EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 612-627
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Separation Axioms in Diframes

Esra Korkmaz1, Rıza Ertürk1,∗

1 Department of Mathematics, Hacettepe University, Ankara, Turkey

Abstract. Ditopological texture spaces are simultaneously generalizations of topological, bitopo-
logical and fuzzy topological spaces, and diframes are generalizations of ditopological texture
spaces. In this paper we define and study the separation axioms in diframe setting.

2010 Mathematics Subject Classifications: 06D22, 54A05

Key Words and Phrases: Diframe, fr-below, cf-below, Urysohn relation

1. Introduction

The concept of ditopological texture spaces grew out of the study of the represen-
tation of lattice-valued topologies by bitopologies. However, as distinct from the theory
of bitopological spaces based on the notion of open sets, it is a structure in which the
open and closed sets play an equal role. Ditopologies are defined on a suitable subfamily
S ⊆ P(S) which is, in fact, a complete, completely distributive lattice with the relation of
inclusion. Ever since the theory was first introduced by L.M. Brown [5], topological con-
cepts, such as separation axioms, compactness and compactifications, have been studied
in a series of papers by L.M. Brown and co-authors [2–4].

This work is a continuation of our previous paper [9]. In that paper, we defined the
notion of diframe by replacing a texturing of a set with a lattice which is both a frame
and a coframe. We also provided a link between the morphisms of the category of texture
spaces (drTex) and the category of frames (Frm). This connection allows us to construct
the category diFrm of diframes and diframe homomorphisms. There are at least two
reasons why the theory of diframes is important. Dropping the complete distributivity
condition, which makes the texture a spatial frame, (that is, a frame isomorphic to the
lattice of open sets, Ω(X), of a set X), we obtain a larger family of lattices. Besides,
diframe theory initiates the frame-theoretical perspective in the theory of ditopological
spaces. It is well-known that the frame (locale) theory is an important area of research
and it translates the (bi)topological concepts into the point-free language [1, 10].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3272

Email addresses: esrakaratas@hacettepe.edu.tr (E. Korkmaz), rerturk@hacettepe.edu.tr (R.
Ertürk)

http://www.ejpam.com 612 c© 2018 EJPAM All rights reserved.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 613

The rest of this paper is structured as follows. In the second section, some basic
concepts and properties of ditopological texture spaces and frames are introduced to make
the paper self-contained. In the third section, we define the separation axioms in the
setting of diframes and we obtain equivalent characterizations of these axioms. Finally,
the conclusion of this paper and some future works are discussed in Section 4.

2. Preliminaries

In this section, we recall some pertinent concepts of ditopological texture spaces,
(co)frames and diframes. We refer to [2, 3] and [4] for ditopological texture spaces, and
to [6] and [10] for lattice and frame theory.

Ditopological Texture Spaces: A texturing on a set S is a point separating, com-
plete, completely distributive lattice S of subsets of S with inclusion relation, which con-
tains S and ∅ and for which arbitrary meet coincides with intersection and finite joins
coincide with the union. The pair (S, S) is known as a texture space, or shortly a texture.

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

(T1) S, ∅ ∈ τ ,

(T2) G1, G2 ∈ τ ⇒ G1 ∩G2 ∈ τ ,

(T3) Gi ∈ τ, i ∈ I ⇒
∨
iGi ∈ τ ,

and the set of closed sets κ satisfies

(CT1) S, ∅ ∈ κ,

(CT2) K1,K2 ∈ κ⇒ K1 ∪K2 ∈ κ,

(CT3) Ki ∈ κ, i ∈ I ⇒
⋂
iKi ∈ κ.

A ditopology can be considered as a representation of lattice-valued topologies by
bitopologies and one can simply infer that it is a structure in which the open and closed
sets play an equal role.

(co)Frames and (co)Locales: Our notation for the theory of (co)frames and (co)locales
is that of [10] and [9]. First we recall the following definitions for a lattice L:

Let L and M be posets. A pair (f, g) of monotone functions f : L→M , g : M → L is
called a Galois adjunction if, for all x ∈ L and y ∈M , f(x) ≤ y iff x ≤ g(y). In this case,
f is called the left adjoint of g (denoted by f = g∗), and g is called the right adjoint of f
(denoted by g = f∗).

Proposition 1. Let (f, g) be a Galois adjunction. Then

(i) f preserves arbitrary join, and g preserves arbitrary meet.

(ii) g is one-one iff f is onto.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 614

(iii) If f is one-one then gf=id, if it is onto then fg=id.

Now let us recall the other required notions for the present paper:
L is called a frame if it is a complete lattice with the property

b ∧ (
∨
A) =

∨
{b ∧ a : a ∈ A}

for any b ∈ L and any subset A ⊆ L.
Dually, M is called a coframe if it is a complete lattice with the property

b ∨ (
∧
A) =

∧
{b ∨ a : a ∈ A}

for any b ∈ L and any subset A ⊆ L.
A frame (resp. coframe) homomorphism is a map between frames (resp. coframes)

preserving arbitrary joins (resp. meets) and finite meets (resp. joins). The category of
frames (resp. co-frames) and frame (resp. co-frame) homomorphisms is denoted by Frm
(resp. coFrm), and the opposite category of Frm (resp. coFrm) is denoted by Loc (resp.
coLoc).

A Heyting algebra is a bounded lattice L equipped with a binary operation→: L×L→
L satisfying

c ≤ a→ b⇔ c ∧ a ≤ b

for all a, b, c ∈ L.
A coHeyting algebra [11] is a bounded lattice M equipped with a binary operation

←: M ×M →M satisfying
a← b ≤ c⇔ a ≤ b ∨ c

for all a, b, c ∈M .
Every complete Boolean algebra is both a Heyting and a coHeyting algebra. The

binary operations are defined by x→ y = x∗ ∨ y and x← y = x∧ y∗, where the exponent
∗ denotes the complement of an element. Both x → 0 and 1 ← x coincide with the
complement present in the Boolean algebra. Any (co)frame is a complete (co)Heyting
algebra, and vice versa, hence each frame (coframe) carries a (co)Heyting operation. The
(co)Heyting operation plays a crucial role in defining a sub(co)locale which is a subobject
of a (co)locale L in the category of (co)Loc.

Given a frame L, a subframe is a subset L′ ⊆ L that is closed under arbitrary join and
finite meets. Dually, a subcoframe is a subset M ′ ⊆ M which is closed under arbitrary
meet and finite joins.

According to [10], a sublocale is a subset S ⊆ L with the following conditions:

(S1) for all N ⊆ S,
∧
N ∈ S,

(S2) x→ s ∈ S for all s ∈ S and x ∈ L.

Similarly, we define a subcolocale of a colocale M as a subset S ⊆ M satisfying the
following conditions:



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 615

(cS1)
∨
N ∈ S for all N ⊆ S,

(cS2) s← x ∈ S for all s ∈ S and x ∈M .

Observe that S ⊆ M is a subcolocale if and only if S is a colocale with the induced
order and the embedding ic : S →M is a morphism of Loc.

The lattice of all sublocales of locale L and the lattice of all subcolocales of colocale
M are denoted by Sl(L) and Scl(M), respectively. Note that these two lattices are both
coframes and hence they satisfy de Morgan’s second law stating that (

∧
i∈I ai)

∗ =
∨
i∈I ai

∗

whenever
∧
i∈I ai exists. (Here ai

∗ denotes the pseudocomplement of ai). All joins and
meets of sublocales (resp. subcolocales) are taken in the lattice Sl(L) (resp. Scl(M)).

Let L be a locale. Then the elements o(a) = {a→ x : x ∈ L} and c(a) =↑ a of Sl(L) are
referred to as open and closed sublocales corresponding a ∈ L, respectively. Dually, given a
coframe M , we define the subcolocales oC(k) = {x← k : k ∈M} = {x ∈M : x← k = x}
and cC(k) =↓ k. The former is referred to as open subcolocale and the latter is referred
to as closed subcolocale corresponding k ∈ M . Unlike subspaces, not every sublocale is
complemented in the lattice of sublocales, however, o(a) and c(a) are complementary pairs
in Sl(L). Similarly, oC(k) is a complement of cC(k) in Scl(M).

There is another way of defining sublocales (resp. subcolocales) by using the notion of
nuclei (resp. conuclei). A nucleus on a frame L is a closure operator v : L→ L preserving
finite meets. For a sublocale S ⊆ L, vS(a) =

∧
{s ∈ S : a ≤ s} is a nucleus, and given a

nucleus v on L, Sv = v(L) is a sublocale. Further we have vSv = v and SvS = S.
A conucleus on a coframe M is a kernel operator t : M → M preserving finite joins.

The subcolocale generated by the conucleus t : M → M is St = t(M). On the other
hand, for a subcolocale S ⊆ M , the corresponding conuclei tS : M → M is defined by
tS(a) = ic∗(a) =

∨
{s ∈ S : s ≤ a}. Moreover, there is a one-one correspondence between

the subcolocales of M and the conuclei defined on M .

Proposition 2. Let M be a coframe. Then

(i) a ≤ b iff cC(a) ⊆ cC(b) iff oC(b) ⊆ oC(a).

(ii)
⋂
i∈I cC(ai) = cC(

∧
i∈I ai).

(iii) cC(a) ∨ cC(b) = cC(a ∨ b).

(iv)
∨
i∈I oC(ai) = oC(

∧
i∈I ai).

(v) oC(a) ∩ oC(b) = oC(a ∨ b.)

See [10, III 6.1.5] for the frame version of the proposition above.
Recall that a diframe is a triple L = (Le, Lfr, Lcf ) in which Le is both a frame and a

coframe, Lfr is a subframe and Lcf is a subcoframe of Le.
A diframe homomorphism is a triple (ϕ,ψ) with the following properties:

(i) ϕ : Le →Me is a frame homomorphism and ϕ[Lfr] ⊆Mfr,



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 616

(ii) ψ : Le →Me is a coframe homomorphism and ψ[Lcf ] ⊆Mcf .

The category of diframes and diframe homomorphisms is denoted by diFrm. The opposite
category of diFrm is called the category of dilocales and denoted by diLoc.

The following examples will be useful in the sequel.

Example 1. (i) Let us see the motivating example: Given a topological space X, de-
note by Ω(X) (resp. C(X)) the lattice of open (resp. closed) sets of X. Then
(P(X),Ω(X),C(X)) is a diframe. For a continuous map f : X → Y , the pair

(f−1, f−1) : (P(Y ),Ω(Y ),C(Y ))→ (P(X),Ω(X),C(X))

is trivially a diframe homomorphism.

(ii) Let Ωreg(R) be the complete Boolean algebra of regular open sets of R (with usual
topology). Let Le = Lcf = Ωreg(R) and Lfr = {(−∞, a) : a ∈ R} ∪ {∅,R}. Then the
triple L = (Le, Lfr, Lcf ) is a diframe.

(iii) Let Le = Ωreg(R), Lfr = {(−∞, a) : a ∈ R} ∪ {∅,R} and Lcf = {(a,∞) : a, b ∈
R} ∪ {∅,R}. Then L = (Le, Lfr, Lcf ) is a diframe.

(iv) If (S, S, τ, κ) is a ditopological space then (S, S, τ, κ) is a diframe.

Now recall the category dfDitop of ditopological texture spaces and bicontinuous
difunctions [3]. We have the following functor E : dfDitop→ diLoc

E((S1, S1, τ1, κ1)
(f,F )−−−→ (S2, S2, τ2, κ2) = (S1, τ1, κ1)

(ϕf ,ψF )
−−−−−→ (S2, τ2, κ2),

where the arrow on the right represents the diLoc morphism corresponding to the diFrm

morphism (S2, τ2, κ2)
(ϕF← ,ψf← )=((ψF )∗,(ϕf )∗)−−−−−−−−−−−−−−−−−→ (S1, τ1, κ1).

A Hutton dispace is a triple (L, τ, κ) where L is a complete, completely distributive
lattice and (τ, κ) is a ditopology. Consider the mappings ϕ : (L1, τ1, κ1) → (L2, τ2, κ2)
preserving arbitrary meets and joins and satisfying ϕ[τ1] ⊆ τ2, ϕ[κ1] ⊆ κ2. The resulting
category is denoted by diH.

By hdiFrm, we shall denote the category of diframes and diframe homomorphism
with ϕ = ψ. Obviously, diH is a full subcategory of hdiFrm, and hdiFrm is a non-full
subcategory of diFrm.

Note that, due to the lack of space, the separation axioms for ditopological texture
spaces is not repeated here. The reader is referred to [4] for a detailed discussion on this
subject.

3. Separation Axioms

In this section, we define the separation axioms on diframes. We also give several
characterizations of these axioms and discuss the relationship between them.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 617

Definition 1. A diframe L = (Le, Lfr, Lcf ) is said to be

(i) T0 if, for all a ∈ Le, there exists cji ∈ Lfr ∪ Lcf , i ∈ I, j ∈ J such that a =∨
j∈J

∧
i∈I c

j
i .

(ii) co-T0 if, for all a ∈ Le, there exists cji ∈ Lfr ∪ Lcf , i ∈ I, j ∈ J such that a =∧
j∈J

∨
i∈I c

j
i .

Note that the axiom T0 is not self-dual, and that T0 and co-T0 are equivalent if Le is
completely distributive.

Remark 1. (i) We say U ⊆ L generates V ⊆ L if V is the smallest subset of L con-
taining U and closed under arbitrary meet and join.

(ii) In a diframe L = (Le, Lfr, Lcf ), Le need not to be generated by Lfr ∪ Lcf . If
Le = P(X), Lfr = Lcf = {∅, X}, Le is not generated by Lfr ∪ Lcf . However,
this property holds for T0 or co-T0 diframes. Indeed, if L is T0, for all a ∈ Le,
a =

∨
j∈J

∧
i∈I c

j
i where cji ∈ Lfr ∪ Lcf . This means that a is an element of the set

generated by Lfr ∪Lcf . The other inclusion is an immediate consequence of the fact
that Le is closed under arbitrary meets and joins.

(iii) If (S, S, τ, κ) is T0 as a diframe, it is not necessarily T0 as a ditopological space.

Definition 2. A diframe L = (Le, Lfr, Lcf ) is said to be

(i) R0 if every element of Lfr can be written as a join of elements from Lcf .

(ii) co-R0 if every element of Lcf can be written as a meet of elements from Lfr.

(iii) T1 if T0 and R0.

(iv) co-T1 if co-T0 and co-R0.

For each property P, the diframe L = (Le, Lfr, Lcf ) is said to be bi-P if it is P and
co-P.

Note that, Kopperman was studied R0 in [8], under the name of “weak symmetry”.

Example 2. Consider the diframe L = (Le, Lfr, Lcf ) of Example 1 (ii). L is R0 since
(−∞, a) =

∨
n∈N(a − n, a) for all a ∈ R. However, L is not co-R0 because the bounded

intervals (a, b) ∈ Lcf can not be expressed as a meet of elements from Lfr.

Here are some statements equivalent to R0 and co-R0.

Proposition 3. Let L = (Le, Lfr, Lcf ) be a diframe.

(i) The following are equivalent:

(a) L is R0.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 618

(b) Every open sublocale associated with the elements of Lfr can be written as a
join of the open sublocales associated with the elements of Lcf , that is,

o(a) =
∨
{o(k) : k ∈ Lcf and k ≤ a} for all a ∈ Lfr.

(c) Every closed sublocale associated with the elements of Lfr can be written as an
intersection of the closed sublocales associated with the elements of Lcf , that is,

c(a) =
⋂
{c(k) : k ∈ Lcf and k ≤ a} for all a ∈ Lfr.

(d) ∀a ∈ Lfr, ∀x, y ∈ Le, a � y → x⇒ k ∈ Lcf ; k ≤ a, y � k → x.

(ii) The following are equivalent:

(a) L is co-R0.

(b) Every open subcolocale associated with the elements of Lcf can be written as a
join of the open subcolocales associated with the elements of Lfr, that is,

oC(k) =
∨
{oC(a) : a ∈ Lfr and k ≤ a} for all k ∈ Lcf .

(c) Every closed subcolocale associated with the elements of Lcf can be written as
an intersection of the closed subcolocales associated with the elements of Lfr,
that is,

cC(k) =
⋂
{cC(a) : a ∈ Lfr and k ≤ a} for all k ∈ Lcf .

(d) ∀k ∈ Lcf , ∀x, y ∈ Le, x← y � k ⇒ a ∈ Lfr; k ≤ a, x← a � y.

Proof. (ii): (a) and (b) are equivalent since the equality
∨
i∈I oC(a) = oC(

∧
i∈I ai)

holds. Similarly, (a) and (c) are equivalent by the property
⋂
i∈I cC(ai) = cC(

∧
i∈I ai).

For (b) implies (d), let x← y � k for k ∈ Lcf and x, y ∈ Le. Then,

oC(k) =
∨
{oC(a) : a ∈ Lfr and k ≤ a} 6⊆ oC(x← y)

and hence there exists an a ∈ Lfr such that k ≤ a and oC(a) 6⊆ oC(x← y), which implies
the existence of an a ∈ Lfr such that k ≤ a and x← a � y.

For the converse, assume contrary that L = (Le, Lfr, Lcf ) does not satisfy (b). Then
there is a k ∈ Lcf such that

oC(k) 6⊆
∨
{oC(a) : a ∈ Lfr and k ≤ a}.

Thus, there exists an x ∈ Le such that x ∈ oC(k) and x /∈ oC(a) for all a ∈ Lfr satisfying
k ≤ a. Now we obtain x ← k = x 6= x ← a, and hence x ← k � x ← a since the
converse inequality is always valid. Thereby, there exists a y ∈ Le such that x ← a ≤ y
and x← k � y. We now obtain x← y ≤ a and x← y � k for all a ∈ Lfr satisfying k ≤ a,
which contradicts with the assumption.

The proof of (i) is omitted since it can be proved in a similar way as above.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 619

Remark 2. The closure of an element a ∈ Le is given by [a] =
∧
{c ∈ Lcf : a ≤ c}, and

the interior by ]a[=
∨
{b ∈ Lfr : b ≤ a}.

Definition 3. A diframe is said to be

(i) R1 if, for all a ∈ Lfr,

a =
∨
j∈J

∧
i∈I

cji =
∨
j∈J

∧
i∈I

[cji ] where cji ∈ Lfr.

(ii) co-R1 if, for all k ∈ Lcf ,

k =
∧
j∈J

∨
i∈I

f ji =
∧
j∈J

∨
i∈I

]f ji [ where f ji ∈ Lcf .

(iii) T2 if R1 and T0

(iv) co-T2 if co-R1 and co-T0.

Note that, R1 was also studied in [8], under the name “pseudo Hausdorff”.

Proposition 4. Every R1 diframe is R0. Dually, every co-R1 diframe is co-R0.

Proof. Straightforward by definitions.

Remark 3. As is well known, a bitopological space (X,T,T∗) is regular if for all G ∈ T

and x ∈ G, there exist a T-open set H and a T∗-closed set F such that x ∈ H ⊆ F ⊆ G,
or equivalently, each G ∈ T can be expressed as follows:

G =
⋃
{H ∈ T : ∃F T∗-closed ; H ⊆ F ⊆ G}

Similarly, the dual space (X,T∗,T) is regular if, for all T∗-closed set F,

F =
⋂
{K T∗-closed : ∃G ∈ T ; F ⊆ G ⊆ K}.

Now define the relations ≺fr and ≺cf on P(X) by declaring that

H ≺fr G iff there exists an F ∈ C(X) such that H ⊆ F ⊆ G

and
F ≺cf K iff there exists a G ∈ Ω(X) such that F ⊆ G ⊆ K.

On the basis of the previous discussion, we introduce the following relations on Le :
We say that a is fr-below b, in symbols a ≺fr b, iff a, b ∈ Lfr and there exists a c ∈ Lcf

such that a ≤ c ≤ b.
Dually, we say that f is cf-below k, in symbols f ≺cf k, iff f, k ∈ Lcf and there exists

an a ∈ Lfr such that f ≤ a ≤ k.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 620

Proposition 5. In a diframe L, the relations ≺fr and ≺cf satisfy the following conditions:

(i) 0 ≺fr a ≺fr 1 for all a ∈ Lfr, and 0 ≺cf k ≺cf 1 for all k ∈ Lcf .

(ii) a ≺fr b implies a ≤ b, and f ≺cf k implies f ≤ k.

(iii) If a ≤ b ≺fr c ≤ d then a ≺fr d. If f ≤ c ≺cf d ≤ k then f ≺cf k.

(iv) For i = 1, 2 if ai ≺fr bi then a1 ∨ a2 ≺fr b1 ∨ b2 and a1 ∧ a2 ≺fr b1 ∧ b2. Moreover,
if fi ≺cf ki then f1 ∨ f2 ≺cf k1 ∨ k2 and f1 ∧ f2 ≺cf k1 ∧ k2.

Clearly, ≺fr and ≺cf are auxiliary relations in the sense of definition I.1.9 in [6].

Definition 4. A diframe is said to be

(i) regular if

a =
∨
{x ∈ Lfr : x ≺fr a} for all a ∈ Lfr.

(ii) co-regular if

c =
∧
{x ∈ Lcf : c ≺cf x} for all c ∈ Lcf .

(iii) T3 if regular and T0.

(iv) co-T3 if co-regular and co-T0.

The following proposition is immediate by definitions:

Proposition 6. (i) A diframe L is regular iff a =
∨
{x ∈ Lfr : [x] ≤ a} for all a ∈ Lfr.

(ii) A diframe L is co-regular iff c =
∧
{x ∈ Lcf : c ≤]x[} for all c ∈ Lcf .

Example 3. Let I = [0, 1] be the unit interval, Le = {[0, r], [0, r) : 0 ≤ r ≤ 1}, Lfr =
{[0, r) : 0 ≤ r ≤ 1} ∪ {I} and Lcf = {[0, r] : 0 ≤ r ≤ 1} ∪ {∅}. Trivially, for [0, r), [0, s) ∈
Lfr, [0, r) ≺fr [0, s) iff r < s.

For each U = [0, r) ∈ Lfr, U =
∨
{[0, r − 1

n) : [0, r − 1
n) ≺fr [0, r)}. Thus, L =

(Le, Lfr, Lcf ) is regular. Similarly, we can show the co-regularity of L.

The proof of the next proposition is quite standard and will therefore be omitted.

Proposition 7. If L = (Le, Lfr, Lcf ) is R0 (R1, regular) and L′cf is a coframe with
Lcf ⊆ L′cf then L′ = (Le, Lfr, L

′
cf ) is R0 (R1, regular). Dually, if L = (Le, Lfr, Lcf ) is

co-R0 (co-R1, co-regular) and L′fr is a frame with Lfr ⊆ L′fr then L′ = (Le, L
′
fr, Lcf ) is

co-R0 (co-R1, co-regular).

Proposition 8. (i) A regular diframe is R1.
(ii) A co-regular diframe is co-R1.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 621

Proof. (i) Given a ∈ Lfr we have a =
∨
i∈I{ci ∈ Lfr : ci ≺fr a} by regularity of L.

Further, if ci ≺fr a then there exists ki ∈ Lcf such that ci ≤ ki ≤ a. Setting J = {j} and

cji = ci, for all i ∈ I, we obtain

a =
∨
i∈I

∧
j∈J

cji ≤
∨
i∈I

∧
j∈J

[cji ] ≤
∨
i∈I

∧
j∈J

kji ≤ a

Thus a =
∨
i∈I

∧
j∈J c

j
i =

∨
i∈I

∧
j∈J [cji ], showing that L is R1.

Proposition 9. (i) Every regular co-R0 diframe is co-R1.
(ii) Every co -regular R0 diframe is R1.

Proof. (i) Let L be regular, co-R0 and let k ∈ Lcf . First we have ai ∈ Lfr such that
k =

∧
i∈I ai. Now, by regularity of L,

ai =
∨
j∈J
{bij ∈ Lfr : ∃fij ∈ Lcf ; bij ≤ fij ≤ ai}

for all i ∈ I. But then,

k ≤
∧
i∈I

∨
j∈J

bij ≤
∧
i∈I

∨
j∈J

]fij [≤
∧
i∈I

∨
j∈J

fij ≤
∧
i∈I

ai ≤ k

and hence k =
∧
i∈I

∨
j∈J fij =

∧
i∈I

∨
j∈J ]fij [. Therefore, L is co-R1.

(ii) Dual to (i), so we omit the details.

Note that complete regularity also has a counterpart in the theory of diframes. But
first we need the following binary relations on Le.

Remark 4. Let D = {k/2n : k, n ∈ N, k = 0, . . . 2n} be the set of dyadic rationals. We can
define a binary relation on Le by setting a ≺≺fr b iff a, b ∈ Lfr and there exists aq ∈ Lfr
(q ∈ D) satisfying

a0 = a, a1 = b, and aq ≺fr ar for q < r.

If a ≺≺fr b then we say that a is completely fr-below b.
Similarly, the dual relation can be defined by setting k ≺≺cf f iff k, f ∈ Lcf and there

exists kq ∈ Lcf (q ∈ D) satisfying

k0 = k, k1 = f, and kq ≺cf kr for q < r.

If k ≺≺cf f then we say that k is completely cf-below f.

The relations ≺≺fr and ≺≺cf have similar properties like those in Proposition 5.

Proposition 10. The relations ≺≺fr and ≺≺cf on Le satisfy the following properties:

(i) 0 ≺≺fr a ≺≺fr 1 for all a ∈ Lfr, and 0 ≺≺cf k ≺≺fr 1 for all k ∈ Lcf .



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 622

(ii) a ≺≺fr b implies a ≤ b. Moreover, f ≺≺cf k implies f ≤ k.

(iii) If a ≤ b ≺≺fr c ≤ d then a ≺≺fr d, and if f ≤ c ≺≺cf d ≤ k then f ≺≺cf k.

(iv) If ai ≺≺fr bi for i = 1, 2 then a1∨a2 ≺≺fr b1∨ b2 and a1∧a2 ≺≺fr b1∧ b2. Similarly,
if fi ≺≺cf ki for i = 1, 2 then f1 ∨ f2 ≺≺cf k1 ∨ k2 and f1 ∧ f2 ≺≺cf k1 ∧ k2.

(v) If a ≺≺fr b then there exists a c ∈ Lfr with a ≺≺fr c ≺≺fr b, that is, the relation
≺≺fr is interpolative. Moreover, it is the largest interpolative relation contained in
≺fr. Similarly, the relation ≺≺cf is interpolative and it is the largest interpolative
relation contained in ≺cf .

Proof. The facts (i)− (iv) are immediate consequences of the definitions.
(v) If a ≺≺fr b then we have aq ∈ Lfr (q ∈ D) with a0 = a, a1 = b and aq ≺fr ar for

q < r. Setting c = a1/2 we obtain a sequence of elements such that x0 = a, x1 = c and
xk/2n = ak/2n+1 . Clearly, xq ≺fr xr for q < r, and consequently a ≺≺fr c. Similarly, we
can find a sequence of elements such that y0 = c, y1 = b and yq ≺fr yr for q < r. Thus
c ≺≺fr b and hence the relation ≺≺fr is interpolative.

Further, ≺≺fr is obviously contained in ≺fr. For the remaining assertion, let ≺ be
any interpolative relation contained in ≺fr. If a ≺ b for a, b ∈ Le then, by induction, we
obtain a sequence of elements with a0 = a, a1 = b and aq ≺ ar for q < r. We also have
“aq ≺ ar ⇒ aq ≺fr ar by assumption. Thus, a ≺≺fr b.

Definition 5. A diframe is said to be

(i) completely regular if

a =
∨
{x ∈ Lfr : x ≺≺fr a} for all a ∈ Lfr.

(ii) completely co-regular if

c =
∧
{x ∈ Lcf : c ≺≺cf x} for all c ∈ Lcf .

(iii) T3 1
2

if completely regular and T0.

(iv) co-T3 1
2

if completely co-regular and co-T0.

As mentioned before, complete (co-) regularity is defined using bicontinuous difunctions
in ditopological spaces. Here, we leave the following questions as open problems:
(1) Can we construct a diframe corresponding to the ditopological unit interval texture
space (I, J, τI, κI) ?
(2) How do we characterize complete regularity by using diframe homomorphisms ?
(3) What is the relation between these two characterizations of completely regularity ?

Proposition 11. (i) A completely regular diframe is regular.
(ii) A completely co-regular diframe is co-regular.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 623

Proof. This is an immediate consequence of the following facts: a ≺≺fr b implies
a ≺fr b, and a ≺≺cf b implies a ≺cf b.

There is another way of characterizing complete regularity of a bitopological space in
terms of a Urysohn relation due to Kopperman [8]. Now we will generalize this idea to
diframes.

We start by recalling the definition of a Urysohn relation [7]. A binary relation C
on a partially ordered set (L,≤) is called a Urysohn relation if it satisfies the following
conditions:

(U1) aC b implies a ≤ b for all a, b ∈ L,

(U2) a ≤ bC c ≤ d implies aC d for all a, b, c, d ∈ L,

(U3) aC b implies the existence of c ∈ L such that aC cC b for all a, b ∈ L (that is, C is
an interpolative relation).

If L is a lattice and C is a Urysohn relation on L, we call the pair (L,C) a Urysohn lattice.
The following are some basic examples of Urysohn relations.

Example 4. (i) Let X be a normal space. For U, V ∈ Ω(X), define a relation C by
setting U C V iff U ⊆ V . Then C is a Urysohn relation.

(ii) The relations ≺fr and ≺cf are not Urysohn since the interpolation property does not
hold. However, ≺≺fr and ≺≺cf are obviously Urysohn relations by Proposition 10.

Proposition 12. Let L = (Le, Lfr, Lcf ) be a diframe.

(i) L is completely regular if and only if there exists a Urysohn relation C on Le satis-
fying the following conditions:

(a) aC b implies [a] ≤]b[,

(b) for every a ∈ Lfr, a =
∨
{x ∈ Lfr : xC a}.

(ii) L is completely co-regular if and only if there exists a Urysohn relation C on Le
satisfying the following conditions:

(a) aC b implies [a] ≤]b[,

(b) for every c ∈ Lcf , c =
∧
{x ∈ Lcf : cC x}.

Proof. Here, we just prove (i), since (ii) can be proven similarly. If L is a completely
regular diframe then ≺≺fr is the desired relation. Indeed, as can be easily checked, it is a
Urysohn relation. Further, the condition (b) is a direct result of the definition. Now let
a ≺≺fr b. Then applying the definitions of ≺≺fr and ≺fr, respectively, we obtain aq ∈ Lfr
and cq ∈ Lcf (q ∈ D) such that

a ≤ . . . aq ≤ cq ≤ ar ≤ . . . ≤ b



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 624

and hence
[a] ≤ . . . ≤ [aq] ≤ [cq] = cq ≤ ar ≤ . . . ≤ b =]b[.

where q < r. Thus, the relation ≺≺fr satisfies (a).
Conversely, suppose that we have a Urysohn relation C on Le satisfying the conditions

(a) and (b). Let xC a for x, a ∈ Lfr. By (U3), there exists yq ∈ Le (q ∈ D) such that

xC . . . yq C yr . . .C a

where q < r. Since ]yq[≤ [yq] ≤]yr[ by (a), we have

x ≺fr . . . ≺fr]yq[≺fr]yr[≺fr . . . ≺fr a.

We now obtain xC a implies x ≺≺fr a. Therefore, for all a ∈ Lfr,

a =
∨
{x ∈ Lfr : xC a} ≤

∨
{x ∈ Lfr : x ≺≺fr a} ≤ a

and hence L = (Le, Lfr, Lcf ) is completely regular.

As is well known, normality is a separation axiom that can be defined purely in terms
of the open and closed sets. In other words, its definition is not based on points, which
makes it easier to discuss them in the point-free context.

Definition 6. A diframe is said to be

(i) normal if, for any c ∈ Lcf and a ∈ Lfr such that c ≤ a, there exists a b ∈ Lfr such
that c ≤ b ≤ [b] ≤ a.

(ii) T4 if normal and T1.

(iii) co-T4 if normal and co-T1.

Remark 5. Normality is self-dual. Hence we can use the equivalent definition:
“for any c ∈ Lcf and a ∈ Lfr such that c ≤ a there exists a k ∈ Lcf such that
c ≤]k[≤ k ≤ a.” This is easily obtained by setting k = [b] in the definition of normality.

Proposition 13. Let C be a binary relation on Le such that “a C b iff [a] ≤]b[”. Then
L = (Le, Lfr, Lcf ) is normal if and only if C is a Urysohn relation on Le.

Proof. Suppose L is a normal diframe. Then we claim that the relation C given in the
proposition satisfies the properties (U1)− (U3). We only prove (U3) since (U1) and (U2)
are straightforward.

Let a C b. Then [a] ≤]b[ and hence, by normality, there exists a c ∈ Lfr such that
[a] ≤]c[= c ≤ [c] ≤]b[. Thus we have aC cC b.

For the converse, let c ≤ a for any c ∈ Lcf and a ∈ Lfr. Then c C a and hence, by
(U3), there exists a b ∈ Le such that cC bC a. Now we have c ≤ [c] ≤]b[≤ b ≤ [b] ≤]a[≤ a.
Setting d =]b[ we obtain c ≤ d ≤ [d] ≤ a. Thus L is a normal diframe.



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 625

Example 5. Normality does not imply regularity. Consider the diframe L of Example 1
(iii). L is normal: Let C ∈ Lcf , A ∈ Lfr with C ⊆ A. Then there are three cases to
consider: (i) C = A = ∅, (ii) C = A = R, (iii) C 6= R, A = R. We may take B = ∅ in
case (i), and B = R in cases (ii) and (iii), showing L is regular. However, L is obviously
not normal.

Proposition 14. (i) Every normal R0 diframe is regular.
(ii) Every normal co-R0 diframe is co-regular.

Proof. (i) Let a ∈ Lfr and set c =
∨
{b ∈ Lfr : b ≺fr a}. Clearly, c ≤ a. On the

other hand, o(a) =
∨
{o(k) : k ∈ Lcf and k ≤ a} since L is R0. Hence, to prove a ≤ c, it is

enough to show that o(a) ⊆ o(c), that is, o(k) ⊆ o(c) for all k ∈ Lcf with k ≤ a. So take
an element k ∈ Lcf such that k ≤ a. Then, by normality, there exists a b ∈ Lfr such that
k ≤ b ≤ [b] ≤ a, yielding b ≺fr a and k ≤ b. Thus k ≤ b ≤ c, and hence o(k) ⊆ o(c), as
required.

Proposition 15. (i) A normal R0 diframe is completely regular.
(ii) A normal co-R0 diframe is completely co-regular.

Proof. We will just prove the first statement and leave the other statement to the
reader. Since each normal R0 diframe is regular it is enough to show that the relations
≺fr and ≺≺fr coincide in a normal diframe. For this, we have to prove that ≺fr is
interpolative. If a ≺fr b then there exists a k ∈ Lcf such that a ≤ k ≤ b. Moreover, by
normality, there is a d ∈ Lfr such that a ≤ k ≤ d ≤ [d] ≤ b . Thus, a ≺fr d ≺fr b, which
means that ≺fr is interpolative. Thus we have, by Proposition 10 (v), ≺fr=≺≺fr .

Corollary 1. We have the following implications in a diframe:

normal and R0 ⇒ completely regular⇒ regular⇒ R0.

normal and co-R0 ⇒ completely co-regular⇒ co-regular⇒ co-R0.

(co-)T4 ⇒ (co-)T3 1
2
⇒ (co-)T3 ⇒ (co-)T2 ⇒ (co-)T1 ⇒ (co-)T0.

We end this section by investigating the image of a diframe with a property P under
a special kind of homomorphism.

Definition 7. A diframe homomorphism (ϕ,ψ) : L→M is called

(i) open (respectively, co-open) if ψ∗(a) ∈ Lfr (resp. ϕ∗(a) ∈ Lfr) for all a ∈Mfr.

(ii) closed (respectively, co-closed) if ψ∗(k) ∈ Lcf (resp. ϕ∗(k) ∈ Lcf ) for all k ∈Mcf .

Proposition 16. Let L and M be diframes and let (ϕ,ψ) : L → M be a one-one onto
diframe homomorphism.

(i) If (ϕ,ψ) is open (resp. co-open) then, for all b ∈Mfr, there exists an a ∈ Lfr such
that ψ(a) = b (resp. ϕ(a) = b).



E. Korkmaz, R. Ertürk / Eur. J. Pure Appl. Math, 11 (3) (2018), 612-627 626

(ii) If (ϕ,ψ) is closed (resp. co-closed) then, for all k ∈ Mcf , there exists an f ∈ Lcf
such that ψ(f) = k (resp. ϕ(f) = k).

Proof. Suppose (ϕ,ψ) : Le → Me is open and b ∈ Mfr. Since ψ is onto, there is an
a ∈ Le with ψ(a) = b. Now, by Proposition 1, we have ψ∗ψ(a) = a = ψ∗(b), and hence
a = ψ∗(b) ∈ Lfr by openness of (ϕ,ψ).

The other cases can be proved similarly.

Remark 6. If ϕ is one-one and onto then, by Proposition 1 (iii), ϕ∗ϕ = 1Le and ϕϕ∗ =
1Me, that is, ϕ−1 = ϕ∗. Similarly, if ψ is one-one and onto then ψ−1 = ψ∗. Thus,
if (ϕ,ϕ) = ϕ : L → M is a one-one onto hdiFrm homomorphism then ϕ∗ = ϕ∗, and
hence the concept of openness (resp., closedness) coincides with co-openness (resp. co-
closedness).

Definition 8. If L and M are diframes, a hdiFrm homomorphism (ϕ,ϕ) = ϕ : L→ M
is called an isomorphism if it is one-one, onto, open and closed,.

Proposition 17. Let L, M be diframes and ϕ : L → M be a hdiFrm isomorphism.
Then, L is bi-R0 (respectively, bi-R1, bi-regular, completely bi-regular, normal) if and only
if M is bi-R0 (respectively, bi-R1, bi-regular, completely bi-regular, normal).

Proof. We will just prove the regularity and the other axioms are left to the interested
reader. Let L be regular and b ∈Mfr. Then, by Proposition 16, there is an a ∈ Lfr such
that ϕ(a) = b and, by regularity of L, a =

∨
{x ∈ Lfr : x ≺fr a}. Moreover, x ≺fr a

implies ϕ(x) ≺fr b by definition of ≺fr. Now we have

b = ϕ(a) = ϕ(
∨
{x ∈ Lfr : x ≺fr a}) ≤

∨
{ϕ(x) ∈Mfr : ϕ(x) ≺fr b)} ≤ b

and hence M is regular.
Conversely, suppose that M is regular and a ∈ Lfr. Then ϕ(a) ∈ Mfr and hence, by

regularity, ϕ(a) =
∨
{x ∈ Mfr : x ≺fr ϕ(a)}. Now if x ≺fr ϕ(a) then, by Proposition 1

together with the closedness of ϕ, we have ϕ∗(x) ≺fr a. But then

a = ϕ∗ϕ(a) = ϕ∗(
∨
{x ∈Mfr : x ≺fr ϕ(a)}) ≤

∨
{ϕ∗(x) ∈ Lfr : ϕ∗(x) ≺fr a} ≤ a

and hence L is regular.

4. Conclusion

In this paper we have studied the separation axioms in diframes and examined the
relations between them. We have defined new binary relations on a diframe and obtained
a characterization of regularity and complete regularity by using these relations. As a
future work, other topological and bitopological structures such as compactness, stability,
join compactness and connectedness, etc. can be constructed on diframes.



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Acknowledgements

The authors thank the referees for valuable comments and suggestions that improved
the quality of this manuscript.

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