JuJeLeLeAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 9, No. 2, 2008 pp. 213-228 Applications of pre-open sets Young Bae Jun, Seong Woo Jeong, Hyeon Jeong Lee and Joon Woo Lee Abstract. Using the concept of pre-open set, we introduce and study topological properties of pre-limit points, pre-derived sets, pre- interior and pre-closure of a set, pre-interior points, pre-border, pre- frontier and pre-exterior. The relations between pre-derived set (resp. pre-limit point, pre-interior (point), pre-border, pre-frontier, and pre- exterior) and α-derived set (resp. α-limit point, α-interior (point), α-border, α-frontier, and α-exterior) are investigated. 2000 AMS Classification: 54A05, 54C08. Keywords: Pre-limit point, Pre-derived set, Pre-interior, Pre-closure, Pre- interior points, Pre-border, Pre-frontier, Pre-exterior. 1. Introduction The notion of α-open set was introduced by Nj̊astad [14]. Since then it has been widely investigated in several literatures (see [1, 3, 4, 5, 6, 7, 9, 10, 12, 15]). In [2], Caldas introduced and studied topological properties of α-derived, α- border, α-frontier, and α-exterior of a set by using the concept of α-open sets. The notion of pre-open set was introduced by Mashhour et al. [8]. In this paper, we introduce the notions of pre-limit points, pre-derived sets, pre-interior and pre-closure of a set, pre-interior points, pre-border, pre-frontier and pre-exterior by using the concept of pre-open sets, and study their topological properties. We provide relations between pre-derived set (resp. pre-limit point, pre-interior (point), pre-border, pre-frontier, and pre-exterior) and α-derived set (resp. α- limit point, α-interior (point), α-border, α-frontier, and α-exterior). 214 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee 2. Preliminaries Through this paper, (X, T ) and (Y, K ) (simply X and Y ) always mean topological spaces. A subset A of X is said to be pre-open [11] (respec- tively, α-open [14] and semi-open [13]) if A ⊂ Int(Cl(A)) (respectively, A ⊂ Int(Cl(Int(A))) and A ⊂ Cl(Int(A))). The complement of a pre-open set (re- spectively, an α-open set and a semi-open set) is called a pre-closed set (respec- tively, an α-closed set and a semi-closed set). The intersection of all pre-closed sets (respectively, α-closed sets and semi-closed sets) containing A is called the pre-closure (respectively, α-closure and semi-closure) of A, denoted by Clp(A) (respectively, Clα(A) and Cls(A)). A subset A is also pre-closed (respectively, α-closed and semi-closed) if and only if A = Clp(A) (respectively, A = Clα(A) and A = Cls(A)). We denote the family of pre-open sets (respectively, α- open sets and semi-open sets) of (X, T ) by T p (respectively, T α and T s). Obviously, we have the following relations. open set (closed set) α-open set (α-closed set) pre-open set (pre-closed set) semi-open set (semi-closed set) ? � ��� H HHj None of these implications is reversible in general. 3. Pre-open sets and α-open sets Definition 3.1 ([11, 14]). A subset A of X is said to be pre-open (respectively, α-open) if A ⊆ Int(ClA) (respectively, A ⊆ Int(Cl(IntA))). The complement of a pre-open set (respectively, an α-open set) is called a pre-closed set (respectively, an α-closed set). The intersection of all pre-closed sets (respectively, α-closed sets) containing A is called the pre-closure (respectively, α-closure) of A, denoted by Clp(A) (respectively, Clα(A)). A subset A is also pre-closed (respectively, α-closed) if and only if A = Clp(A) (respectively, A = Clα(A)). We denote the family of pre-open sets (respectively, α-open sets) of (X, T ) by T p (respectively, T α). Example 3.2. Let T = {∅, X, {a}, {c, d}, {a, c, d}} be a topology on X = {a, b, c, d, e}. Then we have T α = T ∪ {{a, b, c, d}, {a, c, d, e}}, T p = T ∪ {{c}, {d}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, e}, {a, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}}. Applications of pre-open sets 215 4. Applications of pre-open sets Definition 4.1. Let A be a subset of a topological space (X, T ). A point x ∈ X is said to be pre-limit point (resp. α-limit point) of A if it satisfies the following assertion: (∀G ∈ T p( resp. T α)) (x ∈ G ⇒ G ∩ (A \ {x}) 6= ∅). The set of all pre-limit points (resp. α-limit points) of A is called the pre- derived set (resp. α-derived set) of A and is denoted by Dp(A) (resp. Dα(A)). Denote by D(A) the derived set of A. Note that for a subset A of X, a point x ∈ X is not a pre-limit point of A if and only if there exists a pre-open set G in X such that x ∈ G and G ∩ (A \ {x}) = ∅ or, equivalently, x ∈ G and G ∩ A = ∅ or G ∩ A = {x} or, equivalently, x ∈ G and G ∩ A ⊆ {x}. Example 4.2. Let X = {a, b, c} with topology T = {X, ∅, {a}}. Then we have the followings: (i) T p = {X, ∅, {a}, {a, b}, {a, c}} = T α. (ii) If A = {c}, then D(A) = {b} and Dα(A) = Dp(A) = ∅. (iii) If B = {a} and C = {b, c}, then Dp(B) = {b, c}, Dp(C) = ∅ and Dp(B ∪ C) = {b, c}. Theorem 4.3. If a topology T on a set X contains only ∅, X, and {a} for a fixed a ∈ X, then T p = T α. Proof. Let a ∈ X and let A be an element of T p. Then a ∈ A. In fact, if not then A 6⊆ Int(Cl(A)) = Int({a}c) = ∅. Hence A /∈ T p, a contradiction. Now since Int(A) = {a}, we have Int(Cl(Int(A))) = Int(Cl({a})) = Int(X) = X which contains A, that is, A ∈ T α. Note that T α ⊆ T p. Thus T α = T p. � Example 4.4. Let X = {a, b, c, d, e} with topology T = {X, ∅, {a}, {c, d}, {a, c, d}, {b, c, d, e}}. Then T p = {X, ∅, {a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {c, e}, {d, e}, {a, b, c}, {a, b, d}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}, {b, c, d, e}} 216 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee and T α = {X, ∅, {a}, {c, d}, {a, c, d}, {b, c, d}, {c, d, e}, {a, b, c, d}, {a, c, d, e}, {b, c, d, e}}. Consider subsets A = {a, b, c} and B = {b, d} of X. Then D(A) = {b, d, e}, Dp(A) = ∅, Int(A) = {a}, Intp(A) = A, Intα(A) = {a}, Clp(A) = A, Clα(A) = X, Clp(B) = B, Clα(B) = {b, c, d, e}, Int(B) = ∅, Intp(B) = B, Intα(B) = ∅. Example 4.5. Consider a topology T = {X, ∅, {a}, {a, b}, {a, c, d}, {a, b, c, d}, {a, b, e}} on X = {a, b, c, d, e}. Then T p = {X, ∅, {a}, {a, b}, {a, c}, {a, d}, {a, e}, {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e} {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}} = T α. For subsets A = {c, d, e} and B = {b} of X, we have D(A) = {c, d} D(B) = {e}. Dp(A) = ∅ Dp(B) = ∅. Dα(A) = ∅ Dα(B) = ∅. Int(A) = ∅ Int(B) = ∅, Intp(A) = ∅, Intp(B) = ∅, Intα(A) = ∅, Intα(B) = ∅, Clp(A) = {c, d, e}, Clp(B) = {b}, Clα(A) = {c, d, e}, Clα(B) = {b}, Clp({b, d}) = {b, d}, Clα({b, d}) = {b, d}, Int({b, d}) = ∅, Intp({b, d}) = ∅, Intα({b, d}) = ∅. Lemma 4.6. If there exists a ∈ X such that {a} is the smallest element of (T \ {∅}, ⊆), then every non-empty pre-open set contains ⋂ {Gi | Gi ∈ T \ {∅}; i = 1, 2, 3, · · ·}. Proof. If {a} is the smallest element of (T \ {∅}, ⊆), then ⋂ {Gi | Gi ∈ T \ {∅}; i = 1, 2, 3, · · ·} = {a}. Let A be a non-empty pre-open set in X. If a /∈ A, then Cl(A) ⊆ {a} and so A * Int(Cl(A)) ⊆ Int({a}c) = ∅ which is a contradiction. Hence a ∈ A, and so the desired result is valid. � Applications of pre-open sets 217 Theorem 4.7. Let T be a topology on a set X. If there exists a ∈ X such that {a} is the smallest element of (T \ {∅}, ⊆), then T α = T p. Proof. It is sufficient to show that T p ⊆ T α. Let A ∈ T p. If A = ∅, then clearly A ∈ T α. Assume that A 6= ∅. Then a ∈ A by Lemma 4.6. Since {a} ⊆ Int(A), it follows that X = Cl({a}) ⊆ Cl(Int(A)) so that A ⊆ X = Int(X) ⊆ Int(Cl(Int(A))). Hence A is an α-open set. � Theorem 4.8. Let T1 and T2 be topologies on X such that T p 1 ⊆ T p 2 . For any subset A of X, every pre-limit point of A with respect to T2 is a pre-limit point of A with respect to T1. Proof. Let x be a pre-limit point of A with respect to T2. Then (G∩A)\{x} 6= ∅ for every G ∈ T p2 such that x ∈ G. But T p 1 ⊆ T p 2 , so, in particular, (G ∩ A) \ {x} 6= ∅ for every G ∈ T p1 such that x ∈ G. Hence x is a pre-limit point of A with respect to T1. � The converse of Theorem 4.8 is not true in general as seen in the following example. Example 4.9. Consider topologies T1 = {X, ∅, {a}} and T2 = {X, ∅, {a}, {b, c}, {a, b, c}} on a set X = {a, b, c, d}. Then T p 1 = T1 ∪ {{a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}} and T p 2 = T2 ∪ {{b}, {c}, {a, b}, {a, c}, {a, d}, {a, b, d}, {a, c, d}}. Note that T p 1 ⊆ T p 2 and c is a pre-limit point of A = {a, b} with respect to T1, but it is not a pre-limit point of A with respect to T2. Lemma 4.10. If {Ai | i ∈ Λ} is a family of pre-open sets in X, then ⋃ i∈Λ Ai is a pre-open set in X where Λ is any index set. Proof. Straightforward. � In Example 3.2, we see that {a, b, c, e} ∩ {a, b, d, e} = {a, b, e} /∈ T p, which shows that the intersection of two pre-open sets is not pre-open in gen- eral. Thus we know that for any topology T on a set X, T p may not be a topology on X. Proposition 4.11. If I (resp. D) is the indiscrete (resp. discrete) topology on a set X, then I p (resp. D p) is a topology on X. Proof. Straightforward. � 218 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee Theorem 4.12. For any subsets A and B of (X, T ), the following assertions are valid: (1) Dp(A) ⊆ Dα(A). (2) If A ⊆ B, then Dp(A) ⊆ Dp(B). (3) Dp(A) ∪ Dp(B) ⊆ Dp(A ∪ B) and Dp(A ∩ B) ⊆ Dp(A) ∩ Dp(B). (4) Dp(Dp(A)) \ A ⊆ Dp(A). (5) Dp(A ∪ Dp(A)) ⊆ A ∪ Dp(A). Proof. (1) It suffices to observe that every α-open set is pre-open. (2) Let x ∈ Dp(A) and let G ∈ T p with x ∈ G. Then (G ∩ A) \ {x} 6= ∅. Since A ⊆ B, it follows that (G ∩ B) \ {x} 6= ∅ so that x ∈ Dp(B). (3) Straightforward by (2). (4) Let x ∈ Dp(Dp(A)) \ A and let G ∈ T p with x ∈ G. Then G ∩ (Dp(A) \ {x}) 6= ∅. Let y ∈ G ∩ (Dp(A) \ {x}). Then y ∈ G and y ∈ Dp(A), and so G ∩ (A \ {y}) 6= ∅. If we take z ∈ G ∩ (A \ {y}), then x 6= z because x /∈ A. Hence (G ∩ A) \ {x} 6= ∅. Therefore x ∈ Dp(A). (5) Let x ∈ Dp(A ∪ Dp(A)). If x ∈ A, the result is obvious. Assume that x /∈ A. Then G ∩ ((A ∪ Dp(A)) \ {x}) 6= ∅ for all G ∈ T p with x ∈ G. Hence (G ∩ A) \ {x} 6= ∅ or G ∩ (Dp(A) \ {x}) 6= ∅. The first case implies x ∈ Dp(A). If G∩(Dp(A)\{x}) 6= ∅, then x ∈ Dp(Dp(A)). Since x /∈ A, it follows similarly from (4) that x ∈ Dp(Dp(A)) \ A ⊆ Dp(A). Therefore (5) is valid. � In general, in Theorem 4.12, the reverse inclusion of (1), (4) and (5), and the converse of (2) may not be true, and the equality in (3) does not hold as seen in the following example. Example 4.13. (1) Consider the topology T on X = {a, b, c, d, e} described in Example 3.2. For a subset A = {b, c, d} of X, we have Dα(A) = {b, c, d, e} and Dp(A) = {b, e}. This shows that the reverse inclusion of Theorem 4.12(1) is not true. Now let X = {a, b, c, d} with a topology T = {X, ∅, {a}, {d}, {a, b}, {a, d}, {c, d}, {a, b, d}, {a, c, d}}. Then T p = T . For two subsets A = {a, c} and B = {a, b, d} of X, we get Dp(A) = {b} ⊆ {b, c} = Dp(B), but A * B. This shows that the converse of Theorem 4.12(2) is not valid. Now consider two subsets A = {a, b} and B = {b, c, d} of X in Example 3.2. Then Dp(A) = {b, e} = Dp(B), and so Dp(A ∩ B) = ∅ ⊆ Dp(A) ∩ Dp(B). Thus the equality in Theorem 4.12(3) is not valid. (2) Consider a topology T = {X, ∅, {b, c}, {b, c, d}, {a, b, c}} on X = {a, b, c, d}. Then T p = {X, ∅, {b}, {c}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. Let A = {a, b} and B = {a, c} be subsets of X. Then Dp(A) = ∅ = Dp(B), and so Dp(A) ∪ Dp(B) = ∅ ⊂ {a, d} = Dp(A ∪ B). For a subset A = {a, b, c} of X, we have Dp(Dp(A)) = Dp({a, d}) = ∅, Dp(Dp(A)) \ A = ∅ ⊆ Dp(A) = {a, d}, Applications of pre-open sets 219 and so the equality in Theorem 4.12(4) is not valid. Now for a subset B = {b, c} of X, we get Dp(B) = {a, d}, and so B ∪Dp(B) = X and Dp(X) = {a, d} ⊆ X. This shows that Dp(B ∪ Dp(B)) 6= B ∪ Dp(B) = X. Hence the equality in Theorem 4.12(5) is not valid. Theorem 4.14. Let A be a subset of X and x ∈ X. Then the following are equivalent: (i) (∀G ∈ T p) (x ∈ G ⇒ A ∩ G 6= ∅). (ii) x ∈ Clp(A). Proof. (i) ⇒ (ii) If x /∈ Clp(A), then there exists a pre-closed set F such that A ⊆ F and x /∈ F. Hence X \ F is a pre-open set containing x and A ∩ (X \ F ) ⊆ A ∩ (X \ A) = ∅. This is a contradiction, and hence (ii) is valid. (ii) ⇒ (i) Straightforward. � Corollary 4.15. For any subset A of X, we have Dp(A) ⊆ Clp(A). Proof. Straightforward. � Theorem 4.16. For any subset A of X, Clp(A) = A ∪ Dp(A). Proof. Let x ∈ Clp(A). Assume that x /∈ A and let G ∈ T p with x ∈ G. Then (G ∩ A) \ {x} 6= ∅, and so x ∈ Dp(A). Hence Clp(A) ⊆ A ∪ Dp(A). The reverse inclusion is by A ⊆ Clp(A) and Corollary 4.15. � Theorem 4.17. Let A and B be subsets of X. If A ∈ T p and T p is a topology on X, then A ∩ Clp(B) ⊆ Clp(A ∩ B). Proof. Let x ∈ A∩Clp(B). Then x ∈ A and x ∈ Clp(B) = B ∪Dp(B). If x ∈ B, then x ∈ A ∩ B ⊆ Clp(A ∩ B). If x /∈ B, then x ∈ Dp(B) and so G ∩ B 6= ∅ for all pre-open set G containing x. Since A ∈ T p, G ∩ A is also a pre-open set containing x. Hence G ∩ (A ∩ B) = (G ∩ A) ∩ B 6= ∅, and consequently x ∈ Dp(A ∩ B) ⊆ Clp(A ∩ B). Therefore A ∩ Clp(B) ⊆ Clp(A ∩ B). � Example 4.18. Let T = {X, ∅, {b}, {b, c}, {b, c, d}} be a topology on a set X = {a, b, c, d}. Then T p = {X, ∅, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}} which is a topology on X. Let A = {a, b} and B = {b, c} be subsets of X. Then A ∩ Clp(B) = {a, b} 6= X = Clp(A ∩ B). This shows that the equality in Theorem 4.17 is not true in general. Example 4.19. Consider T and T p which are given in Example 4.13(2). Note that T p is not a topology on X. For subsets A = {a, b} and B = {b, c} of X, we have A ∩ Clp(B) = {a, b} * {b} = Clp(A ∩ B). This shows that if T p is not a topology on X then the result in Theorem 4.17 is not true in general. Theorem 4.20. Let A and B subsets of X. If A is pre-closed, then Clp(A ∩ B) ⊆ A ∩ Clp(B). 220 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee Proof. If A is pre-closed, then Clp(A) = A and so Clp(A ∩ B) ⊆ Clp(A) ∩ Clp(B) = A ∩ Clp(B) which is the desired result. � Lemma 4.21. A subset A of X is pre-open if and only if there exists an open set H in X such that A ⊆ H ⊆ Cl(A). Proof. Straightforward. � Lemma 4.22. The intersection of an open set and a pre-open set is a pre-open set. Proof. Let A be an open set in X and B a pre-open set in X. Then there exists an open set G in X such that B ⊆ G ⊆ Cl(B). It follows that A ∩ B ⊆ A ∩ G ⊆ A ∩ Cl(B) ⊆ Cl(A ∩ B). Now since A∩G is open, it follows from Lemma 4.21 that A∩B is pre-open. � Theorem 4.23. Let A and B be subsets of X. If A is open, then A ∩ Clp(B) ⊆ Clp(A ∩ B). Proof. It is by Theorem 4.17 and Lemma 4.22. � Theorem 4.24. If A is a subset of a discrete topological space X, then Dp(A) = ∅. Proof. Let x be any element of X. Recall that every subset of X is open, and so pre-open. In particular, the singleton set G := {x} is pre-open. But x ∈ G and G ∩ A = {x} ∩ A ⊆ {x}. Hence x is not a pre-limit point of A, and so Dp(A) = ∅. � Theorem 4.25. For every subset A of X, we have A is pre-closed if and only if Dp(A) ⊆ A. Proof. Assume that A is pre-closed. Let x /∈ A, i.e., x ∈ X \ A. Since X \ A is pre-open, x is not a pre-limit point of A, i.e., x /∈ Dp(A), because (X \A)∩(A\ {x}) = ∅. Hence Dp(A) ⊆ A. The reverse implication is by Theorem 4.16. � Theorem 4.26. Let A be a subset of X. If F is a pre-closed superset of A, then Dp(A) ⊆ F. Proof. By Theorem 4.12(2) and Theorem 4.25, A ⊆ F implies Dp(A) ⊆ Dp(F ) ⊆ F. � Theorem 4.27. Let A be a subset of X. If a point x ∈ X is a pre-limit point of A, then x is also a pre-limit point of A \ {x}. Proof. Straightforward. � Applications of pre-open sets 221 Definition 4.28 ([2]). Let A be a subset of a topological space X. A point x ∈ X is called an α-interior point of A if there exists an α-open set G containing x such that G ⊆ A. The set of all α-interior points of A is called the α-interior of A and is denoted by Intα(A). Based on the above definition, we give the notion of a pre-interior point. Definition 4.29. Let A be a subset of a topological space X. A point x ∈ X is called a pre-interior point of A if there exists a pre-open set G such that x ∈ G ⊆ A. The set of all pre-interior points of A is called the pre-interior of A and is denoted by Intp(A). Example 4.30. Let (X, T ) be a topological space which is given in Exam- ple 4.4. We know that a is the only pre-interior point of A = {a, b, e}, i.e., Intp(A) = {a}. Theorem 4.31. Let A be a subset of X. Then every α-interior point of A is a pre-interior point of A, i.e., Intα(A) ⊆ Intp(A). Proof. If x is an α-interior point of A, then there exists an α-open set G con- taining x such that G ⊆ A. Since every α-open set is pre-open, it follows that x is a pre-interior point of A. � The following example shows that there exists a pre-interior point of A which is not an α-interior point of A. Example 4.32. In Example 4.4, Intα(A) = {a} and Intp(A) = {a, b, c}. Hence b and c are pre-interior points of A. But they are not α-interior points of A. Proposition 4.33. For subsets A and B of X, the following assertions are valid. (1) Intp(A) is the union of all pre-open subsets of A; (2) A is pre-open if and only if A = Intp(A); (3) Intp(Intp(A)) = Intp(A); (4) Intp(A) = A \ Dp(X \ A). (5) X \ Intp(A) = Clp(X \ A). (6) X \ Clp(A) = Intp(X \ A). (7) A ⊆ B ⇒ Intp(A) ⊆ Intp(B). (8) Intp(A) ∪ Intp(B) ⊆ Intp(A ∪ B). (9) Intp(A ∩ B) ⊆ Intp(A) ∩ Intp(B). Proof. (1) Let {Gi | i ∈ Λ} be a collection of all pre-open subsets of A. If x ∈ Intp(A), then there exists j ∈ Λ such that x ∈ Gj ⊆ A. Hence x ∈ ⋃ i∈Λ Gi, and so Intp(A) ⊆ ⋃ i∈Λ Gi. On the other hand, if y ∈ ⋃ i∈Λ Gi, then y ∈ Gk ⊆ A for some k ∈ Λ. Thus y ∈ Intp(A), and ⋃ i∈Λ Gi ⊆ Intp(A). Accordingly, Intp(A) = ⋃ i∈Λ Gi. (2) Straightforward. 222 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee (3) It follows from (1) and (2). (4) If x ∈ A \ Dp(X \ A), then x /∈ Dp(X \ A) and so there exists a pre-open set G containing x such that G ∩ (X \ A) = ∅. Thus x ∈ G ⊆ A and hence x ∈ Intp(A). This shows that A \ Dp(X \ A) ⊆ Intp(A). Now let x ∈ Intp(A). Since Intp(A) ∈ T p and Intp(A) ∩ (X \ A) = ∅, we have x /∈ Dp(X \ A). Therefore Intp(A) = A \ Dp(X \ A). (5) Using (4) and Theorem 4.16, we have X \ Intp(A) = X \ (A \ Dp(X \ A)) = (X \ A) ∪ Dp(X \ A) = Clp(X \ A). (6) Using (4) and Theorem 4.16, we get Intp(X \ A) = (X \ A) \ Dp(A) = X \ (A ∪ Dp(A)) = X \ Clp(A). (7) Straightforward. (8) and (9) They are by (7). � The converse of (7) in Proposition 4.33 is not true in general as seen in the following example. Example 4.34. Consider a topological space (X, T ) which is described in Example 4.4. Let A = {a, b} and B = {a, c, d} be subsets of X. Then Intp(A) = {a} ⊆ Intp(B) = {a, c, d}. Definition 4.35 ([2]). For any subset A of X, the set bα(A) := A \ Intα(A) is called the α-border of A, and the set Frα(A) := Clα(A) \ Intα(A) is called the α-frontier of A. Definition 4.36. For any subset A of X, the set bp(A) := A \ Intp(A) is called the pre-border of A, and the set Frp(A) := Clp(A) \ Intp(A) is called the pre-frontier of A. Note that if A is a pre-closed subset of X, then bp(A) = Frp(A). Example 4.37. (1) Let (X, T ) be the topological space which is described in Example 4.4. Let A = {a, b, e} be a subset of X. Then Intp(A) = {a}, and so bp(A) = {b, e}. Since A = {a, b, e} is pre-closed, Clp(A) = {a, b, e} and thus Frp(A) = {b, e}. (2) Consider the topological space (X, T ) which is given in Example 3.2. For a subset A = {b, c, d} of X, we have Intp(A) = {c, d} and Clp(A) = {b, c, d, e}. Hence bp(A) = {b} and Frp(A) = {b, e}. Applications of pre-open sets 223 Proposition 4.38. For a subset A of X, the following statements hold: (1) bp(A) ⊆ bα(A). (2) A = Intp(A) ∪ bp(A). (3) Intp(A) ∩ bp(A) = ∅. (4) A is a pre-open set if and only if bp(A) = ∅. (5) bp(Intp(A)) = ∅. (6) Intp(bp(A)) = ∅. (7) bp(bp(A)) = bp(A). (8) bp(A) = A ∩ Clp(X \ A). (9) bp(A) = A ∩ Dp(X \ A). Proof. (1) Since Intα(A) ⊆ Intp(A), we have bp(A) = A \ Intp(A) ⊆ A \ Intα(A) = bα(A). (2) and (3). Straightforward. (4) Since Intp(A) ⊆ A, it follows from Proposition 4.33(2) that A is pre-open ⇔ A = Intp(A) ⇔ bp(A) = A \ Intp(A) = ∅. (5) Since Intp(A) is pre-open, it follows from (4) that bp(Intp(A)) = ∅. (6) If x ∈ Intp(bp(A)), then x ∈ bp(A) ⊆ A and x ∈ Intp(A) since Intp(bp(A)) ⊆ Intp(A). Thus x ∈ bp(A) ∩ Intp(A) = ∅, which is a contradiction. Hence Intp(bp(A)) = ∅. (7) Using (6), we get bp(bp(A)) = bp(A) \ Intp(bp(A)) = bp(A). (8) Using Proposition 4.33(6), we have bp(A) = A \ Intp(A) = A \ (X \ Clp(X \ A)) = A ∩ Clp(X \ A). (9) Applying (8) and Theorem 4.16, we have bp(A) = A ∩ Clp(X \ A) = A ∩ ((X \ A) ∪ Dp(X \ A)) = A ∩ Dp(X \ A). This completes the proof. � Lemma 4.39. For a subset A of X, A is pre-closed if and only if Frp(A) ⊆ A. Proof. Assume that A is pre-closed. Then Frp(A) = Clp(A) \ Intp(A) = A \ Intp(A) ⊆ A. Conversely suppose that Frp(A) ⊆ A. Then Clp(A) \ Intp(A) ⊆ A, and so Clp(A) ⊆ A since Intp(A) ⊆ A. Noticing that A ⊆ Clp(A), we have A = Clp(A). Therefore A is pre-closed. � Theorem 4.40. For a subset A of X, the following assertions are valid: (1) Frp(A) ⊆ Frα(A). (2) Clp(A) = Intp(A) ∪ Frp(A). (3) Intp(A) ∩ Frp(A) = ∅. (4) bp(A) ⊆ Frp(A). 224 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee (5) Frp(A) = bp(A) ∪ (Dp(A) \ Intp(A)). (6) A is a pre-open set if and only if Frp(A) = bp(X \ A). (7) Frp(A) = Clp(A) ∩ Clp(X \ A). (8) Frp(A) = Frp(X \ A). (9) Frp(A) is pre-closed. (10) Frp(Frp(A)) ⊆ Frp(A). (11) Frp(Intp(A)) ⊆ Frp(A). (12) Frp(Clp(A)) ⊆ Frp(A). (13) Intp(A) = A \ Frp(A). Proof. (1) Since Clp(A) ⊆ Clα(A) and Intα(A) ⊆ Intp(A), it follows that Frp(A) = Clp(A) \ Intp(A) ⊆ Clα(A) \ Intp(A) ⊆ Clα(A) \ Intα(A) = Frα(A). (2) Straightforward. (3) Intp(A) ∩ Frp(A) = Intp(A) ∩ (Clp(A) \ Intp(A)) = ∅. (4) Since A ⊆ Clp(A), we have bp(A) = A \ Intp(A) ⊆ Clp(A) \ Intp(A) = Frp(A). (5) Using Theorem 4.16, we obtain Frp(A) = Clp(A) \ Intp(A) = (A ∪ Dp(A)) ∩ (X \ Intp(A)) = (A \ Intp(A)) ∪ (Dp(A) \ Intp(A)) = bp(A) ∪ (Dp(A) \ Intp(A)). (6) Assume that A is pre-open. Then Frp(A) = bp(A) ∪ (Dp(A) \ Intp(A)) = ∅ ∪ (Dp(A) \ A) = Dp(A) \ A = bp(X \ A) by using (5), Proposition 4.38(4), Proposition 4.33(2) and Proposition 4.38(9). Conversely suppose that Frp(A) = bp(X \ A). Then ∅ = Frp(A) \ bp(X \ A) = (Clp(A) \ Intp(A)) \ ((X \ A) \ Intp(X \ A)) = A \ Intp(A) by (4) and (5) of Proposition 4.33, and so A ⊆ Intp(A). Since Intp(A) ⊆ A in general, it follows that Intp(A) = A so from Proposition 4.33(2) that A is pre-open. (7) Using Proposition 4.33(5), we have Clp(A) ∩ Clp(X \ A) = Clp(A) ∩ (X \ Intp(A)) = Clp(A) \ Intp(A) = Frp(A). (8) It follows from (7). Applications of pre-open sets 225 (9) we have Clp(Frp(A)) = Clp(Clp(A) ∩ Clp(X \ A)) ⊆ Clp(Clp(A)) ∩ Clp(Clp(X \ A)) = Clp(A) ∩ Clp(X \ A) = Frp(A). Obviously Frp(A) ⊆ Clp(Frp(A)), and so Frp(A) = Clp(Frp(A)). Hence Frp(A) is pre-closed. (10) This is by (9) and Lemma 4.39. (11) Using Proposition 4.33(3), we get Frp(Intp(A)) = Clp(Intp(A)) \ Intp(Intp(A)) ⊆ Clp(A) \ Intp(A) = Frp(A). (12) We obtain Frp(Clp(A)) = Clp(Clp(A)) \ Intp(Clp(A)) ⊆ Clp(A) \ Intp(A) = Frp(A). (13) We get A \ Frp(A) = A \ (Clp(A) \ Intp(A)) = A ∩ ((X \ Clp(A)) ∪ Intp(A)) = ∅ ∪ (A ∪ Intp(A)) = Intp(A). This completes the proof. � The converses of (1) and (4) of Theorem 4.40 are not true in general as seen in the following example. Example 4.41. In Example 3.2, let A = {a, b, c}. Then Frp(A) = {e} ( {b, c, d, e} = Frα(A), which shows that the reverse inclusion of Theorem 4.40(1) is not valid. Also, Example 4.37(2) shows that the reverse inclusion of Theorem 4.40(4) is not valid in general. Definition 4.42 ([2]). For a subset A of X, Extα(A) = Intα(X \ A) is said to be an α-exterior of A. Definition 4.43. For a subset A of X, the semi-interior of X \ A is called the pre-exterior of A, and is denoted by Extp(A), that is, Extp(A) = Intp(X \ A). Example 4.44. Let (X, T ) be a topological space in Example 4.4. For subsets A = {a, b, c} and B = {b, d} of X, we have Extp(A) = {d, e} and Extp(B) = {a, c, e}. 226 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee Theorem 4.45. For subsets A and B of X, the following assertions are valid. (1) Extα(A) ⊆ Extp(A). (2) Extp(A) is pre-open. (3) Extp(A) = X \ Clp(A). (4) Extp(Extp(A)) = Intp(Clp(A)) ⊇ Intp(A). (5) A ⊆ B ⇒ Extp(B) ⊆ Extp(A). (6) Extp(A ∪ B) ⊆ Extp(A) ∩ Extp(B). (7) Extp(A ∩ B) ⊇ Extp(A) ∪ Extp(B). (8) Extp(X) = ∅, Extp(∅) = X. (9) Extp(A) = Extp(X \ Extp(A)). (10) X = Intp(A) ∪ Extp(A) ∪ Frp(A). Proof. (1) Using Theorem 4.31, we have Extα(A) = Intα(X \ A) ⊂ Intp(X \ A) = Extp(A). (2) It follows from Lemma 4.10 and Proposition 4.33(1). (3) It is straightforward by Proposition 4.33(6). (4) Applying (5) and (7) of Proposition 4.33, we get Extp(Extp(A)) = Extp(Intp(X \ A)) = Intp(X \ Intp(X \ A)) = Intp(Clp(A)) ⊃ Intp(A). (5) Assume that A ⊂ B. Then Extp(B) = Intp(X \ B) ⊆ Intp(X \ A) = Extp(A) by using Proposition 4.33(7). (6) Applying Proposition 4.33(9), we get Extp(A ∪ B) = Intp(X \ (A ∪ B)) = Intp((X \ A) ∩ (X \ B)) ⊆ Intp(X \ A) ∩ Intp(X \ B) = Extp(A) ∩ Extp(B). (7) Using Proposition 4.33(8), we obtain Extp(A ∩ B) = Intp(X \ (A ∩ B)) = Intp((X \ A) ∪ (X \ B)) ⊇ Intp(X \ A) ∪ Intp(X \ B) = Extp(A) ∪ Extp(B). (8) Straightforward. (9) Using Proposition 4.33(3), we have Extp(X \ Extp(A)) = Extp(X \ Intp(X \ A)) = Intp(X \ A) = Extp(A). (10) Straightforward. � Applications of pre-open sets 227 Let (X, T ) be a topological space which is given in Example 4.4. Take A = {d, e}. Then Extα(A) = {a} and Extp(A) = {a, b, c}. Thus the reverse inclusion of Theorem 4.45(1) is not valid. Let A = {b, e} and B = {c, d, e}. Then Extp(B) = {a} ⊆ {a, c, d} = Extp(A). This shows that the converse of (5) in Theorem 4.45 is not valid. Now let A = {d, e} and B = {c}. Then Extp(A ∪ B) = {a} 6= {a, b} = {a, b, c} ∩{a, b, d, e} = Extp(A) ∩ Extp(B) which shows that the equality in Theorem 4.45(6) is not valid. Finally let A = {a, b} and B = {c, d, e}. Then Extp(A ∩ B) = {a, b, c, d, e} and Extp(A) ∪ Extp(B) = {a, c, d, e}. This shows that the equality in Theorem 4.45(7) is not valid. References [1] D. Andrijevic, Some properties of the topology of α-sets, Mat. Vesnik 36 (1984), 1–10. [2] M. Caldas, A note on some applications of α-sets, Int. J. Math. Math. Sci. 2003, no. 2 (2003), 125–130. [3] M. Caldas and J. Dontchev, On spaces with hereditarily compact α-topologies, Acta Math. Hung. 82 (1999), 121–129. [4] S. Jafari and T. Noiri, Contra-α-continuous functions between topological spaces, Iranian Int. J. Sci. 2, no. 2 (2001), 153–167. [5] S. Jafari and T. Noiri, Some remarks on weak α-continuity, Far East J. Math. Sci. 6, no. 4 (1998), 619–625. [6] S. N. Maheshwari and S. S. Thakur, On α-irresolute mappings, Tamkang J. Math. 11 (1980), 209–214. [7] S. N. Maheshwari and S. S. Thakur, On α-compact spaces, Bull. Inst. Math. Acad. Sinica 13 (1985), 341–347. [8] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47–53. [9] H. Maki, R. Devi and K. Balachandran, Generalized α-closed sets in topology, Bull. Fukuoka Univ. Ed. Part III 42 (1993), 13–21. [10] H. Maki and T. Noiri, The pasting lemma for α-continuous maps, Glas. Mat. 23(43) (1988), 357–363. [11] A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, A note on semi-continuity and precontinuity, Indian J. Pure Appl. Math. 13, no. 10 (1982), 1119–1123. [12] A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, α-continuous and α-open mappings, Acta Math. Hungar. 41, no. 3-4 (1983), 213–218. [13] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41. [14] O. Nj̊astad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970. [15] I. L. Reilly and M. K. Vamanamurthy, On α-sets in topological spaces, Tamkang J. Math. 16 (1985), 7–11. [16] J. Tong, On decomposition of continuity in topological spaces, Acta Math. Hungar. 54, no. 1-2 (1989), 51–55. Received May 2007 Accepted February 2008 228 Y. B. Jun, S. W. Jeong, H. J. Lee and J. W. Lee Young Bae Jun (skywine@gmail.com) Department of Mathematics Education (and RINS), Gyeongsang National Uni- versity, Chinju 660-701, Korea Seong Woo Jeong (liveinworld@hanmail.net) Department of Mathematics Education (and RINS), Gyeongsang National Uni- versity, Chinju 660-701, Korea Hyeon Jeong Lee (jfield@hanmail.net) Department of Mathematics Education (and RINS), Gyeongsang National Uni- versity, Chinju 660-701, Korea Joon Woo Lee (jwlee−angel@hanmail.net) Department of Mathematics Education (and RINS), Gyeongsang National Uni- versity, Chinju 660-701, Korea