w-preopen Sets and W -Precontinuity in Weak Spaces

International Journal of Mathematical Analysis Vol. 10, 2016, no. 21, 1009-1017 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6575 w-preopen Sets and W -Precontinuity in Weak Spaces Won Keun Min 1 Department of Mathematics Kangwon National University Chuncheon 200-701, Korea Young Key Kim Department of Mathematics, MyongJi University Youngin 449-728, Korea Copyright c 2016 Won Keun Min and Young Key Kim. This article is 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 purpose of this short note is to introduce the notions of w- preopen set and W -precontinuity in w-spaces, which are generalizations of w-open sets and W -continuity, respectively. The notion of w-preopen set is introduced and studied some basic properties of such the notion. Moreover, w-preopen sets will be invoked to characterize the weak W - continuity. We also introduce W -precontinuous functions defined by w-preopen sets, and investigate its properties. Mathematics Subject Classification: 54A05, 54B10, 54C10, 54D30 Keywords: w-spaces, w-open, W -continuous, w-preopen, W -precontinuous, w-semiopen, W -semicontinuous 1 Corresponding author

1010 Won Keun Min and Young Key Kim 1 Introduction Siwiec [18] introduced the notions of weak neighborhoods and weak base in a topological space. We introduced the weak neighborhood systems defined by using the notion of weak neighborhoods in [13]. The weak neighborhood system induces a weak neighborhood space which is independent of neighborhood spaces [4] and general topological spaces [2]. The notions of weak structure, w-space, W -continuity and W -continuity were investigated in [14]. In fact, the set of all g-closed subsets [5] in a topological space is a kind of weak structure. The one purpose of our research is to generalize w-open sets in w-spaces. Levine [6] introduced the concept of semi-open set in topological spaces and used this to define other new concepts such as semi-closed set and semicontinuity of a function. In the same way, we introduced the notions of w- semiopen set and W -semicontinuity in weak spaces, and investigated some basic properties of such notions in [16]. In this short note, we introduce the notions of w-preopen set and W - precontinuity in w-spaces, which are generalizations of w-open sets and W - continuity, respectively. First, the notion of w-preopen sets is introduce and studied some basic properties of such the notion. Second, we introduce W - precontinuous functions defined by w-preopen sets, and investigate its properties and the relations among W -continuity, W -semicontinuity and W -precontinuity. Finally, we show that if the w-space Y has the property (OP ), then f is a weakly W -continuous function if and only if wc(f 1 (G)) f 1 (wc(g)) for G W P O(Y ). 2 Preliminaries Definition 2.1 ([14]). Let X be a nonempty set. A subfamily w X of the power set P (X) is called a weak structure on X if it satisfies the following: (1) w X and X w X. (2) For U 1, U 2 w X, U 1 U 2 w X. Then the pair (X, w X ) is called a w-space on X. Then V w X is called a w-open set and the complement of a w-open set is a w-closed set. The collection of all w-open sets (resp., w-closed sets) in a w-space X will be denoted by W O(X) (resp., W C(X)). We set W (x) = {U W O(X) : x U}. Let S be a subset of a topological space X. The closure (resp., interior) of S will be denoted by cls (resp., ints). A subset S of X is called a preopen set [10] (resp., α-open set [17], semi-open [6]) if S int(cl(s)) (resp., S int(cl(int(s))), S cl(int(s))). The complement of a preopen set (resp., α- open set, semi-open) is called a preclosed set (resp., α-closed set, semi-closed).

w-preopen sets and W -precontinuity in weak spaces 1011 The family of all preopen sets (resp., α-open sets, semi-open sets) in X will be denoted by P O(X) (resp., α(x), SO(X)). We know the family α(x) is a topology finer than the given topology on X. A subset A of a topological space (X, τ) is said to be: (a) g-closed [5] if cl(a) U whenever A U and U is open in X; (b) gp-closed [7] if pcl(a) U whenever A U and U is open in X; (c) gs-closed [1, 3] if scl(a) U whenever A U and U is open in X; (d) gα-closed [9] if τ α Cl(A) U whenever A U and U is α-open in X where τ α = α(x); (e) gα -closed [8] if τ α Cl(A) int(u) whenever A U and U is α-open in X; (f) gα -closed [8] if τ α Cl(A) int(cl(u)) whenever A U and U is α-open in X; (g) αg-closed [9] if τ α Cl(A) U whenever A U and U is open in X; (h) α g-closed [9] if τ α Cl(A) int(cl(u)) whenever A U and U is open in X. Then the family τ, GO(X), gαo(x), gα O(X), gα O(X), αgo(x) and α go(x) are all weak structures on X. But P O(X), GP O(X) and SO(X) are not weak structures on X. A subfamily m X of the power set P (X) of a nonempty set X is called a minimal structure on X [10] if w X and X w X. Thus clearly every weak structure is a minimal structure. Definition 2.2 ([14]). Let (X, w X ) be a w-space. For a subset A of X, the w-closure of A and the w-interior of A are defined as follows: (1) wc(a) = {F : A F, X F w X }. (2) wi(a) = {U : U A, U w X }. Theorem 2.3 ([14]). Let (X, w X ) be a w-space and A X. (1) x wi(a) if and only if there exists an element U W (x) such that U A. (2) x wc(a) if and only if A V for all V W (x). (3) If A B, then wi(a) wi(b); wc(a) wc(b). (4) wc(x A) = X wi(a); wi(x A) = X wc(a). (5) If A is w-closed (resp., w-open), then wc(a) = A (resp., wi(a) = A). Let f : X Y be a function on w-spaces (X, w X ) and (Y, w Y ). Then f is said to be W -continuous [14] if for x X and V W (f(x)), there is U W (x) such that f(u) V. 3 w-preopen sets; W -precontinuity First, we introduce the notion of w-preopen sets and study some basic properties of such the notion.

1012 Won Keun Min and Young Key Kim Definition 3.1. Let (X, w X ) be a w-space and P X. Then P is called a w-preopen set if P wi(wc(p )). The complement of a w-preopen set is called a w-preclosed set. The family of all w-preopen sets in X will be denoted by W P O(X). Lemma 3.2. Let (X, w X ) be a w-space and A X. Then the following things hold: (1) A is a w-preclosed set if and only if wc(wi(a)) A. (2) Every w-open (resp., w-closed) set is w-preopen (resp., w-preclosed). Remark 3.3. We recall that: Let (X, w X ) be a w-space and S X. Then S is called a w-semiopen set [16] if S wc(wi(s)). The complement of a w-semiopen set is called a w-semiclosed set. Then we have the following implications but the converses are not true in general: See the examples below w-semiopen w-open w-preopen Example 3.4. Let X = {a, b, c, d, e} and w = {, {b}, {a, b}, {c, d}, X}. (1) Let us consider a set A = {a, b, d} in X. Obviously, A is not w-open. Since wc({a, b, d}) = X, A is w-preopen. But since wcwi({a, b, d}) = wc({a, b}) = {a, b, e}, it is not w-semiopen. (2) Let A = {c, d, e}. Then wi(wc(a)) = wi({c, d, e}) = {c, d} and wc(wi(a)) = wc({c, d}) = {c, d, e}. So A is w-semiopen but not w-preopen. Theorem 3.5. Let (X, w X ) be a w-space. w-preopen. Any union of w-preopen sets is Proof. For i J, let A i be w-preopen subset of X. Then A i wi(wc(a i )) wi(wc( A i )). Thus A i wi(wc( A i )), and A i is w-preopen. In general, the intersection of two w-preopen sets is not w-preopen as the next example: Example 3.6. Let X = {a, b, c, d, e} and w = {, {a, b}, {c, d}, X}. Let A = {a, c, d, e} and B = {b, c, d, e}. Note that wc(a) = wc(b) = X; wi(wc(a B)) = wi(wc({c, d, e})) = wi({c, d, e}) = {c, d}. So both A and B are w- preopen, but A B is not w-preopen.

w-preopen sets and W -precontinuity in weak spaces 1013 Let (X, w X ) be a w-space. For A X, the w-pre-closure and the w-preinterior of A, denoted by wpc(a) and wpi(a), respectively, are defined as: wpc(a) = {F X : A F, F is w-preclosed in X}; wpi(a) = {U X : U A, U is w-preopen in X}. Theorem 3.7. Let (X, w X ) be a w-space and A X. Then following hold: (1) wpi(a) A; A wpc(a). (2) If A B, then wpi(a) wpi(b) and wpc(a) wpc(b). (3) A is w-preopen iff wpi(a) = A. (4) F is w-preclosed iff wpc(f ) = F. (5) wpi(wpi(a)) = wpi(a); wpc(wpc(a)) = wpc(a). (6) wpc(x A) = X wpi(a); wpi(x A) = X wpc(a). Proof. Obvious. Lemma 3.8. Let (X, w X ) be a w-space and A X. Then the following things hold: (1) x wpc(a) if and only if A V for every w-preopen set V containing x. (2) x wpi(a) if and only if there exists a w-preopen set U such that U A. Proof. Obvious. We introduce the notion of W -precontinuity defined by w-preopen sets, which is a generalization of W -continuity, and investigate its properties and the relations among W -continuity, W -semicontinuity and W -precontinuity. Definition 3.9. Let f : (X, w X ) (Y, w Y ) be a function on w-spaces X and Y. Then the function f is said to be W -precontinuous if for each point x and each w-open set V containing f(x), there exists a w-preopen set U containing x such that f(u) V. Remark 3.10. We recall that: Let f : (X, w X ) (Y, w Y ) be a function on w-spaces X and Y. Then f is said to be W -semicontinuous [16] if for each x and each w-open set V containing f(x), there exists a w-semiopen set U containing x such that f(u) V. Now, we have the following diagram: W -semicontinuous W -continuous W -precontinuous

1014 Won Keun Min and Young Key Kim Example 3.11. In Example 3.4, (1) let us consider a function f : (X, w) (X, w) defined as follows f(a) = f(b) = b; f(c) = f(e) = e; f(d) = a. Then f is W -precontinuous but neither W -continuous nor W -semicontinuous; (2) let us consider a function f : (X, w) (X, w) defined as follows f(a) = f(b) = e; f(c) = f(e) = b; f(d) = a. Then f is W -semicontinuous but not W -precontinuous. Theorem 3.12. Let f : (X, w X ) (Y, w Y ) be a function on w-spaces X and Y. Then the following statements are equivalent: (1) f is W -precontinuous. (2) f 1 (V ) is a w-preopen set for each w-open set V in Y. (3) f 1 (B) is a w-preclosed set for each w-closed set B in Y. (4) f(wpc(a)) wc(f(a)) for A X. (5) wpc(f 1 (B)) f 1 (wc(b)) for B Y. (6) f 1 (wi(b)) wpi(f 1 (B)) for B Y. Proof. (1) (2) Let V be a w-open set in Y and x f 1 (V ). Then there exists a w-preopen set U containing x such that f(u) V and so U f 1 (V ) for all x f 1 (V ). From Theorem 3.5, f 1 (V ) is w-preopen. (2) (3) Obvious. (3) (4) For A X, f 1 (wc(f(a))) = f 1 ( {F Y : f(a) F and F is w-closed }) = {f 1 (F ) X : A f 1 (F ) and F is w-preclosed } {K X : A K and K is w-preclosed } = wpc(a) This implies that f(wpc(a)) wc(f(a)). (4) (5) For B Y, f(wpc(f 1 (B))) wc(f(f 1 (B))) wc(b). Thus, wpc(f 1 (B)) f 1 (wc(b)). (5) (6) For B Y, f 1 (wi(b)) = f 1 (Y wc(y B)) = X f 1 (wc(y B)) X wpc(f 1 (Y B)) = wpi(f 1 (B)). So, f 1 (wi(b)) wpi(f 1 (B)). (6) (1) Let x X and V be a w-open set containing f(x). Then x f 1 (V ) = f 1 (wi(v )) wpi(f 1 (V )). Since x wpi(f 1 (V )), by Lemma 3.8, there exists a w-preopen set U containing x such that U f 1 (V ). Hence, f is w-precontinuous. Lemma 3.13. Let (X, w X ) be a w-space and A X. Then (1) wc(wi(a)) wc(wi(wpc(a))) wpc(a). (2) wpi(a) wi(wc(wpi(a))) wi(wc(a)).

w-preopen sets and W -precontinuity in weak spaces 1015 Proof. (1) Since wpc(a) is w-preclosed, from Lemma 3.2, it is easily obtained. (2) Since wpi(a) is w-preopen and wpi(a) A, it is obvious. Theorem 3.14. Let f : (X, w X ) (Y, w Y ) be a function on w-spaces X and Y. Then the following statements are equivalent: (1) f is W -precontinuous. (2) f 1 (V ) wi(wc(f 1 (V ))) for each w-open set V in Y. (3) wc(wi(f 1 (F ))) f 1 (F ) for each w-closed set F in Y. (4) f(wc(wi(a))) wc(f(a)) for A X. (5) wc(wi(f 1 (B))) f 1 (wc(b)) for B Y. (6) f 1 (wi(b)) wi(wc(f 1 (B))) for B Y. Proof. (1) (2) and (1) (3) From Theorem 3.12, it is obvious. (3) (4) Let A X. Then by (3), for each w-closed set F in Y, f 1 (F ) is w-preclosed. So we have f(wpc(a))) wc(f(a)) and from Lemma 3.13, f(wc(wi(a))) f(wpc(a))) wc(f(a)). (4) (5) Obvious. (5) (6) For B Y, f 1 (wi(b)) = f 1 (Y wc(y B)) = X (f 1 (wc(y B))) X wc(wi(f 1 (Y B))) = wi(wc(f 1 (B))). So (6) is obtained. (6) (1) Let V be a w-open set in Y. Then f 1 (V ) = f 1 (wi(v )) wi(wc(f 1 (V ))), and so f 1 (V ) is w-preopen. Thus, from Theorem 3.12, f is W -precontinuous. Let f : (X, w X ) (Y, w Y ) be a function in w-spaces X and Y. Then f is said to be weakly W-continuous [15] if for x X and V W (f(x)), there is U W (x) such that f(u) wc(v ). Clearly, every W -continuous function is weakly W -continuous. Theorem 3.15 ([15]). Let f : (X, w X ) (Y, w Y ) be a function on w-spaces X and Y. Then f is weakly W -continuous if and only if wc(f 1 (V )) f 1 (wc(v )) for V W O(Y ). Lemma 3.16. Let (X, w) be a w-space. If G is any w-preopen set in X, then wc(g) = wc(wi(wc(g))). Proof. For any w-preopen set G, wc(wi(wc(g))) wc(wc(g)) = wc(g) wc(wi(wc(g))). It implies wc(g) = wc(wi(wc(g))). Definition 3.17. For a w-space X, X is said to have the operator preserving property (simply, OP P ) if for i J, wi(a i ) = wi( A i ).

1016 Won Keun Min and Young Key Kim Corollary 3.18. Let (X, w) be a w-space. Then X has the operator preserving property (OP P ) if and only if for i J, wc(a i ) = wc( A i ). Proof. From (4) of Theorem 2.4, it follows that wi(a i ) = wi( A i ) if and only if wc(a i ) = wc( A i ) for i J. So the statement is obtained. Theorem 3.19. Let f : (X, w X ) (Y, w Y ) be a function in w-spaces X and Y. If the w-space X has the property (OP P ), then the following holds: f is a weakly W -continuous function if and only if wc(f 1 (G)) f 1 (wc(g)) for V W P O(Y ). Proof. Suppose that f is weakly W -continuous function. Let G be any w- preopen set in Y. Let wi(wc(g)) = V α for V α W O(X); then, obviously G wi(wc(g)) = V α. Now, from the operator preserving property of X and weak W -continuity, it follows that wc(f 1 (G)) wc(f 1 ( V α )) = wc( f 1 (V α )) = wc(f 1 (V α )) f 1 (wc(v α )) = f 1 ( wc(v α )) f 1 (wc( V α )) = f 1 (wc(wi(wc(g)))). Hence, by Lemma 3.16, it implies that wc(f 1 (G)) f 1 (wc(g)). Since every w-open set is w-preopen, from Theorem 3.15, the converse is obtained. References [1] P. Bhattacharyya and B. K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math., 29 (1987), no. 3, 375-382. [2] Á. Csázár, Generalized Topology, Generalized Continuity, Acta Math. Hungar., 96 (2002), 351-357. http://dx.doi.org/10.1023/a:1019713018007 [3] J. Dontchev and H. Maki, On sg-closed sets and semi-λ-closed sets, Q & A in General Topology, 15 (1997), 259-266. [4] D. C. Kent and W. K. Min, Neighborhood Spaces, International Journal of Mathematics and Mathematical Sciences, 32 (2002), no. 7, 387-399. http://dx.doi.org/10.1155/s0161171202202203 [5] N. Levine, Generalized closed sets in topology, Rend. Cir. Mat. Palermo, 19 (1970), 89-96. http://dx.doi.org/10.1007/bf02843888 [6] N. Levine Semi-open sets and semi-continuity in topological spaces, Ams. Math. Monthly, 70 (1963), 36-41. http://dx.doi.org/10.2307/2312781 [7] H. Maki, J. Umehara and T. Noiri, Every topological space is pre-t 1 2, Mem. Fac. Sci. Kochi Univ. Ser. A, 17 (1996), 33-42.

w-preopen sets and W -precontinuity in weak spaces 1017 [8] H. Maki, R. Devi and K. Balachandran, Associated topologies of generalized α-closed sets and α-generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser. A, 15 (1994), 51-63. [9] H. Maki, R. Devi and K. Balachandran, Generalized α-closed maps and α-generalized closed maps, Indian J. Pure Appl. Math., 29 (1998) no. 1, 37-49. [10] H. Maki, On Generalizing Semi-Open and Preopen Sets, Report for Meeting on Topological Spaces Theory and its Applications, August 1996, Yatsushiro College of Technology, 13-18. [11] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. and Phys. Soc. Egypt, 53 (1982), 47-53. [12] W. K. Min, Some Results on Generalized Topological Spaces and Generalized Systems, Acta Math. Hungar., 108 (2005), no. 1-2, 171-181. http://dx.doi.org/10.1007/s10474-005-0218-7 [13] W. K. Min, On Weak Neighborhood Systems and Spaces, Acta Math. Hungar., 121 (2008), no. 3, 283-292. http://dx.doi.org/10.1007/s10474-008-7205-8 [14] W. K. Min and Y. K. Kim On Weak Structures and w-spaces, Far East Journal of Mathematical Sciences, 97 (2015), no/ 5, 549-561. http://dx.doi.org/10.17654/fjmsjul2015 549 561 [15] W. K. Min and Y. K. Kim, Weak W -continuity on Weak Spaces, International Journal of Mathematical Analysis, 9 (2015), no. 31, 1499-1507. http://dx.doi.org/10.12988/ijma.2015.54336 [16] W. K. Min and Y. K. Kim, w-semiopen Sets and W -Semicontinuity in Weak Spaces, submitted. [17] O. Njastad, On some classes of nearly open sets, Pacific Journal of Mathematics, 15 (1964), no. 3, 961-970. http://dx.doi.org/10.2140/pjm.1965.15.961 [18] F. Siwiec, On Defining a Space by a Weak Base, Pacific Journal of Mathematics, 52 (1974), no. 1, 233-245. http://dx.doi.org/10.2140/pjm.1974.52.233 Received: June 12, 2016; Published: July 28, 2016