q is a Lebesgue quasi-uniformity. Let {Aj : j ∈ J} be a τQ-open
cover of X. For each j ∈ J let Bj = X. Then {(Aj, X) : j ∈ J} is an
open dual cover of (X, Q). So there is d ∈ Q such that the open dual cover
{(d[x], d−1[x]) | x ∈ X} refines {(Aj, X) : j ∈ J} which gives the required
result. Similarly we see that Q−1 is a Lebesgue quasi-uniformity. �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 178



Lebesgue quasi-uniformity on textures

Theorem 3.4. Every Lebesgue quasi-uniformity is Q-complete.

Proof. Let B = Bu × Bv be a regular Cauchy bifilter that does not converge
to x. Then for each x ∈ X we have M(x) a τQ-nhd. of x and N(x) a τQ−1 -
nhd. of x such that (M(x), N(x)) /∈ B. Also there exists d ∈ Q such that
{d[x] : x ∈ X} refines M(x). Since B = Bu × Bv is a Cauchy bifilter, d[x] ∈ Bu
for some x ∈ X, which is a contradiction. �

The following result is clear from the above discussion so we omit the proof.

Theorem 3.5. Let Q be a dual covering Lebesgue quasi-uniformity on X.
Then both (X, Q) and (X, Q−1) are Q-complete quasi-uniform spaces.

In the remainder of this section we consider the completeness of Lebesgue
di-uniformities on discrete texture (X, P(X)). We will investigate how the
relation between quasi-uniformity and di-uniformity effects the completeness
of Lebesgue di-uniformities. The reader is referred to [15] for more backround
material, for the benefit of the reader however we will briefly recall the necessary
definitions and results.

In [15] it is shown that di-uniformities on the discrete texture correspond
to quasi uniformities on X. Moreover a direlational uniformity on (X, P(X))
corresponds to a uniformity if and only if it is complemented. Let d ⊆ X×X be
a point relation then u(d) = (d, d←) is a direlation on (X, P(X)) and if Q is a
quasi-uniformity on X, the family u(Q) = {(e, E) | ∃d ∈ Q and u(d) ⊑ (e, E)}
is a direlational uniformity on the discrete texture (X, P(X)).

Proposition 3.6 ([15]). Let Q be a quasi-uniformity on X and Q−1 its conju-
gate. Then the direlational uniformity on (X, P(X), πX) corresponding to Q

−1

is the complement of the direlational uniformity corresponding to Q. That is,

u(Q−1) = u(Q)′.

Theorem 3.7 ([15]). Let Q be a quasi-uniformity on X. Then Q is a unifor-
mity if and only if the corresponding di-uniformity u(Q) on (X, P(X), πX) is
complemented.

We can now tie the completeness of a quasi-uniformity in with the dicom-
pleteness of a di-uniformity on (X, P(X)).

Proposition 3.8. The quasi-uniform space (X, Q) is Q-complete if and only
if the di-uniform discrete texture space (X, P(X), u(Q)) is dicomplete.

Proof. It is similar to the proof of [19, Proposition 2.16]. �

Now we have the following theorems.

Theorem 3.9 ([16, Theorem 2.3]). Let Q be a Lebesgue quasi-uniformity on
X. Then the corresponding di-uniformity u(Q) on (X, P(X), πX) is a Lebesgue
direlational uniformity.

Conversely if U is a Lebesgue direlational uniformity on (X, P(X), πX) then
u−1(U) is a Lebesgue quasi-uniformity on X.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 179



S. Özçağ

Theorem 3.10 ([16, Theorem 2.5]). Let Q be Lebesgue quasi uniformity on X.
Then the complement of the direlational uniformity corresponding to Q, that is

u(Q)′, is a co-Lebesgue direlational uniformity on (X, P(X), πX).

Conversely, if U is the co-Lebesgue direlational uniformity corresponding to

Q−1, then u−1(U′) is a Lebesgue quasi uniformity on X.

We are now in a position to give the promised result.

Theorem 3.11. Let Q be a Lebesgue quasi-uniformity on X. Then the corre-
sponding Lebesgue di-uniformity u(Q) on (X, P(X), πX) is dicomplete.

Proof. Let Q be a Lebesgue quasi-uniformity on X. We know from Theorem
3.9 that the corresponding di-uniformity u(Q) on (X, P(X)) is Lebesgue. Since
every Lebesgue quasi-uniformity is Q-complete by Theorem 3.4, the corre-
sponding Lebesgue di-uniformity u(Q) is dicomplete by Proposition 3.8. �

Theorem 3.12. Let Q be a Lebesgue quasi-uniformity on X. If Q is a unifor-
mity, then the complement of the corresponding Lebesgue di-uniformity u(Q)
on (X, P(X), πX) is dicomplete.

Proof. If Q is a uniformity then Q = Q−1 and u(Q−1) = u(Q)′ = u(Q) by
Proposition 3.6 and Theorem 3.7. Then by Theorem 3.11 the co-Lebesgue
di-uniformity u(Q)′ is dicomplete. �

Remark 3.13. Since every dual covering Lebesgue quasi-uniformity is a Lebesgue
quasi-uniformity it is clear that the last two results hold for the dual covering
Lebesgue quasi-uniformities.

4. Conclusion Remarks

In [10] Hutton gave the definition of uniformities and quasi uniformities
on a Hutton algebra Lx using functions on Lx. A similar representation was
obtained in [18] for di-uniformities and quasi di-uniformities called difunctional
uniformity and difunctional quasi-uniformity [18, Definition 2.7].

For the further studies it would be interesting to investigate the Lebesgue
property on the difunctional uniformities and quasi-uniformities.

Acknowledgements. The author would like to thank the referees for their

helpful suggestions and comments.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 180



Lebesgue quasi-uniformity on textures

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