6_michael.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 1, 2013, 53-58 ISSN 1307-5543 – www.ejpam.com On Semi-open Sets With Respect To an Ideal Friday Ifeanyi Michael Department of Mathematics, Faculty of Science, The University of Western Ontario, London, Ontario, Canada Abstract. We introduce a notion of semi-open sets in terms of ideals, which generalizes the usual notion of semi-open sets. 2010 Mathematics Subject Classifications: 54C10 Key Words and Phrases: Semi-open sets, Semi-closed sets, Generalized closed sets, Ideals 1. Introduction With the impetus given by Levine’s introduction of semi-open sets and generalized closed sets [11, 12], there have been other attempts by some topologists to study closed sets - to- gether with the accompanying topological notions - from different perspectives [see, for ex- ample, 1, 2, 3, 5, 6, 4]. Relevant to the present work is the idea of using topological ideals in describing topological notions, which, for some years now, has been an interesting subject for investigation [see some of the pioneering works in 7, 8, 9]. We recall here that an ideal I on a topological space (X ,τ) is a non-empty collection of subsets of X having the heredity property (that is, if A∈ I and B ⊂ A, then B ∈ I ) and also satisfying finite additivity (that is, if A, B ∈ I , then A∪ B ∈ I ). In this paper, we define semi-open sets with respect to an ideal I , and also study some of their properties. It turns out that our notion of semi-open sets with respect to a given ideal I generalizes both the usual notion of semi-openness [11] and the notion of semi-I -openness considered in [5]; in particular, semi-I -openness implies the usual semi-openness, which in turn implies semi-openness in our sense. Throughout we work with a topological space (X ,τ) (or simply X ), where no separation axioms are assumed. The usual notation cl(A) for the closure, and int(A) for the interior, of a subset A of a topological space (X ,τ), will be used [see 3, 10, 4, for example]. Email address: fmi hael�uwo. a http://www.ejpam.com 53 c© 2013 EJPAM All rights reserved. F. Michael / Eur. J. Pure Appl. Math, 6 (2013), 53-58 54 2. Semi-openness With Respect To an Ideal Let X be a topological space. Recall that a subset A of X is said to be semi-open [11] if there is an open set U such that U ⊆ A⊂ cl(U). This motivates our first definition. Definition 1. A subset A of X is said to be semi-open with respect to an ideal I (written as I -semi-open) if there exists an open set U such that U − A∈ I and A− cl(U) ∈ I . If A ∈ I , then it is easy to see that A is I -semi-open. Moreover, every open set A is semi-open, and every semi-open set B is I -semi-open, for any ideal I on X . Example 1. Consider a topological space (X ,τ); X = {a, b, c} and τ = {∅, {a}, {a, c}, X }. Choose I = {∅, {b}, {c}, {b, c}}, and observe that {b} is I -semi-open; however, {b} is not semi- open in the sense of [11] as there is no open set U such that U ⊂ {b} ⊂ cl(U). Thus, if a set is I -semi-open, it may not be semi-open in the usual sense. For an ideal I that is not countably additive, the concepts of semi-openness and I -semi- openness coincide in the following case. Theorem 1. For an ideal I on a topological space X , the following are equivalent: 1. I is the minimal ideal on X , that is, I = {∅}; 2. The concepts of semi-openness and I -semi-openness are the same. Proof. First suppose that I = {∅}. It suffices to show that whenever a set A is I -semi- open, then it is semi-open in the usual sense. Indeed, if A is I -semi-open, then there is an open set U such that U − A, A− cl(U) ∈ I = {∅}, and so U ⊂ A ⊂ cl(U), proving that A is semi-open. Conversely, suppose that whenever a set A is I -semi-open, then it is semi-open. Let B ∈ I . Then, B is I -semi-open, and by assumption, B is semi-open. Thus, there is an open set V1 such that V1 ⊂ B ⊂ cl(V1). Since B ∈ I and V1 ⊂ B, we have that V1 ∈ I , and so B∪V1 ∈ I . As B∪V1 is I -semi-open, it is semi-open, so that there is an open set V2 for which V2 ⊂ (B ∪ V1) ⊂ cl(V2). Similarly, there is an open set V3 such that V3 ⊂ (B ∪ V1 ∪ V2)⊂ cl(V3). Continuing in this way, we have an infinite collection of open sets V1, V2, V3, . . . , such that B ∪ V1 ∪ V2 ∪ V3 ∪ . . . ∈ I , which is impossible, as the ideal I is not closed under countable additivity. Thus, it must be the case that V1 = ∅ (similarly for the other Vi ’s); therefore, cl(V1) = ∅, and the relations V1 ⊂ B ⊂ cl(V1) then give B = ∅, proving that I = {∅}. Proposition 1. Let I and I ′ be two ideals on a topological space X . 1. If I ⊂ I ′, then every I -semi-open set A is I ′-semi-open; 2. If A is (I ∩I ′)-semi-open, then it is simultaneously I -semi-open and I ′-semi-open. Corollary 1. For a subset A of X and an ideal I on X , recall that IA = {A∩ S|S ∈ I } is also an ideal on X . 1. If a set B is IA-semi-open, then it is I -semi-open. F. Michael / Eur. J. Pure Appl. Math, 6 (2013), 53-58 55 2. If A= ∅, then IA = I∅ = {∅}, the minimal ideal. Thus, if a set C is I∅-semi-open, then C is also I -semi-open. Proposition 2. If A and B are both I -semi-open, then so is their union A∪ B. Proof. Let the given conditions hold. To show that A ∪ B is I -semi-open, we need to produce an open set U such that U − (A∪ B) ∈ I and (A∪ B)− cl(U) ∈ I . Since A and B are both I -semi-open, there are open sets U1 and U2 such that U1 − A∈ I , A− cl(U1) ∈ I , U2 − B ∈ I , B− cl(U2) ∈ I . Choose U = U1 ∪ U2, and observe that (U1 ∪ U2)− (A∪ B) = ((U1− A)− B)∪ ((U2− B)− A) ∈ I . Also, (A∪ B)− cl(U1 ∪ U2) = ((A− cl(U1))− cl(U2))∪ ((B− cl(U2))− cl(U1)) ∈ I . Therefore, by definition, A∪ B is I -semi-open. Proposition 3. Let X be a topological space in which there is an open singleton subset {a} satisfying cl({a}) = X . For any ideal I on X with {a} ∈ I , we have that: 1. Every singleton subset of X is I -semi-open; 2. Every finite subset of X is I -semi-open. Proof. Let the given conditions hold. Suppose that {s} is a singleton subset of X . Since ({a} is open and) {a} − {s} = {a} ∈ I , and {s} − cl({a}) = {s} − X = ∅ ∈ I , it follows that {s} is I -semi-open; this proves (1). To see that (2) holds, let A= {s1, s2, s3, . . . , sn} be a finite subset of X . Since A= {s1} ∪ {s2} ∪ {s3} ∪ . . . ∪ {sn}, the result follows from the fact that each singleton subset {si} (i = 1,2,3, . . . , n) is I -semi-open and a repeated use of Proposition 2 above. Proposition 3 does not hold for any choice of ideal. Example 2. Consider X = {a, b, c}, τ = {∅, {a}, {a, c}, X }, and observe that cl({a}) = X . If we choose the minimal ideal I = {∅} on X , then the singleton subset {b} is not I -semi-open, as there is no open set U satisfying U − {b} ∈ I and {b} − cl(U) ∈ I simultaneously. Proposition 4. Let A and B be subsets of a topological space X such that A is open, A⊂ B, and A is dense in B (that is, B ⊂ cl(A)). Then B is I -semi-open for any ideal I on X . In particular, the conclusion holds in the special case when B = cl(A). Remark 1. If two sets A and B are I -semi-open, then their intersection A∩ B need not be I - semi-open. For example, let X = {a, b, c} be equipped with a topologyσ = {∅, {a}, {c}, {a, c}, X }. Note that cl({a}) = {a, b} and cl({c}) = {b, c}; moreover, the subsets {a, b} and {b, c} are semi- open with respect to the minimal ideal I = {∅}, in view of Proposition 4 above. However, the singleton subset {b} = {a, b}∩{b, c} is not semi-open with respect to the minimal ideal I = {∅}. F. Michael / Eur. J. Pure Appl. Math, 6 (2013), 53-58 56 Obviously, if A, B ∈ I , then A∩ B will be semi-open with respect to the ideal I . For those subsets that are not members of the ideal I , one rather strong condition for their intersection to be semi-open with respect to the ideal I is given below. Proposition 5. Let I be an ideal on a topological space X , where every non-empty open subset of X is dense, and the collection of open subsets of X satisfies the finite intersection property. 1. If A is I -semi-open and A⊂ B, then B is I -semi-open; 2. If A is I -semi-open, then so is A∪ B, for any subset B of X ; 3. If both A and B are I -semi-open, then so is their intersection A∩ B. Proof. (1) Suppose that A is I -semi-open, and that A⊂ B. There is an open set U such that U − A∈ I and A− cl(U) ∈ I . Notice that such an open set U is necessarily non-empty, since we are dealing with those subsets of X that do not belong to the ideal I . Since A ⊂ B, we have that U − B ⊂ U −A∈ I ; moreover, B− cl(U) = B− X = ∅ ∈ I . Thus, B is I -semi-open. (2) Since A⊂ B⇔ A∪ B = B, (2) immediately follows from (1). (3) Suppose that both A and B are I -semi-open. Without loss of generality, suppose that A∩ B 6= ∅; otherwise, A∩ B will be trivially I -semi-open. By assumption, there are open sets U and V such that U − A, A− cl(U) ∈ I and V − B, B − cl(V ) ∈ I . Consider the open set U ∩ V , which is non-empty (by the finite intersection property). Since (U∩V )−(A∩B) = ((U−A)∩V)∪(U∩(V−B)) ∈ I and (A∩B)−cl(U∩V ) = (A∩B)−X = ∅ ∈ I , it follows that A∩ B is I -semi-open. Remark 2. In Example 2, we saw that the singleton subset {b} was not semi-open with respect to the minimal ideal I = {∅}. Notice that the set {a, b} = {a} ∪ {b} is semi-open with respect to I = {∅}, simply because the non-empty open singleton subset {a} is dense in X ; this is an instance of Proposition 5(2) above. Proposition 6. Under the conditions of Proposition 5, we have that A is I -semi-open if and only if cl(A) is I -semi-open. Proof. If A is I -semi-open, then - because A ⊂ cl(A) - so is cl(A), by Proposition 5(2). Conversely, suppose that cl(A) is I -semi-open. Then there is an open set U such that U − cl(A) ∈ I and cl(A)− cl(U) ∈ I . Notice that U is necessarily non-empty; otherwise, we would have cl(U) = ∅, which forces A∈ I , which we don’t want (as we’re dealing with those subsets that do not belong to the ideal I ). To show that A is I -semi-open, consider the open set V = U−cl(A) = U∩(cl(A))c ∈ I , by assumption. We have that V−A= U∩(cl(A))c∩Ac ∈ I , because of the heredity property; moreover, A− cl(V ) = A− cl(U ∩ (cl(A))c) = A− X = ∅ ∈ I . This shows that A is I -semi-open. Theorem 2. The following are equivalent for a subset A of X : REFERENCES 57 1. X − A is I -semi-open. 2. There exists a closed set F such that int(F)− A∈ I and A− F ∈ I . Proof. First suppose that X − A is I -semi-open. Then there exists an open set U such that U − (X − A) ∈ I and (X − A) − cl(U) ∈ I . Since U − (X − A) = A− (X − U) and (X−A)−cl(U) = int(X−U)−A, we have (2) by choosing the closed set X−U as F . Conversely, if we suppose that (2) holds, then the choice of the open set U = X − F shows that X − A is I -semi-open. Definition 2. A subset A of X is said to be semi-closed with respect to I (written as I -semi- closed) if and only if X − A is I -semi-open. Proposition 7. If both A and B are I -semi-closed, then so is their intersection A∩ B. Proof. Let the given conditions hold. There are closed sets F1 and F2 such that int(F1)− A,A− F1 ∈ I and int(F2)− B, B − F2 ∈ I . With F = F1 ∩ F2, we have that int(F1 ∩ F2)− (A∩ B) = ((int(F1)− A)∩ int(F2))∪ (int(F1)∩ (int(F2)− B)) ∈ I , and (A∩ B)− (F1 ∩ F2) = ((A− F1)∩ B)∪ (A∩ (B− F2)) ∈ I ; therefore, A∩ B is I -semi-closed. References [1] K. Al-Zoubi. On generalized ω-closed sets. International Journal of Mathematics and Mathematical Sciences, 2005(13):2011–2021, 2005. [2] S.P. Arya and T.M. Nour. Characterization of s-normal spaces. Indian Journal of Pure and Applied Mathematics, 21(8):717–719, 1990. [3] C. Boonpok. Generalized closed sets in isotonic spaces. International Journal of Mathe- matical Analysis, 5:241–256, 2011. [4] R. Davi, K. Balachandran, and H. Maki. Semi-generalized homeomorphisms and gener- alized semi-homeomorphisms in topological spaces. Indian Journal of Pure and Applied Mathematics, 26(3):271–284, 1995. [5] E. Hatir and T. Noiri. On semi-I -open sets and semi-I -continuous functions. Acta Mathematica Hungarica, 107(4):345–353, 2005. [6] S. Jafari and N. Rajesh. Generalized closed sets with respect to ideals. European Journal of Pure and Applied Mathematics, 4(2):147–151, 2011. [7] D. Jankovic and T.R. Hamlett. Ideals in topology and applications. Lecture Notes in Pure and Applied Mathematics, pages 115–125, 1990. REFERENCES 58 [8] D. Jankovic and T.R. Hamlett. Ideals in topology and the set operator. Bollettino della Unione Matematica Italiana, 7:863–894, 1990. [9] D. Jankovic and T.R. Hamlett. New topologies from old via ideals. American Mathemat- ical Monthly, 97:295–310, 1990. [10] J.L. Kelly. General topology. D van Nostrand Company Incorporated, Princeton, N.J., 1955. [11] N. Levine. Semi-open sets and semi-continuity in topological spaces. American Mathe- matical Monthly, 70:36–41, 1963. [12] N. Levine. Generalized closed sets in topology. Rendiconti del Circolo Matematico di Palermo, 19:89–96, 1970.