EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 533-543
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Rough spaces on covering based rough sets

N. Alharbi1, H. Aydi2,∗, C. Özel1

1 Department of Mathematics, King Abdulaziz University, P.O. Box: 80203 Jeddah 21589,
Saudi Arabia
2 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de
Communication, H. Sousse 4000, Tunisia

Abstract. In this paper, we discuss open (closed) sets, rough interiors, rough closures, continuous
mappings, open (closed) mappings and homeomorphism mappings of covering based rough topology
of Akduman, Özcelik and Özel. We also construct the nano topology on a given covering based
rough set.

2010 Mathematics Subject Classifications: 22A05, 54A05, 03E25

Key Words and Phrases: Covering based rough topology, covering based rough sets, isomor-
phisms of approximations, rough continuity

This paper is produced from the PhD thesis of Ms. Nof Alharbi registered in King
Abdulaziz University.

1. Introduction

The rough set theory was introduced by Pawlak in [2]. It deals with impression, vague-
ness, and uncertainty in data analysis and information systems. The rough set theory
offers ways to find the deciding factors or core from data. The classical rough set theory
(Pawlak version) is based on the equivalence relations. Pawlak defined an approximation
space as an ordered pair < U,R > where U is a non-empty set and R is an equivalence
relation defined on U . Then for any X ∈ P (U), he presented definitions for lower and
upper approximations. Next, he determined rough sets by establishing rough equivalence
relation.

Later in 1988, Pomykala [3] defined corresponding operations on Pawlak rough sets. A
year later, Bryniarski [1] extended classical rough sets given by Pawlak to covering based
rough sets. Moreover, he restricted coverings by some conditions to make operations of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3420

Email addresses: nof20081900@hotmail.com (N. Alharbi),
hassen.aydi@isima.rnu.tn (H. Aydi), cenap.ozel@gmail.com (C. Özel)

http://www.ejpam.com 533 c© 2019 EJPAM All rights reserved.



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 534

Pawlak rough sets, well defined.

• Let U be a non-empty set and C be a family of non-empty subsets of U such that
∪C = U . Then C is called a covering of U . Obviously, a partition is indeed a covering
of U , so that a covering is an extension of a partition.
• Bryniarski also defined the lower and upper approximations and the boundary region
in a similar way as Pawlak. But, instead of using elementary sets (classes), He used the
elements of coverings. Let X ∈ P (U). The ordered pair (X,X) is then the covering
based rough set of X. He also introduced a restricted condition for a covering called an
approximation condition. Since we extend the Pawlak operations to the covering based
rough sets, these operations may not be well defined.

Definition 1. (Bryniarski, 1989) Let U be a set with at least two elements and C be its
covering.
• C satisfies the approximation condition ⇐⇒ for all A,B ⊂ C such that A ⊂ B, there
exists X ⊂ U with A = X and B = X.
• C is minimal ⇐⇒ for all K ∈ C, we have ∪C \ {K} 6= U .
• C is representable ⇐⇒ for all K ∈ C there exists x ∈ K such that for all L ∈ C and
L 6= K, we have x /∈ L. This element x ∈ K is called a representative of K.
• C is a well representable if every K ∈ C has at least two distinct representatives.
• C is called an exact ⇐⇒ for all A ⊂ C,A = {K ∈ C,K ⊂ ∪A}.

Theorem 1. [1] Let C be a covering, then the following conditions are equivalent:

(i) C is minimal;

(ii) C is representable;

(iii) C is exact.

Theorem 2. [1] For a covering C of the universe U , the following conditions are equiva-
lent:

(i) C is well representable;

(ii) C satisfies the approximation condition.

We can say that an ordered pair (U,C) is a covering approximation space if C
satisfies one condition of Theorem 2.

Let (U,C) be a covering approximation space and X,Y ∈ P (U). Then the operations
∨, ∧ and complement on the rough sets are defined in the following:

(i) X ∨ Y = (X ∨ Y ,X ∨ Y );



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 535

(ii) X ∧ Y = (X ∧ Y ,X ∧ Y );

(iii) Xc = (C \X,C \X).

Note that the union and intersection are about classes, not elements in the classes.

A rough set X is exact if X = X. The family of all rough sets of elements belonging
to P (U) is called a rough set of the first order. This family is a distributive lattice
with operations ∨ and ∧. Also, it satisfies De Morgan laws with operations ∨ and ∧
and complement. In addition, the set of all rough exact sets with operations ∨,∧ and
complement forms a Boolean algebra [1].

Proposition 1. [1] If (U,C) is a covering approximation space and A and B are subsets
of U , then

(i) A ⊆ A ⊆ A;

(ii) ∅ = ∅ = ∅;

(iii) U = U = U ;

(iv) A ∨B = A ∨B;

(v) A ∧B ⊆ A ∧B;

(vi) A ∧B = A ∧B.

(vii) A ∨B ⊆ A ∨B;

(viii) If A ⊆ B, then A ⊆ B and A ⊆ B;

(ix) Ac = (A)c, and Ac = (A)c;

(x) A = (A) = A;

(xi) A = (A) = A.

Definition 2. [5] Let (U,C) be a covering approximation space and X = (X,X) be a
covering based rough set. The collection τ consisting of rough subsets of X = (X,X) is
called a covering based rough topology on X = (X,X) if the following conditions are
satisfied:

(i) ∅, X = (X,X) ∈ τ ;

(ii) τ is closed under a finite intersection;

(iii) τ is closed under an arbitrary union.

In 2015, Akduman, Özcelik and Özel [4] constructed a topology on classical rough sets.
They called it a rough topological space. Also, they defined a topology on a given covering
based rough set.



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 536

2. Main results

To define open and closed sets on a covering based rough topology, we need the next
result.

Theorem 3. [1] For every covering approximation space (U,C), there exists a one-to-one
mapping of the set of all pairs (A,B) such that A ⊂ B ⊂ C onto the set of all rough sets
of the first order defined as (A,B) = X ⇐⇒ A = X and B = X, where X ⊆ U .

Let Y ⊆ U be a rough open set in a covering based rough space (X, τ), then there
exist A,B ⊂ U such that A ⊂ B ⊂ C with Y = A, Y = B, A ⊆ X and B ⊆ X.

To define closed sets in X, we define the rough complement of any rough open set with
respect to the rough setX. If Y = (Y , Y ) is a rough open set inX, then Y c = (X\Y ,X\Y )
is a closed set in the covering based rough space (X, τ) such that X\Y ⊆ X and X\Y ⊆ X.

Similarly as the classical rough topology, we can define strong and weak elements for
any covering based rough topology. Then we can apply a neighborhood for strong (resp.
weak) element and strong ( resp. weak) interior and strong (resp. weak) exterior point in
similar way.

However, Bryniarski [1] defined an exact element of a rough set as a nonempty subset
of the set of representatives of some class K ∈ C or the whole class K.

Definition 3. [1] Let < U,C > be an approximation space. An element of a rough set of
the first order is defined as follows: for all X,Y ∈ P (U), Y ∈ Xc ⇐⇒ Y 6= ∅, YC ⊂ XC

and there exists K ∈ C such that Y = {K}.

We are ready to introduce a rough exact neighborhood of an exact element. Let Y be
an exact element in the covering based rough space X and A be a covering based rough
open set in X. Then A is a rough exact neighborhood of Y if YC ⊆ A, that is, the exact
element Y is included in A roughly.

Definition 4. Let (X, τ) be a covering based rough space, where X ⊆ U . For A ⊆ X,

(i) the covering based rough interior of the set A is defined as the union of all
covering based rough open subsets contained in A. It is denoted by cint(A) or A◦;

(ii) the covering based rough closure of the set A is defined as the intersection of
all covering based rough closed subsets containing A. It is denoted by ccl(A);

(iii) the exact interior point of a subset A is an element Y ∈ XC having a rough exact
neighborhood V such that Y ∈ V ⊆ A;

(iv) the exact exterior point of a subset A is an element Y ∈ XC having a rough
exact neighborhood V such that Y ∈ V ⊆ Ac.



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 537

Example 1. Let U = {a, b, c, d, e} and C = {{a, b, e}, {c, d, e}}. Clearly,
⋃
C = U and

the covering C is well representable, so it satisfies an approximation condition. Assume
that X = {a, e, d}, then

X = ∅,

and

X = {{a, b, e}{c, d, e}}.

Define τC = {X = (X,X),= (∅, ∅), {a}, {d}}, then τC is a covering based rough space,
where the approximations of each rough open set are as follows:

{a} = ∅, {a} = {{a, b, e}},

and

{d} = ∅, {d} = {{c, d, e}}.

Exact elements of this covering based rough topology are {a}, {d}, {e}, {a, e}, {a, d},
{e, d}, {a, e, d}, which satisfy the condition in Definition 3.

Proposition 2. Let (U,C) be a covering approximation space and X ⊆ U be such that
(X, τ) is a covering based rough topology. Suppose A and B are subsets of X.

(i) If A ⊆ B, then A◦ ⊆ B◦;

(ii) If A ⊆ B, then ccl(A) ⊆ ccl(B);

(iii) (A ∧B)◦ = A◦ ∧B◦;

(iv) A◦ ∨B◦ ⊆ (A ∨B)◦;

(v) ccl(A ∨B) = ccl(A) ∨ ccl(B);

(vi) ccl(A) ∧ ccl(B) ⊆ ccl(A ∧B).

Proof.

(i) We know that A◦ ⊆ A and B◦ ⊆ B. But A ⊆ B, so

A◦ ⊆ A ⊆ B,

A◦ ⊆ A ⊆ B.

Thus, A◦ ⊆ B. By definition of interior of B as the biggest rough open set contained
in B, A◦ ⊆ B◦.



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 538

(ii) We have A ⊆ ccl(A) and B ⊆ ccl(B). Hence

A ⊆ ccl(A), A ⊆ ccl(A),

B ⊆ ccl(B), B ⊆ ccl(B).

As A ⊆ B, by definition of closure of A, we get

ccl(A) ⊆ ccl(B), ccl(A) ⊆ ccl(B).

Hence ccl(A) ⊆ ccl(B).

Rest items are similar to the classical case.

Subspaces

The inclusion operation on covering based rough sets X and Y is defined by

Y ⊆C X ⇐⇒ Y ⊆ X,

and
Y ⊆ X.

Let (U,C) be a covering approximation space. Suppose that Y and X are elements of
the power set of U such that Y ⊆C X. Our aim is to define topological subspaces of a
covering based rough topology. Following [1], the set of all rough sets of the first order
with operations ∨ and ∧ is a distributive lattice, where

∨i(Vi ∧ Y ) = (∨iVi) ∧ Y,

∧i(Vi ∨ Y ) = (∧iVi) ∨ Y.

That is, Vi and Y are rough sets. Then

∨i(Vi ∧ Y ) = ∨i(Vi) ∧ Y ,

∧i(Vi ∨ Y ) = ∧i(Vi) ∨ Y ,

∨i(Vi ∧ Y ) = ∨i(Vi) ∧ Y ,

and

∧i(Vi ∨ Y ) = ∧i(Vi) ∨ Y .

Therefore, we can define covering based rough subspaces.

Definition 5. Suppose (U,C) be a covering approximation space. Let (X, τ) be a covering
based rough topology with the rough set Y ⊆ X. Define τ

′
= {V ∧Y : V ∈ τ}. Then (Y, τ

′
)

is a covering based rough subspace of (X, τ).

The intersection of Y with all covering rough open sets in τ induces a topology space
on Y .



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 539

Exact base and exact subbase of covering based rough space

A base for a covering based rough space (X, τ) is a collection of covering based rough
open sets, that are, subsets of X satisfying the following fact: for every rough open set
YC = (Y , Y ) and for every exact element Y ∈ YC , there exists a basic open set B = (B,B)
of that collection such that Y ∈ B ⊆ YC , i.e., B ⊆ Y and B ⊆ Y .

Now, we will give a result related to a characterization of the base in the covering
based rough topology on rough sets.

Proposition 3. Let B = {(A,B);A ⊆ B,B ∈ C,A ⊆ X ∧ B ⊆ X} be a collection of
covering based rough open subsets of X. Then B satisfies the following two conditions:

(i) For every Y ∈ XC , there exists (A,B) ∈ B such that Y ∈ (A,B);

(ii) For any (A1, B1), (A2, B2) ∈ B and every element Y ∈ ((A1, B1) ∧ (A2, B2)), there
exists (A,B) ∈ B such that Y ∈ (A,B) ⊆ ((A1, B1) ∧ (A2, B2)).

Therefore B is an exact base for the covering based rough space (X, τ) .

Proof. We need the two following conditions:

(i) Let Y ∈ XC =⇒ Y 6= ∅ ∧ YC ⊆ XC ∧ ∃K ∈ C such that Y = K. We have
Y ⊆ Y = K ⊆ C. So that YC = (Y , Y ) ∈ B, with Y ⊆ X ∧ Y ⊆ X, which is the
desired result.

(ii) Let Y ∈ GC = (G,G) = ((A1, B1) ∧ (A2, B2)), where (A1, B1), (A2, B2) ∈ B. This
implies that Y 6= φ, YC ⊆ GC and there exists B ∈ C such that Y = B. Then
YC = (Y , Y ) with Y ⊆ Y = B. Hence YC ∈ B and Y ∈ YC ⊆ ((A1, B1) ∧ (A2, B2)).

So B is an exact base for the covering based rough topology on the covering based rough
set X.

A family F ⊆ τ is called an exact subbase for a topological space (X, τ) if the family
of all finite intersections of members of F forms a base for (X, τ).

Definition 6. Let F be a family of covering based rough open subsets of XC . Then F is
called an exact covering based rough subbase if the family {Y1C ∧ Y2C ∧ · · · ∧ YnC :
n ∈ N, YiC ∈ F , ∀i ∈ {1, 2, . . . , n}} forms a base for (XC , τ).

Remark 1. Let F be an exact covering based rough subbase. Then it generates a covering
based rough topology on X.

Now, we introduce separation axioms with respect to exact elements for a covering
based rough space (X, τ).
The space (X, τ) is called an exact covering based rough T◦-space if for every exact
distinct elements Y, Y

′ ∈ X, there is a covering based rough open set A of X such that
Y /∈ A 3 Y ′ or Y

′
/∈ A 3 Y .

The space (X, τ) is called an exact covering based rough T1-space if for every exact



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 540

distinct elements Y, Y
′ ∈ X, there are two covering based rough open sets A and B of X

such that Y /∈ A 3 Y ′ and Y
′
/∈ B 3 Y .

The space (X, τ) is called an exact covering based rough T2-space if for every exact
distinct elements Y, Y

′ ∈ X, there are two disjoint covering based rough open sets A and
B of X such that Y /∈ A 3 Y ′ and Y

′
/∈ B 3 Y .

Note that, there is no exact rough T1-space. Also, there is no covering based rough
T1-space. This is true only in the case that the rough set X is a singleton and this singleton
is a class, since exact elements are subset of each other.

Continuous, closed, open and homeomorphism mappings

Here, we will define covering based rough continuous maps between two covering based
rough topological spaces.

Definition 7. Let (U,C) be a covering based approximation space and f : U → U be a
map. Assume that X,Y are covering based rough topological spaces over U . Then the
restriction of f from X to Y is a covering based rough continuous map, if the pre
image of every V ⊆ Y, which is a covering based rough open (resp. closed) set in Y, gives
a subset of X being a covering based rough open (resp. closed) in X.

Definition 8. Let (U,C) and (V,C
′
) be a covering based approximation spaces such that

X ⊆ U and Y ⊆ V . Consider the two rough topological spaces (X, τ), (Y, τ
′
). A mapping

f from X to Y is called a covering based rough continuous if f−1(B) ∈ τ for any
covering based rough open (resp. closed) set B ⊆ Y . That is, if the inverse image of any
subset B of Y being covering based rough open (resp. closed) subset of Y is a subset of X
such that f−1(B) ∈ τ .

Now, we define the concept of a covering based rough closed (resp. open) map. A
covering based rough continuous map f : X → Y is called a covering based rough
closed (resp. open), if for every covering based rough closed (resp. open) set A ⊂ X,
the image f(A) is a covering based rough closed (resp. open) in Y . A covering based
rough map, which is both covering based rough closed and covering based rough open, is
called a covering based rough clopen map.

A covering based rough continuous map f : X → Y is called a homeomorphism if
f maps X onto Y in a one-to-one way and the inverse map f−1 of Y to X is a covering
based rough continuous map. We say that two covering based rough topological spaces X
and Y are homeomorphic if there exists a homeomorphism of X onto Y . Also, a covering
based rough homeomorphism is always an isomorphism, but the converse may be not true.

If there is a covering based rough homeomorphism from X into Y , then there is one-
to-one corresponding between lower (resp. upper) approximations. Then we can say there



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 541

exist isomorphisms fl and fu such that fl : X → Y and fu : X → Y .

If there are lower and upper isomorphisms fl : X → Y and fu : X → Y , then the pair
f = (fl, fu) is not always a homomorphism from X to Y .

The Product of covering based rough topologies

Let U and V be non-empty universes and let C and D be a covering of U and V such
that C and D satisfy the approximation condition. Let (X, τC) and (Y, τD) be covering
based rough topologies. Consider the product cartesian of C and D as follows:

C ×D = {l × k : l ∈ C, k ∈ D}.

Let x, y be two representative elements of l and x
′
, y
′

be two representative elements of
k. Then (x, x

′
), (y, y

′
), (x, y

′
), (y, x

′
) are representative elements of l × k. So that, C ×D

satisfies the approximation condition.

In addition, assume that for every AC ×AD, BC ×BD ⊆ C ×D such that AC ×AD ⊆
BC×BD, there exists a set X×Y ⊆ U×V with AC×AD = X × Y and BC×BD = X × Y .
Now, suppose that X × Y ⊆ U × V . Then define

(i) X × Y = {l × k ∈ C ×D : l × k ⊆ X × Y };

(ii) X × Y = {l × k ∈ X × Y : l × k ∧X × Y 6= φ};

(iii) BN(X × Y ) = X × Y −X × Y .

Let A,B ⊆ U × V . We can define the union, the intersection and the complement of
two sets in U × V as follows:

(i) A ∨B = (A ∨B,A ∨B);

(ii) A ∧B = (A ∧B,A ∧B);

(iii) A
′

= (C ×D \A,C ×D \A).

Definition 9. Let X ⊆ U and Y ⊆ V such that (X, τC) and (Y, τD) are covering based
rough topological spaces on the coverings C and D, respectively. Then the set {A × B :
A ∈ τC , B ∈ τD} is a base for the product covering based rough topology X × Y on C ×D.

Now, let W be a covering based rough open set in the product topology τX×Y . Consider
the pair (X1, X2) ∈ W , where X1, X2 are exact elements in X and Y , respectively. So
there exist WX ∈ τX and WY ∈ τY such that X1 ∈ WX and X2 ∈ WY . This implies that
X1 6= φ 6= X2 and X1 ⊆C WC , X2 ⊆D WY . Then ∃KC ∈ C,KD ∈ D with X1 = {KC} and
X2 = {KD}. Hence (X1, X2) 6= ∅ and X1×X2 ⊆C×D W . There exists KC ×KD ∈ C×D
with W = {KC ×KD}. We can then write (X1, X2) ∈WX ×WY ⊆W , where WX ×WY

is a covering based rough basic open set.



N. Alharbi, H. Aydi, C. Özel / Eur. J. Pure Appl. Math, 12 (2) (2019), 533-543 542

Remark 2. Consider the product covering based rough topology X × Y on C × D. Let
A ⊂ X and B ⊂ Y . Then

(i) cint(A×B) = cint(A)× cint(B);

(ii) ccl(A×B) = ccl(A)× ccl(B).

Properties of Covering Based Rough Topology

Let (U,C) be a covering approximation space. Suppose that X and Y are rough subsets
of U such that Y ⊆ X. Consider the covering based rough topology (X, τ). We generalize
the definition of density from classical rough topology to covering based rough topology.

Definition 10. The rough subset Y of X is dense in the covering based rough topology
(X, τ), if the upper approximation of Y is equal to the upper approximation of X, i.e.,
X = Y .

Generating Covering Based Rough Topology

Let X be a subset of the universe U such that the covering based rough set of X is
XC = (X,X). Let O be a family of subsets of XC that satisfies the base conditions.

Definition 11. The set τ consisting of all possible unions of members of O together
with ∅C = (∅, ∅) is called a generating covering based rough topology and we write
τ =< O >, where O is the base of τ .

Now, let τ1 =< O1 > and τ2 =< O2 > be two generating covering based rough
topologies on the rough set XC . Then we say that τ1 is coarser than τ2 or τ2 is finer than
τ1 and write τ1 ⊆ τ2 if and only if for every AC ∈ τ1 and for every Y ∈ AC , there exists
B ∈ O2 such that Y ∈ B ⊆ AC . In addition, we write τ1 = τ2 if and only if for every
AC ∈ τ1, BC ∈ τ2, Y ∈ AC and Y

′ ∈ BC , there exist K ∈ O1 and K
′ ∈ O2 such that

Y ∈ K ′ ⊆ AC and Y
′ ∈ K ⊆ BC .

Covering Based Nano Space

In this section, we extend the definition of classical nano topology to include a rough
set X determined by a covering C.
Let (U,C) be a covering based approximation space. Similarly, we define the covering
based rough set X ⊆ U as X = (X,X,BN(X)). If Y ⊆ X, then Y ⊆ X,Y ⊆ X and
BN(Y ) ⊆ BN(X). Now, we are constructing a nano topology on the covering based
rough topology.

Definition 12. Let (U,C) be an approximation space and τ be the family of all rough
subsets of X = (X,X,BN(X)) satisfying the following conditions:

(i) X, ∅ ∈ τ ;



REFERENCES 543

(ii) It is closed under finite intersection;

(iii) It is closed under arbitrary union.

Then τ is called a covering based nano topology on a given rough set.

In covering based approximation space, we also find that the nano topology coincides
with the rough topology. The reasons are similar with the ones in a classical approximation
space.

Acknowledgements

The authors wish to thank the Deanship for Scientific Research (DSR) at King Abdu-
laziz University for financially funding this project under grant no. KEP-PhD-2-130-39.

Competing Interests

The authors declare that they have no competing interests.

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