Rough Connected Topologized. Approximation Spaces

Transcription

1 International Journal o Mathematical Analysis Vol no HIARI Ltd Rough Connected Topologized Approximation Spaces M. J. Iqelan Department o Mathematics Faculty o Science Al-Azhar University - Gaza P.O. Box 77 Gaza Palestine Copyright 04 M. J. Iqelan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract The topological view or connectedness is more general and is applied or topologies on discrete sets. Rough thinking is one o the topological connections to uncertainty. The purpose o this paper is to hybridize connectedness and rough set notions to get a new approach or handling uncertainty. The deinition o connectedness in approximation spaces is introduced. Mathematics Subject Classiication: 54C05 54D05 eywords: Topological space Topologized approximation space Rough connected topologized approximation space. Preliminaries This section presents a review o some undamental notions o topological spaces and rough set theory. A topological space [3] is a pair X consisting o a set X and amily o subsets o X satisying the ollowing conditions: (T) and X. (T) is closed under arbitrary union. (T3) is closed under inite intersection.

2 60 M. J. Iqelan Throughout this paper X denotes a topological space the elements o X are called points o the space the subsets o X belonging to are called open sets in the space the complement o the subsets o X belonging to are called closed * sets in the space and the amily o all closed subsets o X is denoted by the amily o open subsets o X is also called a topology or X. A subset A o X in a topological space X is said to be clopen i it is both open and closed in X. A amily is called a base or X i every nonempty open subset o X can be represented as a union o subamily o. Clearly a topological space can have many bases. A amily S is called a subbase i the amily o all inite intersections o S is a base or X. The -closure o a subset A o X is denoted by A and it is given by * A F X : A F and F }. Evidently A is the smallest closed subset o X which contains A. Note that A is closed i A A. The -interior o a subset A o X is denoted by A and it is given by A G X : G A and G}. Evidently A is the largest open subset o X which contained in A. Note that A is open i A A. Motivation or rough set theory has come rom the need to represent subsets o a universe in terms o equivalence classes o a partition o that universe. The partition characterizes a topological space called approximation space X where X is a set called the universe and R is an equivalence relation [5 6]. The equivalence classes o R are also known as the granules elementary sets or blocks. We shall use R x to denote the equivalence class containing x X and X / R to denote the set o all elementary sets o R. In the approximation space X the upper (resp. lower) approximation o a subset A o X is given by RA x X R A resp. RA x X : R A. : x x Pawlak noted [6] that the approximation space X R with equivalence relation R deines a uniquely topological space X where is the amily o all clopen sets in X and X / R is a base o. Moreover the lower ( resp. upper ) approximation o any subset A o X is exactly the interior ( resp. closure ) o A. I R is a general binary relation then the approximation space X R X where is the topology associated is the amily o all open sets in X and S xr : x X is a where xr y X : x R y) [ 4]. deines a uniquely topological space to (i.e. subbase o

3 Rough connected topologized approximation spaces 6 Deinition. []. Let X R be an approximation space with general is the topology associated to. Then the triple X relation R and is called a topologized approximation space. Deinition. []. Let X be a topologized approximation space. I A X then the lower approximation (resp. upper approximation ) o A is deined by R A A (resp. R A A ). Proposition. []. Let X be a topologized approximation space. I A and B are two subsets o X then i) R A A R A. ii) R R and R X R X X. iii) R( A B) R A R B. iv) R( A B) R A R B. v) I A B then R A R B. vi) I A B then R A R B. vii) R( A B) R A R B. viii) R( A B) R A R B. ix) RX A X R A. x) R( X A) X R A.. Rough Connected Topologized Approximation Spaces The present section is devoted to introduce the concept o rough connectedness in approximation spaces with general binary relations. The ollowing deinition introduces the concept o rough disconnected topologized approximation space. Deinition.. Let X be a topologized approximation space. Then is said to be rough disconnected i there are two nonempty subsets A and B o X such that The space X A B X and A R B R A B. is said to be rough connected i it is not rough disconnected. The ollowing deinition introduces concepts o deinability or a subset A X. o X in a topologized approximation space

4 6 M. J. Iqelan Deinition. []. Let X A X. Then be a topologized approximation space and i) A is called totally R deinable (exact) set i R A A R A ii) A is called internally R deinable set i A R A iii) A is called externally iv) A is called R deinable set i A R A R indeinable (rough) set i A R A and A R A. Remark.. Let X A X. be a topologized approximation space and I A is exact set then it is both internally R deinable and externally R deinable set. R A is the largest internally R deinable set contained in A. R A is the smallest externally R deinable set contains A. Lemma.. Let X be a topologized approximation space and A X. Then i) A is exact set i and only i X A is exact. ii) A is internally R deinable (resp. externally R deinable) set i and only i X A is externally R deinable (resp. internally R deinable) set. Proo. i) A is exact set i and only i R A A R A i and only i X R A X A X R A i and only i RX A X A RX A i and only i by Proposition. X A is exact. ii) We shall prove that A is internally R deinable set i and only i X A is externally R deinable and the other case can be proved similarly. Now A is internally R deinable set i and only i A R A i and only i X A X R A i and only i X A RX A i and only i by Proposition. X A is externally R deinable. Proposition.. Let X has a nonempty exact proper subset A then X disconnected. be a topologized approximation space. I X is rough

5 Rough connected topologized approximation spaces 63 Proo. Suppose that A is a nonempty exact proper subset o X. Then by Lemma. we get B X A is also a nonempty exact proper subset o X. Hence A B X and A R B A B R A B. X is rough disconnected. Thus Example.. Let X X a b c d} and ( a a)( b b)( c c)( a d) be a topologized approximation space such that R. Then ar a d} br b} cr c} dr b} c} a d} X b} c} a d} and S * X b} c} a d} b c} a c d} a b d} Hence the space X. is rough disconnected because A a d} is a nonempty exact proper subset o X. Proposition.. I the topologized approximation space X is rough disconnected then there is a nonempty exact proper subset o X. Proo. Let X be a rough disconnected topologized approximation space. Then there exist two nonempty subsets A and B o X such that A B X and A R B R A B. But A R A hence A B. Thus A X B. Also A X R B since A R B and A R B A B X. Hence A R A and B R B. Similarly B R B and A R A. Thereore there exists a nonempty exact proper subset A o X. Theorem.. A topologized approximation space X is rough disconnected i and only i there exists a nonempty exact proper subset o X. Proo. By using Proposition. and Proposition. the proo is obvious. Deinition.3 []. Let X R Y R approximation spaces. Then a mapping R V R V or every subset V o Y in. be two topologized : is called rough continuous i In Deinition.3 inverse image. does not mean the inverse unction but it means the

6 64 M. J. Iqelan Theorem.. Let X R space : be a mapping rom a topologized approximation Y R. Then to a topologized approximation space the ollowing statements are equivalent. i) is rough continuous. ii) The inverse image o each internally R deinable set in. iii) The inverse image o each externally R deinable set in. iv) R A R A v) B R B R R or every subset A o X in. or every subset B o Y in. R Proo. (i)(ii) Let be rough continuous and let V be an internally in. Then R V V and V V R V R V Thereore deinable set in is internally deinable set in is externally R is a subset o X in. By (i) we get. Then deinable set V R V. But R V V. Hence V R V. V is internally R deinable set in. (ii)(i) Let A be a subset o Y in. Since R A A then R A A. Since A (ii) we get R A is internally Hence R A R A A R is internally R deinable set in then by R deinable set in contained in A since R A R deinable set contained in A. Thus R A R A subset A o Y in. Thereore is rough continuous. (ii)(iii) Let F be an externally we get internally Y F is internally is the largest internally or every R deinable set in then by Lemma. R deinable. Thus by (ii) we have Y F R deinable set in. Since Y F X F F R deinable set in. Hence F X is internally R deinable set in. Similarly we can prove (iii)(ii).. is then is externally

7 Rough connected topologized approximation spaces 65 A (ii)(iv) Let A be a subset o X in then R is an externally R deinable set in. Hence R A is internally R deinable set in. Thus by (ii) we get Y Y R A X R A R deinable set in and so R A is externally R A R A since A containing A in. Thus A externally deinable set containing A in. Hence R Thereore R A R A is internally R A R A R A or every subset A in. (iv)(v) Let B be a subset o Y in. Let A X in. By (iv) we get B R A R A R B R B Hence R A R B. Thus R A R B R B Thereore B R B R. or every subset B o Y in.. R deinable set R is the smallest then A is a subset o (v) (ii) Let G be an internally R deinable set in then B Y G is externally deinable set in. Thus by (v) we get R Since B is externally R B R B R. R deinable set then R B B. Thus B B. But B R B then R B B B is externally Since B Y G X G R deinable set in. then X G R deinable set in. Thereore G is internally Example.. Let X R x3 ( x x )( x x)( x x ) R ( y y)( y y)( y y X x} x x} Y y} y y} : ( x ) ( x) y and ( x3) y Y R.. Hence is externally R deinable set in. be two topologized approximation spaces such that X x x } Y y y } R and ) and y3. Then. Deine a mapping such that 3. Hence is rough

8 66 M. J. Iqelan continuous since R V R V illustrated in Table.. or every subset V o Y in as V Y R V ( V ) R ( ) R V V Y Y X X X y } y } y } x } x } x 3 x 3 x y 3 } x } y y 3 y3 y } y } x } x } x } y x x y } y } x } y } X X Table.: R V ( V ) R ( ) V and R V where V R R and are given in Example.. or every subset V o Y in Since is rough continuous then rom Table. we can see that the inverse image o each internally R deinable set in is internally R deinable set in. Example.3. Let X R x3 ( x x )( x x)( x x ) R ( y y)( y y)( y y3 X x} x x} Y y} y y3} : ( x ) y ( x) y and ( x3) y y y 3 Y R be two topologized approximation spaces such that X x x } Y y y } R and ) and. Then y3. Deine a mapping such that 3. Then is not a rough continuous mapping since V } is an internally R deinable set in but V x x } 3 Lemma.. Let X R is not an internally and Y R R deinable set in. be two topologized approximation spaces. I : is a rough continuous mapping then the inverse image o each exact set in is exact set in. Proo. Let A be an exact set in then A is both internally and externally A is both internally R deinable set in. Hence by Theorem. we get

9 Rough connected topologized approximation spaces 67 and externally R deinable set in. Thereore A is an exact set in. Theorem.3. Let X R approximation spaces and let X R onto Y. I connected. Y R be two topologized : be a rough continuous mapping o X Y R is also rough is rough connected then Proo. Assume that Y R is rough disconnected topologized approximation space. Then there exists a nonempty exact proper subset A o Y in. Since is rough continuous mapping rom X onto Y then by Lemma. we get A is a nonempty exact proper subset o X in. Thus is rough disconnected but this is a contradiction. Y R is rough connected. Thereore 3. Conclusions In this paper we used topological concepts to introduce the deinition o rough connected topologized approximation space. We connect rough set notions and topological spaces. Reerences [] M. E. Abd El-Monse A. M. ozae M. J. Iqelan Near approximations in topological spaces Int. Journal o Math. Analysis 4 (6) (00) [] M. J. Iqelan On topological structures and uncertainty Ph.D. Thesis Tanta Univ. Egypt 00. [3] J. elley General topology Van Nostrand Company 955. [4] E. Lashin A. ozae A. Abo hadra and T. Medhat Rough set theory or topological spaces International Journal o Approximate Reasoning 40 (005) [5] T. Y. Lin Topological and uzzy rough sets in: R. Slowinski (Ed.) Decision Support by Experience Application o the Rough Sets Theory luwer Academic Publishers 99 pp

10 68 M. J. Iqelan [6] Z. Pawlak Rough Sets Theoretical Aspects o Reasoning about Data luwer Academic Boston Received: October 04; Published: November 04