Journal of Software Engineering and Applications, 2013, 6, 1-6
doi:10.4236/jsea.2013.67B001 Published Online July 2013 (http://www.scirp.org/journal/jsea)
New Topological Approaches for Data Granulation
A. S. Salama, O. G. Elbarbary
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.
Received March, 2013
ABSTRACT
Data granulation is a good tool of decision making in various types of real life applications. The basic ideas of data
granulation have appeared in many fields, such as interval analysis, quantization, rough set theory, Dempster-Shafer
theory of belief functions, divide and conquer, cluster analysis, machine learning, databases, information retrieval, and
many others. In this paper, we initiate some new topological tools for data granulation using rough set approximations.
Moreover, we define some topological measures of data granulation in topological I formation systems. Topological
generalizations using δβ-open sets and their applications of information granulation are developed.
Keywords: Knowledge Granulation; Topological Spaces; Rough Sets; Data Mining; Decision Making; Fuzzy Sets
1. Introduction
Granulation of the universe involves the decomposition
of the universe into parts. In other words, the grouping
individual elements or objects into classes, based on
offering information and knowledge [7,14,21,40,44,45].
Elements in a granule are pinched together by indiscerni-
bility, similarity, proximity or functionality [42,43]. One
can thus form a granulated view of the universe. The
theory of rough sets can be used for constructing a
granulated vision of the universe and for interpreting,
representing, and processing concepts in the granulated
universe. It offers a more actual model of granular
computing. The starting point of the theory of rough sets
is the indiscernibility of objects or elements in a universe
of concern [15-22]. The customary approach for model-
ing indiscernibility of objects is through an equivalence
relation defined based on their attribute values with
reference to an information system [16]. Two objects are
comparable if they have exactly the same description.
The induced granulation is a part of the universe, i.e., A
family of pairwise disjoint subsets. It is studied widely in
mathematics under the name of quotient set. The notion
of indiscernibility can be generalized by any general
binary relation.
The original rough set theory was based on an
equivalent relation on a finite universe U. For practice
use, there have been some extensions on it. One
extension is to replace the equivalent relation by a
arbitrary binary relation ; the other direction is to study
rough set via topological method [8,14]. In this work, we
construct topology for a family covering rough sets.
After that, we study the relationship among upper
approximations based on this topological space so that
we can study data granulation by method of topology.
In [40] Y.Y. Yao addressed four operators on a
knowledge base, which are sufficient for generating new
knowledge structures. Also, they addressed an axiomatic
definition of knowledge granulation in knowledge bases.
Rough set theory, proposed by Pawlak in the early 1980s
[17-20], is an expansion of set theory for the study of
intelligent systems characterized by inexact, uncertain or
insufficient information. Moreover, this theory may serve
as a new mathematical tool to soft computing besides
fuzzy set theory [42-45], and has been successfully
applied in machine learning, information sciences, expert
systems, data reduction, and so on [30-38,40]. In recent
times, lots of researchers are interested to generalize this
theory in many fields of applications [1-6,16,22].
But, partition or equivalence relation is still limiting
for many applications. To study this matter, several
interesting and having an important effect generalization
to equivalence relation have been proposed in the past,
such as tolerance relations, similarity relations [51],
topological bases and subbases [1,2,6,22,23,27-29] and
others [3,4,16,20,25,26,33]. Particularly, some researchers
have used coverings of the universe of discourse for
establishing the generalized rough sets by coverings
[11,15]. Others [9-12,16,27] combined fuzzy sets with
rough sets in a successful way by defining rough fuzzy
sets and fuzzy rough sets. Furthermore, another group
has characterized a measure of the roughness of a fuzzy
set making use of the concept of rough fuzzy sets
[9,10,14,15,21,41,42]. They also suggested some possi-

New Topological Approaches for Data Granulation 2
ble real world applications of these measures in pattern
recognition and image analysis problems [13,24,30].
Topology is a significant and interesting area of
mathematics, whose study introduces you to new
concepts ( semi-open, pre open, δβ-open sets and others)
and theorems, which are very useful in many applica-
tions .Topological notions like semi-open, preopen,

open sets are as basic to mathematicians of today as
sets and functions were to those of last century
[14,15,17]. Then, we think the topological structure will
be so important base for knowledge extraction and
processing.
The topology induced by binary relations on the
universes of information systems is used to generalize
the basic rough set concepts. The suggested topological
operations and structure open up the way for applying
affluent more of topological facts and methods in the
process of granular computing. In particular, the notion
of topological membership function is introduced that
integrates the concept of rough and fuzzy sets [17,42-45].
2. Essentials of Rough Set Approximations
under General Binary Relations
For any approximation space (,)
A
UR, where is
an equivalence relation, lower and upper approximations
of a subset
R
X
U, namely ()RX and ()RX are
defined as follows:
(){ :[]}
R
RXx UxX ,
(){ :[]}
R
RXx UxX

 .
The lower and upper approximations have the
following properties:
For every ,
X
YU from the approximation space
(,)
A
UR we have:
1. ()(),
2.()( ),
3.()( ),
4.()()(),
RXX RX
RURU U
RR
RXYRXRY





5.()()( ),
6.()()( ),
7.()()( ),
8. ()( ),
9. ()(),
10.( ( ))( ( ))( ),
11.(())(())(),
12.,()()
() ().
RX YRXRY
RX YRXRY
RX YRXRY
RX RX
RX RX
RRXRRX RX
RRX RRXRX
IfXYthenRXR Y
andRXR Y












The equality in all properties happens when
() ()RX RXX. The proof of all these properties can
be found in [17-20].
Furthermore, for a subset
X
U, a rough membership
function is defined as follows: []
() []
R
XR
x
X
xx  ,
where
X
denotes the cardinality of the set
X
. The
rough membership value ()
X
x

may be interpreted as
the conditional probability that an arbitrary element
belongs to
X
given that the element belongs to []
R
x
.
Based on the lower and upper approximations, the
universe can be divided into three disjoint regions,
the positive , the negative and the
boundary , where:
U
()POSX
()BND X()NEG X
() ()
() ()
()() ()
POSXRX
NEG XUR X
BND XR XR X



Considering general binary relations in [18,52] is an
extension to the classical lower and upper approxima-
tions of any subset
X
of U. {: }
x
RxX

 is the
base generated by the general relation defined in [18,52].
The general forms based on

are defined as follows:
() {:,}
x
RXBBB X



,
() {:,}
x
RXBB BX 



,
where {:
xBxB}



.
For data granulation by any binary relation, in [8] a
rough membership function is defined as follows:
()
()
x
Xx
X
x

 
 .
3. Data Granulation Using Equivalence and
General Binary Relations
By generalizing equivalence relations to general binary
relations, one may obtain a different granulation of the
universe. For any kind of relations, a pair of rough set
approximation operators, known as lower and upper ap-
proximation operators can be defined in many ways
[17-20].
Let be an equivalence relation on finite
non-empty universe U. The equivalence class,
R
RUU
{:(,[]) }
x
yU xyR

 consists of all elements
equivalent to
x
, and is also the equivalence class con-
taining
x
. The relation R induces a partition of the uni-
verse namely U/{[]:
R
x xU}UR

.
The partition is regularly known as the quotient
set and provides a granulated view of the universe under
the equivalence classes of U. Naturally speaking, the
available knowledge only allows us to talk about an
equivalence class as a single unit. In other words, under
the granulated view, we consider an equivalence class as a
whole instead of individuals. The pair
/RU
(,)
A
UR

is
referred to as an approximation space, indicating the in-

New Topological Approaches for Data Granulation 3
tended application of the partition for approxima-
tion. Each equivalence class is called a simple granule.
/UR
A topological space [1,2,25] is a pair (,)X

consisting of a set
X
and a family

of subset of
X
satisfying the following conditions:
(1) ,X

 ,
(2)

is closed under arbitrary union,
(3)

is closed under finite intersection.
The pair (,)X

is called a topological space. The
elements of
X
are called points . The subsets of
X
belonging to

are called open sets. The complement of
the open subsets are called closed sets. The family

of
all open subsets of
X
is also called a topology for
X
.
is called () ={ai}clAFnd F
:F Xs closedA

-
closure of a subset
A
X.
Obviously, is the smallest closed subset of ()cl A
X
which contains
A
. Note that
A
is closed iff
=()
A
cl A. ()=intA {:ais openGX ndG
}GA
is called the

-interior of a subset
A
X. Manifestly,
is the union of all open subsets of
()int A
X
which
contained in
A
. Make a note of that
A
is open iff
=()
A
int A. is called the ()= ()(clA int)AbA

-
boundary of a subset
A
X.
For any subset
A
of the topological space (,)X

,
, and are closure, interior, and
boundary of
()cl A()int A()bA
A
respectively. The subset
A
is exact if
()bA=

, otherwise
A
is rough. It is clear that
A
is
exact iff . In Pawlak space a subset ()cl A=int()A
A
X has two possibilities either rough or exact.
In later years a number of generalizations of open sets
have been considered [22,23]. We talk about some of
these generalizations concepts in the following defini-
tions.
Let be a finite universe set and is any binary
relation defined on , and be the set of all el-
ements which are in relation to certain elements
U R
)xU(rR
x
in
from right for all U
x
U
;

,)
, in symbols
where
() {}xxR U,xrR
{:(, }
x
RyxyRxyU.
Let

be the general knowledge base (topological
base) using all possible intersections of the members of
. The component that will be equal to any union of
some members of
(rR x)

must be misplaced.
4. Topological Granulation of Topological
Information Systems
Let (,)
A
UR be an approximation space where is
any binary relation defined on . Then we can define
two new approximations as follows:
R
U
()( ())
X
XR
 
 RX  ,
()( ())
X
XR
 
 RX

.
The topological lower and the topological upper ap-
proximations have the following properties:
For every ,
X
YU and every approximation space
(,)
A
UR

we have:
1. () ()
X
XX


  ,
2. () ()UU U


  ,
3. () ()




,
4. ()()()
X
YX

  Y

,
5. ()()()
X
YX

Y


  ,
6. ()()()
X
YX

 Y

,
7. ()()()
X
YX

Y




 ,
8. () ()
X
X

  ,
9. () ()
X
X




 ,
10. (()) ()
X
X
 
  ,
11. (()) ()
X
X
 



,
12. ,()()
I
fXY thenXY

 
() ().and XY


 
Given that topological lower and topological upper
approximations satisfy that:
() ()()()RXXXXRXU



 
this enables us to divide the universe into five dis-
joint regions (granules) as follows:
U
1. () ()POS XR X
 

,
2. () ()()POSXXRX
 



 ,
3. ()() ()BND XXX

 ,
4. ()() ()NEGXRXX   ,
5. () ()NEG XURX


 .
The following theorems study the properties and rela-
tionships among the above regions namely boundary,
positive and negative regions.
Theorem 4.1 let (,, )
R
ISU A


be a topological in-
formation system and for any subset
X
U we have:
1. ()()BND XX





,
2. ()()BND XNEG X



 ,
3. () ()()
X
XBND

 X,
4. (), ()
X
NEG X




and ()BND X

 are dis-
joint granules of .
U
Proof: directly.
Theorem 4.2 let (,, )
R
ISU A


be a topological in-
formation system and for any subsets ,
X
YU we
have:
1. ()BND U



,
2. ()( )BNDXBNDUX



 ,
3. (())(BNDBND XBND X)



 ,

New Topological Approaches for Data Granulation 4
()() (BND XYBND XBNDY)

 
Proof: (1) and (2) is obvious, by definitions.
(())
(()())
(() ())
((() ())
() ()
()()
()()
BNDBND X
BNDXU X
XUX
UXUX
XUX
BND XBND XY
)
X
YUXY

 
 











 

 

 

Theorem 4.3 let (,, )
R
ISU A

 a topological infor-
mation system and for any subset ,
X
YU we have:
1. ()UNEG

  ,
2. ()()NEGXUX


 ,
3. ()XNEGX

  ,
4. (())NEGUNEGXNEGX()

  ,
()() (NEGXYNEGXNEGY)

 
()()(NEGXYNEG XNEGY)

 
Proof: (1), (2), (3) and (4) are obvious.
()
() (()(
(())(())
() ()
() ()
(())(())
(() ())
() ().
NEGXY
UXYU XY
UXUY
NEGXNEG Y
NEGXNEG Y
UXUY
UXY
UXY NEGXY





))










 
 
 

 
 
 

Example 4.1 let 1234567
be the
universe of 7 patients have data sheets shown in Table 1
with possible dengue symptoms. If some experts give us
the general relation
{,,,,,,}U uuuuuuu
R
defined among those patients as
follows:
1117223 3
36 4455 6677
{( ,),( ,),(,),(,),
(,),(,),(,),(,),( ,)}.
R uuuuuuuu
uu uuuuuu uu

Table 1. Patients information system.
Conditional Attributes ( C) Decision (D)
Patients (U) Temperature Flu Headache Dengue
u1 Normal NoNo No
u2 High NoNo No
u3 Very High NoNo Yes
u4 High NoYes Yes
u5 Very High NoYes Yes
u6 High Yes Yes Yes
u7 Very High Yes Yes Yes
The topological knowledge base will take the follow-
ing form:
172364567
{{,},{ },{ ,},{ },{},{ },{ }}uuuuuuuuu


.
For some patients 237 {, , }
X
uuu

the upper and
lower approximations based on the topological
knowledge base are given by:
12 3 67
(){, ,,,}RX uuuuu

, and 27 {,}Ruu

.
By using the lower and upper approximations, the
granules of universe are three disjoint regions as follows:
27
() (){,}POSXRXuu
 

,
136
()() (){,,}BNDXRXRXu uu 

 ,
45
() (){,}NEGXURXuu


 .
According to the topological knowledge base we can
easily see that:
1237
(){,,,}
X
uuuu

 , 237 () {,,}
X
uuu


.
Then we have the following granules of the universe:
1. (){2,7}POSXu u


,
2. (){3}POSXu


,
3. (){1}BND Xu


,
4. () {6}NEGXu


,
5. (){4,5}NEGXu u


5. Conclusions and Application Notes
The rough set approach to approximation of sets leads to
useful forms of granular computing that are part of com-
putational intelligence. The basic idea underlying the
rough set approach and their topological generalizations
to information granulation are to discover to what extent
a given set of objects (these objects can be pixels of an
image) approximates another set of objects of interest.
Objects of definite universe are compared by considering
their descriptions. The recent generalization of rough set
theory has led to the introduction of topological rough set
approaches [24-26,35] and a consideration of the affini-
ties (topological nearness) of objects.
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