Chapter 5 TOPOLOGICAL ASPECTS OF YAO S ROUGH SET In this chapter, we introduce the concept of transmissing right neighborhood via transmissing expression of a relation R on domain U, and then we study various topological properties of Yao s rough set defined through right neighborhood. 5.1 Introduction Rough set was introduced by Pawlak [44] in 1982 as a tool to deal with vagueness and uncertainty of imprecise data. Considerable amount of works have been done on fundamental results of rough set. Partitioning of a set with an equivalence relation is the core concept behind Pawlak s rough set theory. But this is too restrictive to deal with different real life situations. To handle such type of circumstances, several interesting and meaningful extensions of Pawlak rough set theory like, covering based, relation based and neighborhood based rough sets ([30], [31], [32], [33] etc.) have been introduced where covering or cover (it is a finite family of non empty subsets Contents of this chapter has appeared as a paper entitled Topological Properties of Yao s Rough Set in International Journal of Mathematical and Computer Sciences 7(2), 2011, 44 46. 56
5.2. BASIC CONCEPTS 57 of the universe whose union is the universe), binary relation and right neighborhood respectively are used instead of the equivalence relation. Y.Y. Yao [64] introduced a new type of rough set based on right neighborhoods and E.F. Lashin et al. [26] have generated a topology for the rough set defined by Yao considering the family S = {xr x U} as subbase for the topology, where R is a binary relation on a finite universe U and xr is the right neighborhood of the element x defined as xr = {y xry}. The family S as the subbase for the induced topology τ will be denoted by S R = {xr x U}. In [29], Z. Li has introduced the concept of transmissing expression of a relation, and obtained several interesting results on topological concepts of rough sets. Topology has its own theory and own significance. Rough set theory combined with topology is expected to provide us with new area to study. 5.2 Basic concepts If a relation is reflexive then the right neighborhood of an element will contain the element. Let us call this right neighborhood as reflexive right neighborhood. We review below the lower and upper approximations as defined in [64] and the notion of transmissing expression of a relation as introduced in [29]. Definition 5.2.1. [64] Let U be the universe and R U U be a binary relation. For two elements x, y U, we say that y is R-related to x if xry. Then the set r R (x) = {y U xry} is called the right neighborhood of x U. Definition 5.2.2. [64] Let U be the universe and R U U be a binary relation. For any X U, the lower and upper approximations of X are respectively X =
5.2. BASIC CONCEPTS 58 r R (x) and X = (X c ) c. If X = X then X said to be an exact set and otherwise r R (x) X a rough set. Example 5.2.3. Let U = {a, b, c, d} and R = {(a, b), (a, c), (b, d), (c, a), (d, b)} be a binary relation on U. If X = {a, c}, then X = {a} and X = {a, c}. Definition 5.2.4. Let R be a binary relation on X, then (i) R is called a similarity relation on X if R is both reflexive and transitive; (ii) R is called a tolerance relation on X if R is both reflexive and symmetric. Definition 5.2.5. [29] Let R and R S be two binary relations on X and A X. For all x, y X, we define xr S y iff xry or { v 1, v 2,..., v n } A such that xrv 1, v 1 Rv 2,..., v n Ry. Then R S is called the transmissing expression of R on A. If R S is the transmissing expression of R on X then R S is called transmissing expression of R. Example 5.2.6. In Example 5.2.3., arb and brd ar s d, arc and cra ar s a, brd and drb br s b, cra and arc cr s c, drb and brd dr s d. Here R s is not a transmissing expression of R, since for all x, y U we do not have xr s y. Proposition 5.2.7. [29] Let R be a reflexive relation on X and R S be the transmissing expression of R. Then for all x, y X, xr S y if and only if { v 1, v 2,..., v n } X such that xrv 1, v 1 Rv 2,..., v n Ry. We refer to [41] for various concepts of a topological space.
5.3. TRANSMISSING RIGHT NEIGHBORHOOD 59 5.3 Transmissing right neighborhood In this section, we introduce transmissing right neighborhood of an element by considering a reflexive relation. Then we examine a few topological notions like, compactness, connectedness etc. for the transmissing right neighborhood of an element. Let U be a universal set and R be a reflexive binary relation on A U. Suppose τ R = { X U X = X} is a collection of subsets of U. Then it can be easily shown that (U, τ R ) is a topological space. Example 5.3.1. Let U = {a, b, c, d} and R = {(a, a), (b, b), (c, c), (d, d), (a, c), (b, d), (a, d), (c, d)} be a reflexive binary relation on U. Now r R (a) = {a, c, d}, r R (b) = {b, d}, r R (c) = {c, d} and r R (d) = {d}. Then τ R = { X U X = X} = {φ, {d}, {b, d}, {c, d}, {a, c, d}, {b, c, d}, U} is a topology on U and (U, τ R ) is a topological space. Proposition 5.3.2. If R is a reflexive relation on U, then for all A U, A = A A c = A c. Proof. Let A = A. It will suffice to prove that A c A c. Let y A c y / A y / r R (x) x such that y r R (x) but r R (x) A r R (x) A x such that y r R (x) A c y r R (x) y A c A c A c. r R (x) A c Proposition 5.3.3. Let R be a reflexive relation on U, then the topological space (U, τ R ) has the property that A U is open if and only if A is closed. Proof. A U be an open set A = A A c = A c A c is an open set. A is a closed set. Definition 5.3.4. Let R S be the transmissing expression of the reflexive relation R on U. Then the transmissing right neighborhood of x X is defined as r RS (x) = {y U xr S y}.
5.3. TRANSMISSING RIGHT NEIGHBORHOOD 60 Lemma 5.3.5. Let R be a reflexive relation on U and for each x U, let L x = {y U y r RS (x)}. Then (i) x L x ; (ii) L x τ R ; (iii) {L x } is an open neighborhood base at x ; (iv) L x is compact subset of (U, τ R ); (v) B R = {L x x X} is a base of (U, τ R ). Proof. (i) Since R is a reflexive binary relation, x L x. (ii) It is sufficient to show L x L x. Let y L x. Then { v 1, v 2,..., v n } U such that xrv 1, v 1 Rv 2,..., v n Ry y r R (v n ). For z U, z r R (v n ) v n Rz. So, xrv 1, v 1 Rv 2,..., v n R z xr S z z r (x) z L R S x r R (v n ) L x y r R (v n ) y L x L x L x. r R (v n) L x (iii) Here we prove that for each B τ R and x B, L x B. Let y L x y r (x) y r R S R(x) or { v 1, v 2,..., v n } U such that v 1 r R (x), v 2 r R (v 1 ),..., y r R (v n ). Now x B = B = r R (t), for some t U. r R (t) B If y r R (x), then we claim y B. For otherwise, y B c y B c y r R (m) B c r R (m) for some m U If y is in some right neighborhood of m then that right neighborhood is contained in B c. i.e., y r R (x) r R (x) B c. But xrx x r R (x) and r R (x) B c x B c, which is a contradiction. Hence we have y B. If { v 1, v 2, v 3,..., v n } U such that v 1 r R (x), v 2 r R (v 1 ),..., y r R (v n ), then by the above argument we have, v 1 r R (x) v 1 B; v 2 r R (v 1 ) v 2 B;... ; y r R (v n ) y B.
5.3. TRANSMISSING RIGHT NEIGHBORHOOD 61 (iv) Let {G λ λ Λ} be an open covering of L x x G λi for some λ i Λ. Then by (iii) L x G λi. Hence L x is a compact subset of (U, τ R ). (v) Similar to that of (iii). Theorem 5.3.6. (U, τ R ) = (U, τ R S ). Proof. From Lemma 5.3.4.(vi), we have B R = {L x x X} is a base of (U, τ R ). We need to show that B R = {L x x X} is also a base of (U, τ R ). For this, we have to S show that (i) L x τ R S, and (ii) For any x G τ RS, x L x G. (i) We have r (x) L R S x. Let y L x y r R (x). For each z U, S r RS (x) L x z r (x) z L R S x y r (x) L R S x r (x). R S r RS (x) L x r RS (x) L x (ii) Suppose x G τ. Now for each y L R S x y r R (x). Since x G = S r (x), by (iii) of Lemma 5.3.4, y G, i.e., x L R S x G. r RS (x) G Theorem 5.3.7. If (U, τ R ) is a topological space induced by a reflexive relation R, then (i) (U, τ R ) is a first countable space; (ii) (U, τ R ) is a locally compact space. Proof. Follows from (iii) and (v) of Lemma 5.3.4. respectively. Theorem 5.3.8. If (U, τ R ) is a topological space induced by a tolerance relation R, and R S be the transmissing expression of R, then (i) cl({x}) = L x, where cl({x}) is the closure of {x}; (ii) L x is a connected branch that contains x;
5.3. TRANSMISSING RIGHT NEIGHBORHOOD 62 (iii) L x is a separable subset of (U, τ R ). Proof. Let (U, τ R ) be a topological space induced by a tolerance relation R and R S is the transmissing expression of R R S is an equivalence relation. Then L x = {y U y r R S (x)} is an equivalence class for each x U, i.e., L x = [x] R S. (i) Suppose there exists y cl({x}) such that y / L x. Then [x] R S [y] RS = φ for an open neighborhood L y of y, {x} L y = φ y / cl({x}) which contradicts our supposition. Hence we must have y cl({x}) y L x cl({x}) L x. On the other hand, let y L x y [x] R S L x = L y. Suppose G is a neighborhood of y. So, L y G L y G φ L x G φ {x} G φ y cl({x}) L x cl({x}). (ii) First we shall show that L x = {y U y r R (x)} is a connected set. Let A S be a non-empty clopen subset of L x. So, there exists y A L x = [x] R S. Then y L y L y = A [y] R S = [x] RS = L y = L x = A. Hence L x is a connected set. Next, let C x be any connected branch that contains x. We shall show L x = C x. If not, L x is an open and closed proper subset of C x, which contradicts the fact that C x is a connected set. Hence L x = C x. (iii) Form (i) we have {x} is a countable dense subset of L x. Hence L x is a separable subset of (U, τ R ). Theorem 5.3.9. If R is a tolerance relation, then (i) (U, τ R ) is a regular space; (ii) (U, τ R ) is a normal space; (iii) (U, τ R ) is a locally connected space; (iv) (U, τ R ) is a locally separable space.
5.4. CONCLUSION 63 Proof. (i) Let A be a closed subset of U and x A c. Then A and A c are also two open subsets of U such that A A and x A c. Hence (U, τ R ) is a regular space. (ii) If A and B are two disjoint closed subsets of U then they are also two disjoint open subsets of U. Hence (U, τ R ) is a normal space. (iii) We have from (iii) of Lemma 5.3.4, every open neighborhood of x contains the open neighborhood L x of x which is connected. (iv) Follows from (iii) of Theorem 5.3.7. 5.4 Conclusion Here a study on various topological structures of rough sets is undertaken by introducing transmissing right neighborhood. We have considered the rough set defined by Yao as his definition of rough set is based on the concept of neighborhood, and taken those sets whose lower approximation is equal to itself, to get a topology on the universe of discourse. Then it was found that the set L x, which is the set of the elements belonging to transmissing right neighborhoods of x, is an open set satisfying notions like, compactness and connectedness etc.