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@ Appl. Gen. Topol. 17, no. 1(2016), 17-35doi:10.4995/agt.2016.4085
c© AGT, UPV, 2016

Convergence in graded ditopological

texture spaces

Dedicated to the memory of Lawrence Michael Brown

Ramazan Ekmekçİa and Rıza Ertürkb

a
Çanakkale Onsekiz Mart University, Faculty of Arts and Sciences, Department of Mathematics,

17100 Çanakkale, Turkey. (ekmekci@comu.edu.tr)
b
Hacettepe University, Faculty of Science, Department of Mathematics, 06800 Beytepe, Ankara,

Turkey. (rerturk@hacettepe.edu.tr)

Abstract

Graded ditopological texture spaces have been presented and discussed

in categorical aspects by Lawrence M. Brown and Alexander Šostak in

[6]. In this paper, the authors generalize the structure of difilters in
ditopological texture spaces defined in [11] to the graded ditopologi-
cal texture spaces and compare the properties of difilters and graded

difilters.

2010 MSC: 54A05; 54A20; 06D10.

Keywords: texture; graded ditopology; graded difilter; fuzzy topology.

1. Introduction

The concept of fuzzy topological space was defined in 1968 by C.Chang as
an ordinary subset of the family of all fuzzy subsets of a given set[7]. As a
more suitable approach to the idea of fuzzyness, in 1985, Šostak and Kubiak
independently redefined fuzzy topology where a fuzzy subset has a degree of
openness rather than being open or not [12, 10] (for historical developments
and basic ideas of the theory of fuzzy topology see [13]).

In classical topology the notion of open set is usually taken as primitive with
that of closed set being auxiliary. However, since closed sets are easily obtained

Received 19 August 2015 – Accepted 09 January 2016

http://dx.doi.org/10.4995/agt.2016.4085


R. Ekmekçi and R. Ertürk

as the complements of open sets they often play an important, sometimes dom-
inating role in topological arguments. A similar situation holds for topologies
on lattices where the role of set complement is played by an order reversing
involution. It is the case, however, that there may be no order reversing in-
volution available, or that the presence of such an involution is irrelevant to
the topic under consideration. To deal with such cases it is natural to consider
a topological structure consisting of a priori unrelated families of open sets
and of closed sets. This was the approach adapted from the beginning for the
topological structures on textures, originally introduced as a point-based repre-
sentation for fuzzy sets [1, 2]. Such topological structures were given the name
of a dichotomous topology, or ditopology for short. They consist of a family τ
of open sets and a generally unrelated family κ of closed sets. Hence, both the
open and the closed sets are regarded as primitive concepts for a ditopology.

A ditopology (τ, κ) on the discrete texture (X, P(X)) gives rise to a bitopo-
logical space (X, τ, κc). This link with bitopological spaces has had a powerful
influence on the development of the theory of ditopological texture spaces, but
it should be emphasized that a ditopology and a bitopology are conceptually
different. Indeed, a bitopology consists of two separate topological structures
(complete with their open and closed sets) whose interrelations we wish to
study, whereas a ditopology represents a single topological structure.

Ditopological texture spaces were introduced by L. M. Brown as a natural
extention of the work of the second author on the representation of lattice-
valued topologies by bitopologies in [9]. The concept of ditopology is more
general than general topology, bitopology and fuzzy topology in Chang’s sense.
An adequate introduction to the theory of texture spaces and ditopological
texture spaces may be obtained from [1, 2, 3, 4, 5].

Recently, L. M. Brown and A. Šostak have presented the concept ”graded
ditopology” on textures as an extention of the concept of ditopology to the
case where openness and closedness are given in terms of a priori unrelated
grading functions [6]. The concept of graded ditopology is more general than
ditopology and smooth topology. Two sorts of neighbohood structure on graded
ditopological texture spaces are presented and investigated by the authors in
[8].

The aim of this work is to generalize the structure of difilters in ditopological
texture spaces defined by S. Özçağ, F. Yıldız and L. M. Brown in [11] to the
graded ditopological texture spaces which is introduced by L. M. Brown and
A. Šostak in [6]. Furthermore we compare the properties of difilters and graded
difilters. The material in this work forms a part of the first named author’s
Ph.D. Thesis, currently being written under the supervision of the second name
author Dr. Rıza Ertürk.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 18



Convergence in graded ditopological texture spaces

2. Preliminaries

We recall various concepts and properties from [3, 4, 5] under the following
subtitle.
Ditopological Texture Spaces: Let S be a set. A texturing S on S is a
subset of P(S) which is a point separating, complete, completely distributive
lattice with respect to inclusion which contains S, ∅ and for which meet

∧

coincides with intersection
⋂

and finite joins
∨

with unions
⋃

. The pair (S, S)
is then called a texture or a texture space.

In general, a texturing of S need not be closed under set complementation,
but there may exist a mapping σ : S → S satisfying σ(σ(A)) = A and A ⊆
B ⇒ σ(B) ⊆ σ(A) for all A, B ∈ S. In this case σ is called a complementation
on (S, S) and (S, S, σ) is said to be a complemented texture.

For a texture (S, S), most properties are conveniently defined in terms of
the p − sets

Ps =
⋂

{A ∈ S | s ∈ A}

and the q − sets

Qs =
∨

{A ∈ S | s 6∈ A} =
∨

{Pu | u ∈ S, s 6∈ Pu}.

A texture (S, S) is called a plain texture if it satisfies any of the following
equivalent conditions:

(1) Ps * Qs for all s ∈ S
(2) A =

∨

i∈I Ai =
⋃

i∈I Ai for all Ai ∈ S, i ∈ I

Recall that M ∈ S is called a molecule in S if M 6= ∅ and M ⊆ A∪B, A, B ∈ S
implies M ⊆ A or M ⊆ B. The sets Ps, s ∈ S are molecules, and the texture
(S, S) is called ”simple” if all molecules of S are in the form {Ps | s ∈ S}. For
a set A ∈ S, the core of A (denoted by A♭) is defined by

A
♭ =

⋂

{

⋃

{Ai | i ∈ I} |{Ai | i ∈ I} ⊆ S, A =
∨

{Ai | i ∈ I}
}

.

Theorem 2.1 ([3]). In any texture space (S, S), the following statements hold:

(1) s 6∈ A ⇒ A ⊆ Qs ⇒ s 6∈ A
♭ for all s ∈ S, A ∈ S.

(2) A♭ = {s | A * Qs} for all A ∈ S.
(3) For Aj ∈ S, j ∈ J we have (

∨

j∈J Aj)
♭ =

⋃

j∈J A
♭
j.

(4) A is the smallest element of S containing A♭ for all A ∈ S.
(5) For A, B ∈ S, if A * B then there exists s ∈ S with A * Qs and

Ps * B.
(6) A =

⋂

{Qs | Ps * A} for all A ∈ S.
(7) A =

∨

{Ps | A * Qs} for all A ∈ S.

Example 2.2. (1) If P(X) is the powerset of a set X, then (X, P(X)) is the
discrete texture on X. For x ∈ X, Px = {x} and Qx = X \ {x}. The mapping
πX : P(X) → P(X), πX(Y ) = X \ Y for Y ⊆ X is a complementation on the
texture (X, P(X)).
(2) Setting I = [0, 1], J = {[0, r), [0, r] |r ∈ I} gives the unit interval texture

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 19



R. Ekmekçi and R. Ertürk

(I, J ). For r ∈ I, Pr = [0, r] and Qr = [0, r). And the mapping ι : J → J ,
ι[0, r] = [0, 1 − r), ι[0, r) = [0, 1 − r] is a complementation on this texture.
(3) The texture (L, L, λ) is defined by L = (0, 1], L = {(0, r] | r ∈ [0, 1]},
λ((0, r]) = (0, 1 − r]. For r ∈ L, Pr = (0, r] = Qr.
(4) S = {∅, {a, b}, {b}, {b, c}, S} is a simple texturing of S = {a, b, c}. Pa =
{a, b}, Pb = {b}, Pc = {b, c}. It is not possible to define a complementation on
(S, S).
(5) If (S, S), (V, V) are textures, the product texturing S ⊗ V of S × V consists
of arbitrary intersections of sets of the form (A × V ) ∪ (S × B), A ∈ S, B ∈ V,
and (S × V, S ⊗ V) is called the product of (S, S) and (V, V). For s ∈ S, v ∈ V ,
P(s,v) = Ps × Pv and Q(s,v) = (Qs × V ) ∪ (S × Qv).

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 ⇒

∨

i
Gi ∈ τ

and the set of closed sets κ satisfies

(CT1) S, ∅ ∈ κ
(CT2) K1, K2 ∈ κ ⇒ K1 ∪ K2 ∈ κ
(CT3) Ki ∈ κ, i ∈ I ⇒

⋂

i
Ki ∈ κ.

Hence a ditopology is essentially a ”topology” for which there is no priori
relation between the open and closed sets.

Let (τ, κ) be a ditopology on (S, S).

(1) If s ∈ S♭, a neighborhood of s is a set N ∈ S for which there exists
G ∈ τ satisfying Ps ⊆ G ⊆ N * Qs.

(2) If s ∈ S, a coneighborhood of s is a set M ∈ S for which there exists
K ∈ κ satisfying Ps * M ⊆ K ⊆ Qs.

If the set of nhds (conhds) of s is denoted by η(s) (µ(s)) respectively, then
(η, µ) is called the dinhd system of (τ, κ).

Theorem 2.3 ([5]). For a ditopology (τ, κ) on (S, S) let the families η(s),
s ∈ S♭ and µ(s), s ∈ S be defined as above.
(1) For each s ∈ S♭ we have η(s) 6= ∅ and these families satisfy the following
conditions:

(i) N ∈ η(s) ⇒ N * Qs
(ii) N ∈ η(s), N ⊆ N′ ∈ S ⇒ N′ ∈ η(s)
(iii) N1, N2 ∈ η(s), N1 ∩ N2 * Qs ⇒ N1 ∩ N2 ∈ η(s)
(iv) (a) N ∈ η(s) ⇒ ∃N⋆ ∈ S, Ps ⊆ N

⋆ ⊆ N, so that N⋆ * Qt ⇒ N⋆ ∈
η(t), ∀t ∈ S♭

(b) For N ∈ S and N * Qs, if there exists N⋆ ∈ S, Ps ⊆ N⋆ ⊆ N,
which satisfies N⋆ * Qt ⇒ N⋆ ∈ η(t), ∀t ∈ S♭, then N ∈ η(s).

Moreover, the sets G in τ are characterized by the condition that G ∈ η(s) for
all s with G * Qs.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 20



Convergence in graded ditopological texture spaces

(2) For each s ∈ S we have µ(s) 6= ∅ and these families satisfy the following
conditions:

(i) M ∈ µ(s) ⇒ Ps * M
(ii) M ∈ µ(s), M ⊇ M′ ∈ S ⇒ M′ ∈ µ(s)
(iii) M1, M2 ∈ µ(s) ⇒ M1 ∪ M2 ∈ µ(s)
(iv) (a) M ∈ µ(s) ⇒ ∃M⋆ ∈ S, M ⊆ M⋆ ⊆ Qs, so that Pt * M⋆ ⇒ M⋆ ∈

µ(t), ∀t ∈ S
(b) For M ∈ S and Ps * M, if there exists M⋆ ∈ S, M ⊆ M⋆ ⊆ Qs,

which satisfies Pt * M⋆ ⇒ M⋆ ∈ µ(t), ∀t ∈ S, then M ∈ µ(s).

Moreover, the sets K in κ are characterized by the condition that K ∈ µ(s) for
all s with Ps * K.

Conversely, if η(s), s ∈ S♭ and µ(s), s ∈ S are non-empty families of sets in
S which satisfy conditions (1) and (2) above, respectively, then there exists a
ditopology (τ, κ) on (S, S) for which η(s) (µ(s)) are the families of nhds (resp.,
conhds) of the ditopology (τ, κ).

Difilters on Textures: [11] Let (S, S) be a texture.

(1) F ⊆ S is called a S-filter or a filter on (S, S), if F 6= ∅ and satisfies:
(F1) ∅ 6∈ F
(F2) F ∈ F, F ⊆ F

′ ∈ S ⇒ F ′ ∈ F, and
(F3) F1, F2 ∈ F ⇒ F1 ∩ F2 ∈ F.

(2) G ⊆ S is called a S-cofilter or a cofilter on (S, S), if G 6= ∅ and satisfies:
(CF1) S 6∈ G
(CF2) G ∈ G, G ⊇ G

′ ∈ S ⇒ G′ ∈ G, and
(CF3) G1, G2 ∈ G ⇒ G1 ∪ G2 ∈ G.

(3) If F is a S-filter and G is a S-cofilter then F × G is called a S-difilter
or a difilter on (S, S).

A difilter F × G on (S, S) is called regular if it satisfies following equivalent
conditions:

(1) F ∩ G = ∅.
(2) (Fi, Gi) ∈ F × G, i = 1, 2, ..., n ⇒

⋂n
i=1 Fi *

⋃n
i=1 Gi.

(3) A * B for all A ∈ F and B ∈ G.

Example 2.4. (1) For a plain texture (S, S) and ditopology (τ, κ) on (S, S),
then η(s) × µ(s) is a regular S-difilter for all s ∈ S♭ = S on (S, S).

(2) Let (S, S, τ, κ) be a ditopological texture space. Then the families

η∗(s) = {A ∈ S | ∃Gk ∈ τ : Gk * Qs, 1 ≤ k ≤ n and G1 ∩ ... ∩ Gn ⊆ A}, s ∈ S
♭

µ∗(s) = {A ∈ S | ∃Fk ∈ κ : Ps * Fk, 1 ≤ k ≤ n and A ⊆ F1 ∪ ... ∪ Fn}, s ∈ S

form a regular difilter η∗(s) × µ∗(s) on (S, S).

Definition 2.5 ([11]). Let (S, S, τ, κ) be a ditopological texture space, F a
S-filter and G a S-cofilter. (1) We say F converges to a point s ∈ S♭, and
write F → s if η∗(s) ⊆ F; G converges to a point s ∈ S, and write G → s if

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 21



R. Ekmekçi and R. Ertürk

µ∗(s) ⊆ G. The difilter F × G is said to be diconvergent if F → s and G → s′

for some s, s′ ∈ S satisfying Ps′ * Qs.
(2) F is called prime if A1, A2 ∈ S, A1 ∪ A2 ∈ F ⇒ A1 ∈ F or A2 ∈ F. G is

called prime if B1, B2 ∈ S, B1 ∩ B2 ∈ G ⇒ B1 ∈ G or B2 ∈ G.

Proposition 2.6 ([11]). The following are equivalent for a regular difilter F×G
on (S, S).

(1) F × G is maximal.
(2) F ∪ G = S.
(3) F is a prime S-filter and G = S \ F.
(4) G is a prime S-cofilter and F = S \ G.

It is obtained in [11] that there exist one to one correspondences among the
set of maximal regular difilters on (S, S), the set of prime filters on (S, S) and
the set of prime cofilters on (S, S).
Graded Ditopological Texture Spaces: [6] Let (S, S), (V, V) be textures
and consider T , K : S → V satisfying

(GT1) T (S) = T (∅) = V
(GT2) T (A1) ∩ T (A2) ⊆ T (A1 ∩ A2) ∀A1, A2 ∈ S
(GT3)

⋂

j∈J T (Aj) ⊆ T (
∨

j∈J Aj) ∀Aj ∈ S, j ∈ J

and

(GCT1) K(S) = K(∅) = V
(GCT2) K(A1) ∩ K(A2) ⊆ K(A1 ∪ A2) ∀A1, A2 ∈ S
(GCT3)

⋂

j∈J K(Aj) ⊆ K(
⋂

j∈J Aj) ∀Aj ∈ S, j ∈ J.

Then T is called a (V, V)-graded topology, K a (V, V)-graded cotopology and
(T , K) a (V, V)-graded ditopology on (S, S). The tuple (S, S, T , K, V, V) is
called a graded ditopological texture space. For v ∈ V we define

T v = {A ∈ S | Pv ⊆ T (A)}, K
v = {A ∈ S | Pv ⊆ K(A)}.

Then (T v, Kv) is a ditopology on (S, S) for each v ∈ V . That is, if (S, S, T , K, V, V)
is any graded ditopological texture space, then there exists a ditopology (T v, Kv)
on the texture space (S, S) for each v ∈ V

If (S, S, σ) is a complemented texture and (T , K) is a (V, V)-graded ditopol-
ogy on (S, S), then (K ◦ σ, T ◦ σ) is also a (V, V)-graded ditopology on (S, S).
(T , K) is called complemented if (T , K) = (K ◦ σ, T ◦ σ).

Example 2.7. Let (S, S, τ, κ) be a ditopological texture space and (V, V) the
discrete texture on a singleton. Take (V, V) = (1, P(1)) (The notation 1 denotes
the set {0}) and define τg : S → P(1) by τg(A) = 1 ⇔ A ∈ τ. Then τg is a
(V, V)-graded topology on (S, S). Likewise, κg defined by κg(A) = 1 ⇔ A ∈ κ
is a (V, V)-graded cotopology on (S, S). Then (τg, κg) is called the graded
ditopology on (S, S) corresponding to ditopology (τ, κ).

Therefore graded ditopological texture spaces are more general than ditopo-
logical texture spaces.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 22



Convergence in graded ditopological texture spaces

Graded Dineighborhood Systems: [8] Let (S, S) and (V, V) be two texture
spaces. For any mapping K : S → V, we use the notation vK to denote the
family {A ∈ S : K(A) * Qv} for all v ∈ V , and thus for each v ∈ V , we have
vK ⊆ Kv.

Definition 2.8 ([8]). Let (T , K) be a (V, V)-graded ditopology on texture
(S, S) and N : S♭ → VS, M : S → VS mappings where N(s) = Ns : S → V for
each s ∈ S♭ and M(s) = Ms : S → V for each s ∈ S. Then the mapping Ns is
called a ”graded neighborhood system of s” if

(2.1) vNs = {A ∈ S : T (B) * Qv and Ps ⊆ B ⊆ A * Qs for some B ∈ S}

for each v ∈ V ♭. The mapping Ms is called a ”graded coneighborhood system
of s” if

(2.2) vMs = {A ∈ S : K(B) * Qv and Ps * A ⊆ B ⊆ Qs for some B ∈ S}

for each v ∈ V ♭. The mapping N (M) is called a ”graded neighborhood
system” (”graded coneighborhood system”) of graded ditopological texture
space (S, S, T , K, V, V) if Ns (Ms) is a graded neighborhood system for each
s ∈ S♭ (graded coneighborhood system for each s ∈ S) respectively. (N, M) is
called a ”graded dineighborhood system” of graded ditopological texture space
(S, S, T , K, V, V) if N is a graded neighborhood system and M is a graded
coneighborhood system of graded ditopological texture space (S, S, T , K, V, V).

Proposition 2.9 ([8]). For the above notations, (N, M) is a graded dinhd
system of a graded ditopological texture space (S, S, T , K, V, V) iff

(2.3) Ns(A) =

{

sup{T (B) : Ps ⊆ B ⊆ A * Qs, B ∈ S}, A * Qs
∅, A ⊆ Qs

for each s ∈ S♭, A ∈ S and

(2.4) Ms(A) =

{

sup{K(B) : Ps * A ⊆ B ⊆ Qs, B ∈ S}, Ps * A
∅, Ps ⊆ A

for each s ∈ S, A ∈ S.

Theorem 2.10 ([8]). Let (T , K) be a (V, V)-graded ditopology on a texture
(S, S). If (N, M) is the graded dinhd system of the graded ditopological texture
space (S, S, T , K, V, V), then the following properties hold for all A, A1, A2 ∈ S:
(1) For each s ∈ S♭;

(N1) Ns(A) 6= ∅ ⇒ A * Qs
(N2) Ns(∅) = ∅ and Ns(S) = V
(N3) A1 ⊆ A2 ⇒ Ns(A1) ⊆ Ns(A2)
(N4) A1 ∩ A2 * Qs ⇒ Ns(A1) ∧ Ns(A2) ⊆ Ns(A1 ∩ A2)
(N5) Ns(A) ⊆ sup{

∧

s′∈B♭ Ns′(B) : Ps ⊆ B ⊆ A * Qs, B ∈ S}

(2) For each s ∈ S;
(M1) Ms(A) 6= ∅ ⇒ Ps * A
(M2) Ms(S) = ∅ and Ms(∅) = V

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 23



R. Ekmekçi and R. Ertürk

(M3) A1 ⊆ A2 ⇒ Ms(A2) ⊆ Ms(A1)
(M4) Ms(A1) ∧ Ms(A2) ⊆ Ms(A1 ∪ A2)
(M5) Ms(A) ⊆ sup{

∧

s′∈(S\B) Ms′(B) : Ps * A ⊆ B ⊆ Qs, B ∈ S}

3. Graded Difilters and Convergence

Definition 3.1. Let (S, S) and (V, V) be textures.

(1) A mapping F : S → V is called a (V, V)-graded filter on (S, S) if F
satisfies:

(GF1) F(∅) = ∅
(GF2) A1 ⊆ A2 ⇒ F(A1) ⊆ F(A2)
(GF3) F(A1) ∧ F(A2) ⊆ F(A1 ∩ A2)

(2) A mapping G : S → V is called a (V, V)-graded cofilter on (S, S) if G
satisfies:

(GCF1) G(S) = ∅
(GCF2) A1 ⊆ A2 ⇒ G(A2) ⊆ G(A1)
(GCF3) G(A1) ∧ G(A2) ⊆ G(A1 ∪ A2)
(3) If F is a (V, V)-graded filter and G (V, V)-graded cofilter on (S, S) then

the pair (F, G) is called a (V, V)-graded difilter on (S, S).

Proposition 3.2. The following are equivalent for a (V, V)-graded difilter
(F, G) on (S, S).

(1) F ∧ G = ∅ i.e. F(A) ∧ G(A) = ∅ for all A ∈ S.
(2) ∀n ∈ N,

∧n
i=1(F(Ai) ∧ G(Bi)) 6= ∅ ⇒

⋂n
i=1 Ai *

⋃n
i=1 Bi, for all

Ai, Bi ∈ S
(3) F(A) ∧ G(B) 6= ∅ ⇒ A * B, for all A, B ∈ S

Proof. (1) ⇒ (2) : Let n ∈ N,
∧n

i=1(F(Ai) ∧ G(Bi)) 6= ∅ for all Ai, Bi ∈ S and
suppose that

⋂n
i=1 Ai ⊆

⋃n
i=1 Bi. Then, from (1) we get

∅ = F(
n
⋂

i=1

Ai) ∧ G(

n
⋂

i=1

Ai) ⊇ F(

n
⋂

i=1

Ai) ∧ G(

n
⋃

i=1

Bi)

⊇

n
∧

i=1

F(Ai) ∧

n
∧

i=1

G(Bi) =

n
∧

i=1

(F(Ai) ∧ G(Bi))

which contradicts with
∧n

i=1(F(Ai) ∧ G(Bi)) 6= ∅.
(2) ⇒ (3) : Clear.
(3) ⇒ (1) : If we assume that F ∧ G 6= ∅ then F(A) ∧ G(A) 6= ∅ for some
A ∈ S. Thus we obtain that A 6⊆ A by (3) but this is a contradiction. So we
have F ∧ G = ∅. �

Definition 3.3. A (V, V)-graded difilter (F, G) on (S, S) is called regular if it
satisfies the equivalent conditions in the previous proposition.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 24



Convergence in graded ditopological texture spaces

Example 3.4. (1) Let (S, S) be a texture space and F × G a (regular) difilter
on it. Then the mappings F, G : S → P(1) defined by

F(A) =

{

{0}, A ∈ F
∅, A 6∈ F

and

G(A) =

{

{0}, A ∈ G
∅, A 6∈ G

for all A ∈ S, form a (regular) (1, P(1))-graded difilter on (S, S).
On the other hand, if (F, G) is a (regular) (1, P(1))-graded difilter on (S, S)

then the families defined by

F = {A ∈ S : F(A) = {0} }, G = {A ∈ S : G(A) = {0} }

form a (regular) difilter F × G on (S, S).
Besides, if (F, G) be a (regular) (V, V)-graded difilter on a texture (S, S)

then the families

F
v = {A ∈ S | Pv ⊆ F(A)}, G

v = {A ∈ S | Pv ⊆ G(A)}

form a (regular) difilter Fv × Gv on (S, S) for each v ∈ V .
(2)Analogous with the dinhd-difilter situation; if (N, M) is a graded dinhd
system of a graded ditopological texture space (S, S, T , K, V, V) then Ms is a
(V, V)-graded cofilter on (S, S) but in general for s ∈ S♭, Ns is not a (V, V)-
graded filter on (S, S). If (S, S) is plain then Ns is a (V, V)-graded filter on
(S, S) for each s ∈ S♭ = S. As a result, if (S, S) is plain then (Ns, Ms) is a
(V, V)-graded difilter on (S, S) for each s ∈ S♭ = S.
(3) Let (S, S, T , K, V, V) be a graded ditopological texture space. If the map-
pings ∀s ∈ S♭, N∗s : S → V, ∀s ∈ S, M

∗
s : S → V are defined by

N
∗
s (A) =

{

sup{
⋂n

k=1 T (Bk) : Bk * Qs, B1 ∩ ... ∩ Bn ⊆ A forBi ∈ S}, A * Qs
∅, A ⊆ Qs

and

M∗s (A) =

{

sup{
⋂n

k=1 K(Bk) : Ps * Bk, A ⊆ B1 ∪ ... ∪ Bn forBi ∈ S}, Ps * A
∅, Ps ⊆ A

for all A ∈ S then (N∗s , M
∗
s ) is a regular (V, V)-graded difilter on (S, S) for

each s ∈ S♭.

Proposition 3.5. For the above notations, if the texture (S, S) is plain then
Ns = N

∗
s and Ms = M

∗
s for each s ∈ S

♭ = S.

Proof. Let A ∈ S, s ∈ S. A ⊆ Qs implies N
∗
s (A) = Ns(A) = ∅ so, let it be

A * Qs, and suppose that Ns(A) * N∗s (A) for some A ∈ S. Then there exists
v ∈ V such that Ns(A) * Qv and Pv * N∗s (A). Considering Ns(A) * Qv,
we have Ps ⊆ B ⊆ A * Qs and T (B) * Qv for some B ∈ S. Since (S, S)
is plain, we have B * Qs. So, T (B) ⊆ N∗s (A) and considering T (B) * Qv
we get N∗s (A) * Qv and Pv ⊆ N

∗
s (A) which contradicts with Pv * N

∗
s (A).

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 25



R. Ekmekçi and R. Ertürk

Thus, Ns(A) ⊆ N
∗
s (A). Now we assume that N

∗
s (A) * Ns(A) for some A ∈ S.

Then there exists v ∈ V such that N∗s (A) * Qv and Pv * Ns(A). Considering
N∗s (A) * Qv, there exist B1, B2, ..., Bn ∈ S such that Bk * Qs for 1 ≤ k ≤ n,
B1∩B2∩...∩Bn ⊆ A and

⋂n
k=1 T (Bk) * Qv. Since

⋂n
k=1 T (Bk) ⊆ T (

⋂n
k=1 Bk)

we have T (
⋂n

k=1 Bk) * Qv and so, Pv ⊆ T (
⋂n

k=1 Bk). Moreover, since Bk *
Qs for 1 ≤ k ≤ n and (S, S) is plain, we get Ps ⊆ (B1 ∩B2 ∩...∩Bn) ⊆ A * Qs.
Thus, it is obtained that Pv ⊆ T (

⋂n
k=1 Bk) ⊆ Ns(A) which is a contradiction.

Therefore we get N∗s (A) ⊆ Ns(A) and so, Ns = N
∗
s . Similarly it can be shown

that Ms = M
∗
s . �

Definition 3.6. Let F be a (V, V)-graded filter and G a (V, V)-graded co-
filter on a graded ditopological texture space (S, S, T , K, V, V). We say that F
converges to s and write that F → s if N∗s ⊆ F. Also we say that G converges
to s and write that G → s if M∗s ⊆ G.

For s, s′ ∈ S, the graded difilter (F, G) is called diconvergent if Ps′ * Qs,
F → s and G → s′. In this case, s (s′) is called (co-)limit of (F, G).

Proposition 3.7. If (F, G) is a (V, V)-graded difilter on (S, S, T , K, V, V) then

(a) F → s ⇔ ”A * Qs ⇒ T (A) ⊆ F(A)”
(b) G → s ⇔ ”Ps * A ⇒ K(A) ⊆ G(A)”

Proof. (a) Let F → s and A * Qs. Since F → s, we have N∗s ⊆ F and
N∗s (A) ⊆ F(A). Considering A * Qs we obtain that T (A) ⊆ N

∗
s (A) and so

T (A) ⊆ F(A). On the other hand, if we suppose that ”A * Qs ⇒ T (A) ⊆
F(A)” then N∗s (A) ⊆ F(A) and so we get F → s.

The proof of (b) is similar. �

Proposition 3.8. Let the texture (S, S) be plain. If (F, G) is a graded difilter
on (S, S, T , K, V, V) then the following are equivalent:

(a) (F, G) is diconvergent.
(b) ∃s ∈ S : (Ns, Ms) ⊆ (F, G)

Proof. (a) ⇒ (b): If (F, G) is diconvergent then there exists s, s′ ∈ S such that
Ps′ * Qs and (N∗s , M

∗
s′) ⊆ (F, G). Since (S, S) is plain, from Proposition 3.5

we have (Ns, Ms′) = (N
∗
s , M

∗
s′). Besides, since Ps′ * Qs, Ps * A ⊆ B ⊆ Qs ⇒

Ps′ * A ⊆ B ⊆ Qs′ for all A, B ∈ S and so we have Ms ⊆ Ms′. Thus we get
Ns = N

∗
s ⊆ F, Ms ⊆ Ms′ = M

∗
s′ ⊆ G and hence (Ns, Ms) ⊆ (F, G).

(b) ⇒ (a): Since (S, S) is plain we get Ps * Qs and by Proposition 3.5 we have
(Ns, Ms) = (N

∗
s , M

∗
s ). Therefore (F, G) is diconvergent. �

Definition 3.9. Let (S, S, T , K, V, V) be a graded ditopological texture space,
A ∈ S and v ∈ V . The set

⋂

{B ∈ S|A ⊆ B, Pv ⊆ K(B)} ∈ S

is called v-closure of A and denoted by [A]v. The set
∨

{B ∈ S|B ⊆ A, Pv ⊆ T (B)} ∈ S

is called v-interior of A and denoted by ]A[v.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 26



Convergence in graded ditopological texture spaces

Note that for each v ∈ V , [A]v (]A[v) is the closure (the interior) of A in the
ditopological texture space (S, S, T v, Kv).

Proposition 3.10. Let (F, G) be a regular graded difilter on a graded ditopo-
logical texture space (S, S, T , K, V, V) and s ∈ S.

(a) F → s ⇒ ”∀A ∈ S, v ∈ G(A) ⇒]A[v⊆ Qs”
(b) G → s ⇒ ”∀A ∈ S, v ∈ F(A) ⇒ Ps ⊆ [A]

v”

Proof. (a) Let F → s, and suppose that there exists v ∈ G(A) such that
]A[v* Qs for a set A ∈ S. Then B ⊆ A, Pv ⊆ T (B) and B * Qs for
some B ∈ S. Considering F → s and B * Qs, from proposition 3.7 we get
T (B) ⊆ F(B). So, Pv ⊆ T (B) ⊆ F(B) ⊆ F(A) and considering v ∈ G(A) we
get F(A) ∩ G(A) 6= ∅ which contradicts with the regularity of (F, G).

The proof of (b) is similar. �

Definition 3.11. Let (F, G) be a regular graded difilter on a graded ditopo-
logical texture space (S, S, T , K, V, V).

(1) s ∈ S is called a cluster point of F if for all A ∈ S, v ∈ F(A) ⇒ Ps ⊆
[A]v

(2) s ∈ S is called a cluster point of G if for all A ∈ S, v ∈ G(A) ⇒]A[v⊆ Qs
(3) For s, s′ ∈ S, Ps * Qs′, if s is a cluster point of F and s′ is a cluster

point of G then (F, G) is called diclustering in (S, S, T , K, V, V).

Corollary 3.12. On a graded ditopological texture space (S, S, T , K, V, V), each
diconvergent regular graded difilter is diclustering.

Proof. If (F, G) is a diconvergent regular graded difilter on (S, S, T , K, V, V)
then there exist s, s′ ∈ S such that F → s, G → s′ and Ps′ * Qs. Considering
Proposition 3.10., s is a cluster point of G and s′ a cluster point of F. Since
Ps′ * Qs, (F, G) is diclustering. �

Definition 3.13. Let (F, G) be a (V, V)-graded difilter on (S, S). F is called
prime if F(A1 ∪ A2) ⊆ F(A1) ∪ F(A2) and G is called prime if G(A1 ∩ A2) ⊆
G(A1) ∪ G(A2) for all A1, A2 ∈ S.

Example 3.14. Let S = {1, 2}, S = {∅, {1}, {2}, S}, V = {a, b, c} and
V = P(V ). In this case, (S, S) and (V, V) are texture spaces. If the map-
pings F, G : S → V are defined by

F(∅) = ∅, F({1}) = {a}, F({2}) = {b}, F(S) = {a, b}
G(∅) = {a, b}, G({1}) = {b}, G({2}) = {a}, G(S) = ∅

then (F, G) is a regular (V, V)-graded difilter. Moreover, F and G are prime.

The structure of graded difilter is more general than the structure of difilter.
Most of the properties of difilters can be generalized to the graded case and
it can be expected that graded difilters satisfy these generalized properties.
But this is not possible in each case. For instance, the statements (1) − (4)
in Proposition 2.6. are equivalent for difilters however the generalizations of

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 27



R. Ekmekçi and R. Ertürk

these statements are not always equivalent for graded difilters as in the next
example.

Definition 3.15. Let (F, G) be a (V, V)-graded difilter on (S, S). (F, G) is
called a maximal (V, V)-graded difilter on (S, S) if whenever (F′, G′) is a (V, V)-
graded difilter on (S, S) and (F, G) ⊆ (F′, G′) then we have (F, G) = (F′, G′) is
hold.

Example 3.16. Let S = {1, 2}, S = {∅, {1}, {2}, S}, V = {a, b, c} and
V = {∅, {b}, {c}, {b, c}, V }. In this case, (S, S) and (V, V) are plain texture
spaces. If the mappings F, G : S → V are defined by

F(∅) = ∅, F({1}) = {b}, F({2}) = {c}, F(S) = V
G(∅) = V , G({1}) = {c}, G({2}) = {b}, G(S) = ∅

then (F, G) is a regular (V, V)-graded difilter. Moreover, (F, G) is a maxi-
mal regular (V, V)-graded difilter but F ∨ G 6= V . (Example for (1) ; (2) in
Proposition 3.17.)

Proposition 3.17. Let (F, G) be a regular (V, V)-graded difilter on (S, S). For
the statements

(1) (F, G) is a maximal regular (V, V)-graded difilter
(2) F ∨ G = V (i.e. ∀A ∈ S, F(A) ∨ G(A) = F(A) ∪ G(A) = V )
(3) F is prime and G = V \ F
(4) G is prime and F = V \ G

the following implications are hold:
(1) ⇐ (2) ⇔ (3) ⇔ (4), (1) ; (2)

Proof. (1) ⇐ (2): Let (F, G) be a regular (V, V)-graded difilter on (S, S) and
F ∨ G = V . From regularity of (F, G) we have F ∩ G = ∅. Considering
F ∨ G = F ∪ G = V we get G = V \ F and F = V \ G. If (F′, G′) is a
regular (V, V)-graded difilter on (S, S) and (F, G) ⊆ (F′, G′) then considering
F′ ∩ G′ = ∅ we get

F(A) = V \ G(A) ⊇ V \ G′(A) ⊇ F′(A) ⊇ F(A)

G(A) = V \ F(A) ⊇ V \ F′(A) ⊇ G′(A) ⊇ G(A).

Thus, (F, G) = (F′, G′) is obtained. So, (F, G) is a maximal regular (V, V)-
graded difilter on (S, S).
(2) ⇔ (3): If F ∨ G = V then from regularity of (F, G), we have F ∩ G = ∅ and
so G = V \ F. Suppose that F is not prime. Then, there exist A1, A2 ∈ S such
that F(A1 ∪ A2) * F(A1) ∪ F(A2). Thus it is obtained that

G(A1) ∩ G(A2) = V \ F(A1) ∩ V \ F(A2) = V \ (F(A1) ∪ F(A2))

* V \ F(A1 ∪ A2) = G(A1 ∪ A2)

which contradicts with (GCF3). Therefore F is prime.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 28



Convergence in graded ditopological texture spaces

On the other hand, if G = V \ F then F ∨ G = F ∪ G = V is obtained.
(2) ⇔ (4): Similar as (2) ⇔ (3).
(1) ; (2): Example 3.16. �

Proposition 3.18. A texture space (V, V) is discrete if and only if Pv * A ⇒
Pv ∩ A = ∅ for all v ∈ V , A ∈ V.

Proof. Let Pv * A ⇒ Pv ∩ A = ∅ for all v ∈ V , A ∈ V and suppose that
(V, V) is not discrete. Then there exists v ∈ V such that Pv 6= {v}. So, t ∈ Pv,
t 6= v for some t ∈ V . Since textures are point separating lattices, we have
v 6∈ Pt and Pv * Pt. Considering {v} ⊆ Pv ∩ Pt 6= ∅, if we take A = Pt in the
hypothesis then we get a contradiction. Thus, (V, V) is discrete. On the other
hand, if (V, V) is discrete then Pv = {v} and so Pv * A ⇒ Pv ∩ A = ∅ for all
v ∈ V , A ∈ V. �

The generalizations of the equivalent statements in Proposition 2.6. to the
graded case are equivalent if (V, V) is discrete as in the next theorem. Hence
the concepts studied and results obtained in this paper are more general than
those in [11].

Theorem 3.19. If (F, G) is a regular (V, V)-graded difilter on (S, S) and (V, V)
is discrete then the statements (1) − (4) in Proposition 3.17 are equivalent.

Proof. It is sufficient to show the implication (1) ⇒ (2). Let (F, G) be a
maximal regular (V, V)-graded difilter on (S, S) and suppose that F ∨ G 6= V .
Then there exists A0 ∈ S such that F(A0) ∪ G(A0) 6= V and so there exists
v ∈ V such that Pv = {v} * F(A0) ∪ G(A0). There are two cases: ”∀A, B ∈
S, Pv * F(A) ∩ G(B)” or ”∃A, B ∈ S : Pv ⊆ F(A) ∩ G(B)”.

Case 1: ”∀A, B ∈ S, Pv * F(A) ∩ G(B)”. In particular, if we take A = S
and B = ∅ then Pv * F(S) ∩ G(∅). So, there are two subcases: Pv * F(S) or
Pv * G(∅).

Case 1.1: ”Pv * F(S)”. Since from (GF2) F(A) ⊆ F(S) for all A ∈ S, we
have Pv * F(A) for all A ∈ S. If we define a mapping F∗ : S → V by

F
∗(B) =







∅, B = ∅
F(B), A0 * B
F(B) ∪ Pv, A0 ⊆ B

then (F∗, G) is a regular (V, V)-graded difilter on (S, S):
(GF1) F∗(∅) = ∅
(GF2) Let B1, B2 ∈ S and B1 ⊆ B2. If B1 = ∅ then we have F∗(B1) = ∅ ⊆
F∗(B2). If B1 6= ∅ then B2 6= ∅ and so,

(a) If A0 ⊆ B1 then A0 ⊆ B2 and so F
∗(B1) = Pv ∪ F(B1) ⊆ Pv ∪ F(B2) =

F∗(B2).
(b) If A0 * B1 then F∗(B1) = F(B1) ⊆ F(B2) ⊆ F∗(B2).

(GF3) Let B1, B2 ∈ S. If B1 = ∅ or B2 = ∅ then we have F∗(B1) ∧ F∗(B2) =
∅ ⊆ F∗(B1 ∩ B2). If B1, B2 6= ∅ then

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 29



R. Ekmekçi and R. Ertürk

(a) If A0 * B1, B2 then A0 * B1 ∩ B2 and F∗(B1) ∧ F∗(B2) = F(B1) ∩
F(B2) ⊆ F(B1 ∩ B2) = F

∗(B1 ∩ B2).
(b) If A0 ⊆ B1, B2 then A0 ⊆ B1∩B2 and F

∗(B1)∧F
∗(B2) = (Pv ∪F(B1))∩

(Pv ∪F(B2)) = Pv ∪(F(B1)∩F(B2)) ⊆ Pv ∪F(B1 ∩B2) = F
∗(B1 ∩B2).

(c) Without loss of generality let A0 * B1 and A0 ⊆ B2. Then A0 *
B1 ∩ B2. Considering that Pv * F(A) for all A ∈ S and V is discrete
we have F(B1) ∩ Pv = ∅. Thus we get

F
∗(B1) ∧ F

∗(B2) = F(B1) ∩ (Pv ∪ F(B2)) = (F(B1) ∩ Pv) ∪ (F(B1) ∩ F(B2))

= F(B1) ∩ F(B2) ⊆ F(B1 ∩ B2) = F
∗(B1 ∩ B2).

Let B ∈ S. Then since (F, G) is regular;

(a) If A0 ⊆ B then, since V is discrete, G(B) ⊆ G(A0) and Pv * F(A0) ∪
G(A0) it is obtained that Pv * G(B) and so we have Pv ∩ G(B) = ∅.
Therefore we get F∗(B)∩G(B) = (Pv ∪F(B))∩G(B) = (Pv ∩G(B))∪
(F(B) ∩ G(B)) = ∅, i.e. F∗(B) ∩ G(B) = ∅.

(b) If A0 * B then we get F∗(B) ∩ G(B) = F(B) ∩ G(B) = ∅.
Therefore (F∗, G) is a regular (V, V)-graded difilter on (S, S). However,

(F, G) $ (F∗, G) (at least F∗(A0) = Pv ∪ F(A0) 6= F(A0)) and this contradicts
with the maximality of (F, G). Hence, the implication (1) ⇒ (2) is satisfied.

Case 1.2: ”Pv * G(∅)”. Since from (GCF2) G(A) ⊆ G(∅) for all A ∈ S,
we have Pv * G(A) for all A ∈ S. If we define a mapping G∗ : S → V by

G
∗(B) =







∅, B = S
G(B), B * A0
G(B) ∪ Pv, B ⊆ A0

then (F, G∗) is a regular (V, V)-graded difilter on (S, S) (it can be shown like
in case 1.1).

However, (F, G) $ (F, G∗) (at least G∗(A0) = Pv ∪G(A0) 6= G(A0)) and this
contradicts with the maximality of (F, G). Hence, the implication (1) ⇒ (2) is
satisfied.

Case 2: ”∃C, D ∈ S, Pv ⊆ F(C) ∩ G(D)”. If we suppose that A0 = S then
since C ⊆ A0 we have Pv ⊆ F(C) ⊆ F(A0) which contradicts with Pv * F(A0).
Similarly if we suppose that A0 = ∅ then since A0 ⊆ D we have Pv ⊆ G(D) ⊆
G(A0) which contradicts with Pv * G(A0). Thus we get that A0 6= ∅, S.

Now, we show that

∀A, B ∈ S : Pv ⊆ F(A) ∩ G(B) ⇒ A0 ∩ A * B

or

∀A, B ∈ S : Pv ⊆ F(A) ∩ G(B) ⇒ A * A0 ∪ B.

Contrary, if we assume that there exist A1, B1, A2, B2 ∈ S such that ”Pv ⊆
F(A1)∩G(B1), A0∩A1 ⊆ B1” and ”Pv ⊆ F(A2)∩G(B2), A2 ⊆ A0∪B2” then we
have Pv ⊆ F(A1)∩F(A2) ⊆ F(A1 ∩A2) and Pv ⊆ G(B1)∩G(B2) ⊆ G(B1 ∪B2)
and so F(A1 ∩ A2) ∩ G(B1 ∪ B2) 6= ∅. Since (F, G) is regular, from proposition
3.2. (3) we get A1 ∩ A2 * B1 ∪ B2. Hence there exists s ∈ A1 ∩ A2 such that

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 30



Convergence in graded ditopological texture spaces

s 6∈ B1 ∪ B2. Considering s ∈ A2, s 6∈ B2 and A2 ⊆ A0 ∪ B2 we have s ∈ A0.
Now, considering s ∈ A0, s ∈ A1 and A0 ∩ A1 ⊆ B1 we have s ∈ B1 which
contradicts with s 6∈ B1 ∪ B2. Therefore we have again two subcases.

Case 2.1:”∀A, B ∈ S : Pv ⊆ F(A) ∩ G(B) ⇒ A0 ∩ A * B”. If we define a
mapping F′ : S → V by

F
′(B) =







∅, B = ∅
F(B), B 6= ∅ and ”Pv ⊆ F(A) for all A ∈ S ⇒ A0 ∩ A * B”
F(B) ∪ Pv, B 6= ∅ and ”∃A ∈ S : Pv ⊆ F(A) and A0 ∩ A ⊆ B”

then (F′, G) is a regular (V, V)-graded difilter on (S, S):
(GF1) F′(∅) = ∅
(GF2) Let B1, B2 ∈ S and B1 ⊆ B2. If B1 = ∅ then we have F′(B1) = ∅ ⊆
F′(B2). If B1 6= ∅ then B2 6= ∅ and so,

(a) Let ”∃A ∈ S : Pv ⊆ F(A) and A0 ∩A ⊆ B1”. Then A0 ∩A ⊆ B1 ⊆ B2
and so F′(B1) = Pv ∪ F(B1) ⊆ Pv ∪ F(B2) = F

′(B2).
(b) Let ”Pv ⊆ F(A) ⇒ A0 ∩ A * B1 for all A ∈ S”. Then F′(B1) =

F(B1) ⊆ F(B2) ⊆ F
′(B2).

(GF3) Let B1, B2 ∈ S. If B1 = ∅ or B2 = ∅ then we have F′(B1) ∧ F′(B2) =
∅ ⊆ F′(B1 ∩ B2). If B1, B2 6= ∅ then

(a) Let ”∃A1 ∈ S : Pv ⊆ F(A1), A0 ∩ A1 ⊆ B1” and ”∃A2 ∈ S : Pv ⊆
F(A2), A0 ∩ A2 ⊆ B2”. Then A1 ∩ A2 ∈ S, A0 ∩ (A1 ∩ A2) ⊆ B1 ∩ B2
and so, F′(B1) ∧ F

′(B2) = (Pv ∪ F(B1)) ∩ (Pv ∪ F(B2)) = Pv ∪ (F(B1) ∩
F(B2)) ⊆ Pv ∪ F(B1 ∩ B2) = F

′(B1 ∩ B2).
(b) Let ”Pv ⊆ F(A) ⇒ A0 ∩ A * B1” and ”Pv ⊆ F(A) ⇒ A0 ∩ A * B2”.

Then F′(B1) ∧ F
′(B2) = F(B1) ∩ F(B2) ⊆ F(B1 ∩ B2) ⊆ F

′(B1 ∩ B2).
(c) Without loss of generality let ”∃A1 ∈ S : Pv ⊆ F(A1), A0 ∩ A1 ⊆ B1”

and ”Pv ⊆ F(A) ⇒ A0 ∩ A * B2”. Since Pv ⊆ F(B2) implies the
contradiction A0 ∩B2 * B2 we have Pv * F(B2). Then Pv ∩F(B2) = ∅
because V is discrete. Moreover, since ”Pv ⊆ F(A) ⇒ A0 ∩ A * B2 we
have ”Pv ⊆ F(A) ⇒ A0 ∩A * B1 ∩B2 and so F′(B1 ∩B2) = F(B1 ∩B2).
Thus we get

F
′(B1) ∧ F

′(B2) = (Pv ∪ F(B1)) ∩ F(B2) = (Pv ∩ F(B2)) ∪ (F(B1) ∩ F(B2))

= F(B1) ∩ F(B2) ⊆ F(B1 ∩ B2) = F
′(B1 ∩ B2).

Let B ∈ S. If B = ∅ then F′(B) ∩ G(B) = ∅. So, assume that B 6= ∅. Then
since (F, G) is regular;

(a) If ”∃A ∈ S : Pv ⊆ F(A) and A0 ∩ A ⊆ B” then, because of the
implication of case 2.1. we have Pv * G(B). Since V is discrete we get
Pv ∩G(B) = ∅. Therefore we get F′(B)∩G(B) = (Pv ∪F(B))∩G(B) =
(Pv ∩G(B))∪(F(B)∩G(B)) = ∅∪(F(B)∩G(B)) = F(B)∩G(B) = ∅.
That is F′(B) ∩ G(B) = ∅.

(b) If ”Pv ⊆ F(A) ⇒ A0 ∩ A * B” then we get F′(B) ∩ G(B) = F(B) ∩
G(B) = ∅.

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R. Ekmekçi and R. Ertürk

Therefore (F′, G) is a regular (V, V)-graded difilter on (S, S). However, since
Pv ⊆ F(C) and A0∩C ⊆ A0 we have F

′(A0) = Pv ∪F(A0) and since Pv * F(A0)
we get F′ 6= F. Thus (F, G) $ (F′, G) and this contradicts with the maximality
of (F, G). Hence, the implication (1) ⇒ (2) is satisfied.

Case 2.2:”∀A, B ∈ S : Pv ⊆ F(A) ∩ G(B) ⇒ A * A0 ∪ B”. If we define a
mapping G′ : S → V by

G
′(A) =







∅, A = S
G(A), A 6= S and ”Pv ⊆ G(B) for all B ∈ S ⇒ A * A0 ∪ B”
G(A) ∪ Pv, A 6= S and ”∃B ∈ S : Pv ⊆ G(B) and A ⊆ A0 ∪ B”

then (F, G′) is a regular (V, V)-graded difilter on (S, S) (it can be shown like in
case 2.1). However, (F, G) $ (F, G′) and this contradicts with the maximality
of (F, G). Hence, the implication (1) ⇒ (2) is satisfied. �

Corollary 3.20. If (F, G) is a maximal regular (V, V)-graded difilter on (S, S)
and (V, V) is discrete then F(S) = G(∅) = V .

Proof. Considering theorem 3.19., since (F, G) is maximal regular we get F ∨
G = V . Suppose that F(S) 6= V or G(∅) 6= V . Then we have F(S)∪G(S) 6= V
or F(∅) ∪ G(∅) 6= V which contradicts with F ∨ G = V . �

It is obtained in [11] that if F is a prime filter on a texture (S, S) then S \F
is a prime cofilter and F × (S \ F) is a maximal regular difilter on the same
texture. However the generalization of this statement is not true for graded
difilters. In example 3.14., F is a prime (V, V)-graded filter on (S, S) and the
texture (V, V) is discrete but V \F isn’t a prime (V, V)-graded cofilter on (S, S)
since F(S) 6= V and (V \ F)(S) 6= ∅. Since (V, V) is discrete, by corollary 3.20.
we get that there is no maximal regular (V, V)-graded difilter on (S, S) whose
first component is F.

Proposition 3.21. Let (F, G) be a regular (V, V)-graded difilter on (S, S).
Then there exists a maximal regular (V, V)-graded difilter (FM , GM) on (S, S)
such that (F, G) ⊆ (FM , GM ).

Proof. Let (Fj, Gj)j∈J be a chain of regular (V, V)-graded difilters on (S, S)
which satisfies (F, G) ⊆ (Fj, Gj) for all j ∈ J. If the mappings F

′, G′ : S → V
are defined by F′ =

∨

j∈J Fj and G
′ =

∨

j∈J Gj then (F
′, G′) is a regular (V, V)-

graded difilter on (S, S):
(GF1) F′(∅) =

∨

j∈J Fj(∅) = ∅
(GF2) Let A1, A2 ∈ S and A1 ⊆ A2. Since Fj(A1) ⊆ Fj(A2) for each j ∈ J we
get F′(A1) =

∨

j∈J Fj(A1) ⊆
∨

j∈J Fj(A2) = F
′(A2).

(GF3) Let A1, A2 ∈ S. Consider the sets Ji1 = {j ∈ J | (Fi, Gi) ⊆ (Fj, Gj)}
and Ji2 = {j ∈ J | (Fi, Gi) % (Fj, Gj)} for each i ∈ J. Since (Fj, Gj)j∈J is a

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Convergence in graded ditopological texture spaces

chain we have Ji1 ∪ Ji2 = J for each i ∈ J and so we get

F
′(A1) ∧ F

′(A2) =
∨

i∈J

Fi(A1) ∧
∨

j∈J

Fj(A2) =
∨

i∈J

(Fi(A1) ∧
∨

j∈J

Fj(A2))

=
∨

i∈J

(
∨

j∈J

(Fi(A1) ∧ Fj(A2)))

=
∨

i∈J

(
∨

j∈J
i1

(Fi(A1) ∧ Fj(A2)) ∨
∨

j∈J
i2

(Fi(A1) ∧ Fj(A2)))

⊆
∨

i∈J

(
∨

j∈J
i1

(Fj(A1) ∧ Fj(A2)) ∨
∨

j∈J
i2

(Fi(A1) ∧ Fi(A2)))

⊆
∨

i∈J

(
∨

j∈J
i1

(Fj(A1 ∩ A2)) ∨
∨

j∈J
i2

(Fi(A1 ∩ A2)))

=
∨

i∈J

Fi(A1 ∩ A2) = F
′(A1 ∩ A2).

Thus F′ is a (V, V)-graded filter on (S, S) and similarly it can be shown that
G′ is a (V, V)-graded cofilter on (S, S).

Now, we use the similar method as in (GF3) to show that (F′, G′) is regular.
So, consider the sets Ji1, Ji2 for each i ∈ J as above. Let A ∈ S. Since (Fj, Gj)
is regular for each j ∈ J we obtain that

F
′(A) ∩ G′(A) =

∨

i∈J

Fi(A) ∩
∨

j∈J

Gj(A) =
∨

i∈J

(Fi(A) ∩
∨

j∈J

Gj(A))

=
∨

i∈J

(
∨

j∈J

(Fi(A) ∩ Gj(A)))

=
∨

i∈J

(
∨

j∈J
i1

(Fi(A) ∩ Gj(A)) ∨
∨

j∈J
i2

(Fi(A) ∩ Gj(A)))

⊆
∨

i∈J

(
∨

j∈J
i1

(Fj(A) ∩ Gj(A)) ∨
∨

j∈J
i2

(Fi(A) ∧ Gi(A))) = ∅.

Therefore (F′, G′) is an upper bound for the chain (Fj, Gj)j∈J in the set
Z = {(U, R) | (U, R) is a regular (V, V)−graded difilter on (S, S) and (F, G) ⊆
(U, R)}. By Zorn’s Lemma the set Z has a maximal element (FM , GM ). Hence
(F, G) ⊆ (FM , GM ) and we obtain that (FM , GM) is maximal in the set of all
regular (V, V)-graded difilters on (S, S). �

Proposition 3.22. Let (V, V) be a discrete texture space and (F, G) a maximal
regular graded difilter on (S, S, T , K, V, V). Then (F, G) is diconvergent if and
only if it is diclustering.

Proof. If (F, G) is diconvergent then it is diclustering by Corollary 3.12. On
the other hand, let (F, G) be diclustering. Namely let s be cluster point of
F, s′ cluster point of G where Ps * Qs′. Then v ∈ F(A) ⇒ Ps ⊆ [A]v and
v ∈ G(A) ⇒]A[v⊆ Qs′ for all A ∈ S.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 1 33



R. Ekmekçi and R. Ertürk

Let A ∈ S with A * Qs′ and v 6∈ F(A). By Theorem 3.19. we have
F(A) ∪ G(A) = V . So, considering regularity of (F, G) we get v ∈ G(A) and
so ]A[v⊆ Qs′. Hence v 6∈ T (A) because v ∈ T (A) implies A =]A[

v⊆ Qs′

which contradicts with A * Qs′. Thus we have v 6∈ F(A) ⇒ v 6∈ T (A), i.e.
T (A) ⊆ F(A). Therefore, considering Proposition 3.7. we get F → s′. Using
similar method, it can be obtained that G → s. So (F, G) is diconvergent. �

Proposition 3.23. Let (S, S, T , K, V, V) be a graded ditopological texture space.
Then, for the staements

(a) Every regular graded difilter on (S, S, T , K, V, V) is diclustering.
(b) Every maximal regular graded difilter on (S, S, T , K, V, V) is diconver-

gent.

the implication (b) ⇒ (a) and in case of (V, V) is discrete, (a) ⇒ (b) are hold.

Proof. (a) ⇒ (b): Let (F, G) be a maximal regular graded difilter on (S, S, T , K, V, V).
From (a), (F, G) is diclustering. Considering that (V, V) is discrete and Propo-
sition 3.22. (F, G) is diconvergent.
(b) ⇒ (a): Let (F, G) be a regular graded difilter on (S, S, T , K, V, V). Consider-
ing Proposition 3.21., there exists a maximal regular graded difilter (FM , GM )
on (S, S, T , K, V, V) with (F, G) ⊆ (FM , GM). From (b) we have FM → s,
GM → s′ and Ps′ * Qs for some s, s′ ∈ S. Considering GM → s′ and Proposi-
tion 3.10. we have v ∈ F(A) ⇒ v ∈ FM (A) ⇒ Ps′ ⊆ [A]

v and so we get that s′

is a cluster point of F. Similarly it can be obtained that s is a cluster point of
G. Thus (F, G) is diclustering. �

4. Conclusion

Filters and their convergence are convenient tools for topological spaces as
filter convergence can describe some of topological concepts. In this work,
graded difilters are introduced and their convergence which characterizes inte-
rior, closure of sets, etc. is investigated. This new sturucture is helpful to deal
with the theory of graded ditopological texture spaces. Moreover, the relations
between difilters and graded difilters are studied.

As expected, graded difilters are based on graded dinhd systems. Graded
dinhd systems are not graded difilters in general so, the method used in [11] is
used to define the convergence of graded difilters. Obviously, graded difilters
are more general than difilters and naturally some properties of difilters are
not valid for graded difilters in general (see Prop. 3.17. and Theorem 3.19.).

Clearly, graded difilters and their convergence can be useful to define and
investigate the concept of compactness on graded ditopological texture spaces.

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

helpful suggestions and comments.

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Convergence in graded ditopological texture spaces

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