New Notion of ((-open Sets in Topological Spaces

International Journal of Computer Applications (0975 – 8887) Volume 177 – No.3, November 2017 New Notion of -open Sets in Topological Spaces Bishnupada Debnath Department of Mathematics, Tripura University, Pin-799022, Tripura, India ABSTRACT Nakaoka and Oda ([1] and [2]) initiated the notion of maximal open (resp. minimal closed) sets in topological spaces. Thereafter, in 2005, Cao,Ganster, Reilly and Steiner [4] introduced -open (resp. -closed) sets in general topology. In the present work, the author introduces new classes of open and closed sets called maximal -open sets, minimal -open sets, maximal -closed sets, minimal -closed sets, -semi maximal open and -semi minimal closed and investigate some of their fundamental properties. Keywords -open, -open, maximal (resp. minimal) -open, maximal (resp. minimal) -closed, -semi maximal open and semi minimal closed sets. 1. INTRODUCTION In 1970, Norman Levine [5] introduced the notion of generalized closed sets and its dual open sets in topological spaces. After him many authors concentrated in this directions and defined various types of generalized closed sets in that spaces. Generalized open sets play a very important role in General Topology and they are now the research topics of many topologists worldwide. Indeed a significant theme in General Topology and Real Analysis concerns the variously modified forms of continuity, separation axioms etc by utilizing generalized open sets. One of the most wellknown notions and also an inspiration source is the notion of -open and -open sets introduced by N. V. Velicko [3] in 1968. Since the collection of -open sets in a topological space (X,) forms a topology  on X, then the union of two -open sets is of course -open. Moreover,  =  if and only if (X,) is regular. F. Nakaoka and N. Oda in [1] and [2] introduced the notion of maximal open sets and minimal closed sets. Thereafter, J. Cao, M. Ganster, I. Reilly and M. Steiner in [4] introduced -open and -closed sets. The main purpose of the present paper is to introduce the concept of a new class of open sets called maximal -open sets, minimal -closed sets, -semi maximal open and -semi minimal closed sets. We also investigate some of their fundamental properties. 2. PRELIMINARY NOTE Throughout the work ordered pairs (X,) and (Y,) (or X and Y ) will denote topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of X. We denote the interior and the closure of a set A by Int(A) and Cl(A), respectively.Let us recall the following definitions which are useful in the sequel. Definition 2.1 [3]. Let A be a subset of a space X. A point xX is called a -cluster point of A if ACl(U)  , for every open set U of X containing x. The set of all -cluster points of A is called the -closure of A, denoted by Cl(A). Definition 2.2 [3]. A subset A of X is called -closed if A = Cl(A). The complement of a -closed set is called -open set in X. We denote the collection of all -open (respectively, -closed) sets by -O(X,) (respectively, -C(X,)). Definition 2.3 [3]. Let A be a subset of a space X. A point xX is called a -cluster point of A if AU  , for every regular open set U of X containing x. The set of all -cluster points of A is called the -closure of A, denoted by Cl(A). Definition 2.4 [3]. A subset A of X is called -closed if A = Cl(A). The complement of a -closed set is called -open set in X. We denote the collection of all -open (respectively, -closed) sets by -O(X,) (respectively, -C(X,)). It is very well-known that the families of all -open (resp. open) subsets of (X,) are topologies on X which we shall denote by  (resp.). From the definitions it follows immediately that     . The space (X, ) is also called the semi-regularization of (X,). A space (X,) is said to be semi-regular if  = and (X,) is regular iff  = . It is easily seen that one always has, A  Cl(A)  Cl(A)  Cl(A)  A , where, (X, ). A denotes the closure of A with respect to Definition 2.5 [1]. A proper nonempty open set U of X is said to be a maximal open set if any open set which contains U is either X or U. Definition 2.6 [2]. A proper nonempty closed set V of X is said to be a minimal closed set if any closed set contained in V is either  or V. The family of all maximal open (resp. minimal closed) sets will be denoted by MaO(X) (resp. MiC(X)). We define MaO(X, x) = {U : x U MaO(X) } and MiC(X,x) = {V : x  V  MiC(X)}. Definition 2.7[4]. A subset A of a topological space (X,τ), is called -closed [2] if Cl(A) ⊆ U whenever A ⊆ U and U is -open in X. The complement of -closed is -open set in X. 3. MAXIMAL -OPEN AND MINIMAL -CLOSED SETS In this section the notion of maximal -open set and minimal -closed sets are introduced and investigate some fundamental results with example and counter examples. Definition 3.1. (a) A proper nonempty -open set A of X is said to be a maximal -open set if any -open set U which contains A is either X or A (i.e. U = A or U = X, whenever, A  U). 33 International Journal of Computer Applications (0975 – 8887) Volume 177 – No.3, November 2017 (b) A proper nonempty -open set A of X is said to be a minimal -open set if any -open set U which is contained in A is either A or  (i.e. U = A or U = , whenever, U  A). Definition 3.2. (a) A proper nonempty -closed set B of X is said to be a maximal -closed set if any -closed subset U which contains B is either X or B (i.e. U = B or U = X, whenever, U B). (b) A proper nonempty -closed set B of X is said to be a minimal -closed set if any -closed set V which is contained in B is either  or B (i.e. V = B or V = , whenever, V B). The family of all maximal -open, minimal -open, maximal -closed and minimal -closed sets will be denoted by MaO(X), MiO(X), MaC(X) and MiC(X) respectbly.We define MaO(X, x) = {A: xA  MaO(X)}, MiO(X, x) = {A: xA  MiO(X)}, MaC(X, x) = {F : x  F  MaC(X)} and MiC(X, x) = {F : x  F  MiC(X)}. Theorem 3.3. Let A be a proper nonempty subset A of X. Then A is a maximal -open set if and only if X\A is a minimal -closed set. Proof: Let A be a maximal -open set. Then A  X or A  A. Hence,  X\A or X\A  X\A: Therefore by Definition 3.2, X\A is a minimal -closed set. Conversely, let X\A be a minimal -closed set. Then,   X\A or X\A  X\A. Hence A  X or A  A which implies that A is a maximal -open set. Remark 3.4. The following example shows that (1) maximalopen sets and maximal -open sets are independent. (2) Maximal -open sets and maximal -open sets are independent. (3) Maximal -open sets and maximal open sets are independent. That is, Maximal -Open Maximal -Open     Maximal Open   Maximal -Open. Example 3.5. Let us consider the topological space (X,) such that X = {a, b, c, d} and  = {X,, {c}, {c,d}, {a, b}, {a, b,c}}. Throughout careful computation we find that O(X,) = {X,, {c,d}, {a, b}} = O(X,); O(X,) = {X,, {c,d}, {a, b}, {a,b,c}, {a,c,d},{a,b,d},{b,c,d}}. Clearly, (1) {c,d} is maximal open which is not maximal open and {b,c,d} is maximal -open which is not maximal open set in X. (2) {c,d} is maximal -open which is not maximal -open and {b,c,d} is maximal -open which is not maximal -open set in X. (3) {c,d} is maximal -open which is not maximal -open and {b,c,d} is maximal -open which is not maximal -open set in X. Theorem 3.6. In any topological space (X,), if A be a maximal -open set and B be a -open set of (X,). Then either A  B = X or B  A. Proof. Let A be a maximal -open set and B be a -open set of (X,). If AB = X, then we are done. But if AB  X, then we have to prove that B  A. Now A  B  X means B  A  B and A  A  B. Therefore, we have, A  A  B and A is maximal -open, then by definition, A  B = X or A  B = A but A  B  X, then A  B = A which implies B  A. Theorem 3.7. In any topological space (X,), if A and B be maximal -open sets of (X,). Then either AB = X or B = A. Proof. Let A and B be maximal -open sets of (X,). If A  B = X, then we are done. But if A  B  X, then we have to prove that B = A. Now A  B  X means A  A  B and B  A  B. Now, since, A  A  B and A is a maximal open set, then by definition, A  B = X or A  B = A but A  B  X, therefore, A  B = A which implies B  A. Similarly, if B  A  B we obtain, A  B. Therefore A = B. Theorem 3.8. In any topological space (X,), if F be a minimal -closed set and G be a -closed set of (X,). Then either F  G =  or F  G. Proof. Let F be a minimal -closed set and G be a -closed set of (X,). If FG = , then there is nothing to prove. But if FG  , then we have to prove that F G: Now if FG  , then F G  F and F  G  G. Since F  G  F and given that F is minimal -closed, then by definition F  G = F or F  G = . But F  G  , then F  G = F which shows that F  G. Theorem 3.9. In any topological space (X,), if F and G be minimal -closed sets of (X,). Then either F  G =  or F = G. Proof. Let F and G be two minimal -closed sets of (X,). If F  G = , then there is nothing to prove. But if F  G  , then we have to prove that F = G. Now, if F  G  , then F  G  F and F  G  G. Since F  G  F and given that F is minimal -closed, then by definition F  G = F or F  G = . But F  G  , then F  G = F which implies that F  G. Similarly, if F  G  G and given that G is minimal closed, then by definition F  G = G or F  G = . But F  G  , then F  G = G which implies G  F. Hence, F = G. Theorem 3.10. In any topological space (X,), (a) Let A be a maximal -open set of (X,) and x an element of X\A. Then for any -open set B containing x, X\A  B. (b) Let A be a maximal -open set of (X,). Then, either of the following (i) and (ii) holds. (i) For each xX\A and each -open set B containing x, B = X. (ii) There exists a -open set B such that X\A  B and B  X. (c) Let A be a maximal -open set of (X,). Then, either of the following (i) and (ii) holds. (i) For each x X\A and each -open set B containing x, we have X\A  B. (ii) There exists a -open set B such that X\A = B  X. Proof. (1) Since x X\A, we have B  A for any -open set B containing x.Then, A  B = X by Theorem 3.6. Therefore, X\A  B. 34 International Journal of Computer Applications (0975 – 8887) Volume 177 – No.3, November 2017 (2) If (i) does not hold, then there exists an element x of X\A and a -open set B containing x such that B  X. By (1), we have, X\A  B. (2) Let us assume that H  (   H)  , then there exists (3) If (ii) does not hold, then, by (1), we have X\A  B for each x X\A and each -open set B containing x. Hence, we have X\A  B.  H) = . Theorem 3.11. Let A,B,C be maximal -open sets such that A  B. If A  B  C, then either A = C or B = C. Proof. Given that A  B  C. If A = C, then there is nothing to prove. But if A C, then we have to prove B = C, Using Theorem 3.7, we have, B  C = B  [C  X] = B  [C  (A  B)] = B  [(C A)  (C  B)] = (B  C  A)  (B  C  B) = (A  B)  (C  B) [since, A  B  C] = (A  C)  B = X  B = B, [since A  C = X]. This implies B  C also from the definition of maximal open set it follows that B = C. Theorem 3.12. Let A, B,C be maximal -open sets which are different from each other. Then (A  B)  (A  C). Proof. Let (A B)  (A C). Then, (A  B)  (C  B)  (A C)  (C  B). Hence, (A  C)  B  C  (A B). Since by Theorem 3.7, A  C = X. We have X  B  C  X which implies B  C. From the definition of maximal -open set it follows that B = C. Contradiction to the fact that A, B and C are different from each other. Therefore (A  B)  (A  C). Theorem 3.13. (1) Let F be a minimal -closed set of X. If x  F, then F  G for any -closed set G containing x. (2) Let F be a minimal -closed set of X. Then F =  {G : xGC(X)} for any element x of F. Proof. (1) Let F MiC(X,x) and GC(X,x) such that F  G. This implies that F  G  F and F  G  . But since F is minimal -closed, by Definition 3.2, F  G = F which contradicts the relation F  G  F. Therefore, F  G. (2) By (1) and the fact that F is -closed containing x, we have, F   { G : GC(X, x)}  F. Therefore, we have the result. Theorem 3.14. (1) Let H and H (, index set) be minimal -closed sets of (X,). If H   H , then there exists  such that H = H. (2) Let H and H () be minimal -closed sets of (X,). If H  H , for any , then H  (   H) = . Proof. (1) Let H and H () be minimal -closed sets  H We have to prove that H  H  . Since if H  H = , then H  X\H and hence H    such that H  H X\H which is a contradiction. Now as H  H  , then H  H  H and H  H  H. Since H  H  H and given that F is minimal -closed, then by definition H  H = H or H  H = . But H  H  , then H  H = H which implies H  H. Similarly, if H  H  H and given that H is minimal -closed, then by definition H  H = H or H  H = . But H  H  . then H  H = H which implies H  H. Then H = H.    such that H  H  . By Theorem 3.9. we have H = H which is a contradiction to the fact H  H. Hence H  ( 4. -semi-maximal open sets and - semi-minimal closed sets This section introduces the notion of -semi-maximal open set and -semi-minimal closed sets and investigate some of their properties with examples. Definition 4.1. A subset G in a topological space X is said to be -semi-maximal open if there exists a maximal -open set U such that U  G  Cl(U). The complement of a semi-maximal open set is called a -semi-minimal closed set. Remark 4.2. Every maximal -open (resp. minimal closed) set is -semi-maximal open (resp. -semi-minimal closed). Example 4.3. Let us consider the topological space (X,) such that X = {a, b, c, d} and  = {X,, {c,d}, {a, b}}. We observe that O(X,) = {X,, {c,d}, {a, b}} = O(X,); O(X,) = {X,, {c,d}, {a, b}, {a,b,c}, {a,c,d},{a,b,d},{b,c,d}}. Clearly, G = {a,b,c} is maximal open set. Then G is a -semi-maximal open set, since there exists maximal -open set U = {a,b,c} such that U  G Cl(U) = X. Also it is clear that the complement of G = {d} is minimal -closed set and hence it is -semi-minimal closed set. The collection of all -semi-maximal open (resp. -semiminimal closed) set of X is denoted by SMaO(X) (resp. SMiC(X)). Theorem 4.4. If G is a -semi-maximal open set of X and G  M  Cl(G), then M is a -semi-maximal open set of X. Proof. Since G is -semi-maximal open, there exists a maximal -open set U such that U  G  Cl(U). Then U  G  M  Cl(G)  Cl(U). Hence, U  M  Cl(U). Thus M is -semi-maximal open. Theorem 4.5. A subset B of a topological space X is -semiminimal closed if and only if there exists a minimal -closed set G in X such that Int(G)  B  G. Proof. Suppose B is -semi-minimal closed in X. By Definition 4.1, X\B is -semi-maximal open in X. Therefore, there exists a maximal -open set U such that U  X\B  Cl(U), which implies Int(X\U) = X\Cl(U)  B  X\U.Let us take G = X\U, so that G is a minimal -closed set such that Int(G)  B  G. Conversely, suppose that there exists a minimal -closed set G in X, such that Int(G)  B  G. Hence X \ G  X \ B  X \ Int(G) = Cl(X \ G). So, there exists a maximal -open set U = X \ G such that U  X \ B  Cl(U), which implies that X \ B is -semi-maximal open in X. It follows that B is -semiminimal closed. Theorem 4.6. If G is -semi-minimal closed in (X,) and if Int(G)  F  G, then F is also -semi -minimal closed in X. 35 International Journal of Computer Applications (0975 – 8887) Volume 177 – No.3, November 2017 Proof. Let G be a -semi-minimal closed set of X. Then there exists a minimal -closed set H in X, such that Int(H)  G  H. Hence, Int(H)  Int(G)  F  G  H. It follows Int(H)  F  H. Therefore, F is a -semi-minimal closed set in X. Theorem 4.7. Let (Y,Y) be an open subspace of (X,) and A  Y and A is a -semi-maximal open set of X, then A is also a -semi-maximal open set of Y. Proof. Since A is a -semi-maximal open set of X, there exists a maximal -open set U such that U  A  Cl(U). Hence, U is a subset of Y. Since U is maximal -open in X, Y  U = U is maximal -open in Y and U = Y  U  Y  A  Y  Cl(U)  U  A  ClY(U). Hence A is -semimaximal open in Y. Theorem 4.8. If G is a -semi-maximal open set of topological spaces (X,X1X2) ( = 1, 2), then G1 G2 is a semi-maximal open set in the Product Space (X1 X2, X1X2). Proof. Let G be a -semi-maximal open set of topological spaces X ( = 1, 2), then there exists a maximal -open set U such that U  G  ClX (U), for each . Therefore, U1 U2  G1 G2  ClX1(U1)  ClX2(U2) = ClX1X2 (U1  U2). Hence G1 G2 is -semi-maximal open in (X1 X2, X1X2). 5. CONCLUSION In this work, the concept of maximal -open sets, minimal -closed sets, -semi maximal open and -semi minimal closed sets which are fundamental results for further research on topological spaces are introduced and aimed in IJCATM : www.ijcaonline.org investigating the properties of these new notions of open sets with example, counter examples and some of their fundamental results are also established. Hope that the findings in this paper will help researcher enhance and promote the further study on topological spaces to carry out a general framework for their applications in separation axioms, connectedness, compactness etc. and also in practical life. 6. ACKNOWLEDGMENTS The author wishes to thank the reviewers valuable comments towards the publication of the work and also thanks to the learned referee for their valuable suggestions which improved the paper to a great extent. 7. REFERENCES [1] F. Nakaoka and N. Oda, “Some applications of minimal open sets”, Int. J. 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