1_hatir.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 2, 2009, (172-181) ISSN 1307-5543 – www.ejpam.com On Hausdorff Spaces Via Ideals and Semi-I-irresolute Functions E. Hatir1∗ and T. Noiri2 1 Selçuk Üniversitesi, Eğitim Fakültesi, 42090, Meram-Konya, Turkey 2 2949 Shiokita-shi, Kumamoto-ken 869-5142, Japan Abstract. We introduce the notion of semi-I-Hausdorff spaces which is weaker than Hausdorff spaces and independent both I-Hausdorff and quasi-I-Hausdorff. AMS subject classifications: Primary 54C08, 54H05; Secondary 54C10 Key words: I-Hausdorff, quasi-I-Hausdorff, semi-I-Hausdorff, Semi-I-irresolute, Semi-I-open set. 1. Introduction In [4], Dontchev has introduced and studied I-Hausdorff spaces. In [13] , Nasef has improved I-Hausdorff spaces and defined quasi-I-Hausdorff spaces. In [5] , the ∗Corresponding author. Email addresses: hatir10�yahoo. om (E. Hatir), t.noiri�nifty. om (T. Noiri) http://www.ejpam.com 172 c© 2009 EJPAM All rights reserved. E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 173 present authors defined the notion of semi-open sets via ideals to obtain decomposi- tion of continuity. In the present paper, we introduce the notion of semi-I-Hausdorff spaces which is weaker than Hausdorff spaces and independent both I-Hausdorff and quasi-I-Hausdorff spaces. Using semi-I-irresolute [6] functions, we also investigate its relation with semi-I-Hausdorff spaces. 2. Preliminaries Throughout this paper, (X ,τ) (simply X ) denotes a topological space on which no separation axiom is assumed unless explicitly stated. For a subset A of a topological space X , the closure and the interior of A in X are denoted by Cl(A) and Int(A), respectively. A nonempty collection I of subsets on a topological space (X ,τ) is called a topological ideal on (X ,τ) if it satisfies the following two conditions: (1) if A ∈ I and B ⊂ A, then B ∈ I (heredity); (2) if A ∈ I and B ∈ I , then A∪ B ∈ I (finite additivity). If I is a proper ideal, that is, X /∈ I , then {A : X − A∈ I} is a filter, hence proper ideals are sometimes called dual filters. By (X ,τ, I), we will denote an ideal topological space which means a topological space (X ,τ) with an ideal I on X . No separation property is assumed on X . For a space (X ,τ, I) and a subset A of X , A∗(I) = � x ∈ X : U ∩ A /∈ I for each neighborhood U of x is called the local function of A with respect to I and τ [7] . We simply write A∗ instead of A∗(I) in case there is no chance for confusion. The simplest ideals are {∅} and ℘(X ) which satisfy {∅} ⊂ I ⊂ ℘(X ), for any ideal I on X . Note that Cl∗(A) = A∪A∗ defines a Kuratowski closure operator for a topology τ∗(I) (also denoted by τ∗ when there is no chance for confusion) finer than τ. Definition 2.1. A subset A of an ideal topological space (X ,τ, I) is said to be semi- open [10] (resp. β − open [1] , semi-I-open [5] , I-open [9] , quasi-I-open [2]) if A ⊂ Cl(Int(A)) (resp. A ⊂ Cl(Int(Cl(A))), A ⊂ Cl∗(Int(A)), A ⊂ Int(A∗), A ⊂ E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 174 Cl(Int(A∗))). For a subsets defined above, the following diagram holds: DIAGRAM I open −→ semi-I-open −→ semi-open ↓ I − open −→ quasi-I-open −→ β − open Definition 2.2. A space (X ,τ) is said to be semi-Hausdorff [11] (resp. β−Hausdorff [12]) if for every two different points x , y of X , there exist disjoint semi-open (resp. β − open) sets U, V of X such that x ∈ U and y ∈ V. Definition 2.3. An ideal topological space (X ,τ, I) is called I-Hausdorff [4] (resp. quasi-I-Hausdorff [13]) if for every two different points x , y of X , there exist disjoint I-open sets (resp. quasi-I-open) U, V of X such that x ∈ U and y ∈ V. 3. Semi-I-Hausdorff Spaces Definition 3.1. An ideal topological space (X ,τ, I) is called semi-I-Hausdorff if for each two distinct points x 6= y, there exist semi-I-open sets U and V containig x and y, respectively such that U ∩ V = ∅. Then the points x and y are said to be semi − I − separated. Theorem 3.1. For an ideal topological space (X ,τ, I), the following statements hold: 1. Every Hausdorff space is semi-I-Hausdorff. 2. Every semi-I-Hausdorff space is semi-Hausdorff. Proof. This follows from the definition of semi-I -open sets. E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 175 For ideal topological spaces, the following diagram holds: DIAGRAM II Hausdorff −→ semi-I-Hausdorff −→ semi-Hausdorff ↓ I-Hausdorff −→ quasi-I-Hausdorff −→ β −Hausdorff Remark 3.1. (1) It is shown in Example 2.3 and 2.4 of [4] that Hausdorffness and I-Hausdorffness are independent of each other. (2) In the following examples, it will be shown that semi-I-Hausdorffness is indepen- dent to quasi-I-Hausdorffness and to I-Hausdorffness. Example 3.1. Let X be the real line with the "rigth-ray" topology τ that is the nontrivial open sets are the form (x,∞), where x is any real number. Let I be the ideal of all finite subsets of X. Then the ideal topological space (X ,τ, I) is an I-Hausdorff space which is not Hausdorff � 4, Example 2.3 � . However, this space is not even semi-Hausdorff because, every nonempty semi-open set has the nonempty interior. Example 3.2. Let X = {a, b} , τ be the discrete topology on X and I = ℘(X ). Then Dontchev [4] showed that the space is Hausdorff, but it is not I-Hausdorff. Moreover, Nasef [13] showed that the space is not even quasi-I-Hausdorff. Example 3.3. Let X = {a, b, c} , τ = {∅, X , {a} , {b} , {a, b}} and I = {∅} . Then (X ,τ, I) is a semi-I-Hausdorff space which is not Hausdorff. If we take I = ℘(X ), then (X ,τ, I) is semi-Hausdorff, but it is neither semi-I-Hausdorff nor quasi-I-Hausdorff. Theorem 3.2. Let (X ,τ, I) be an ideal topological space. 1. Let I = {∅} . Then (X ,τ, I) is semi-I-Hausdorff (resp. quasi-I-Hausdorff) if and only if it is semi-Hausdorff (resp. β−Hausdorff). E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 176 2. Let I = ℘(X ). Then (X ,τ, I) is Hausdorff if and only if it is semi-I-Hausdorff. Proof. (1) Let I = {∅} . Then A∗ = Cl(A) and Cl∗(A) = Cl(A) for every subset A of X . Therefore, we have SIO(X ,τ) = SO(X ,τ) (resp. QIO(X ,τ) = β(X ,τ)) and hence (X ,τ, I) is semi-I-Hausdorff (resp. quasi-I-Hausdorff) if and only if semi- Hausdorff (resp. β−Hausdorff), where QIO(X ,τ) denotes the set of all quasi-I-open sets. (2) Let I = ℘(X ). Then A∗ = ∅ and Cl∗(A) = A for every subset A of X . Let A ∈ SIO(X ,τ), then A⊂ Cl∗(Int(A)) = Int(A) and hence A is open in (X ,τ). Therefore, (X ,τ, I) is Hausdorff if and only if it is semi-I-Hausdorff. Definition 3.2. An ideal topological space (X ,τ, I) is called semi-I-complete (resp. quasi- I-complete [13]) if τ∗ = SIO(X ,τ) (resp. τ∗ = QIO(X ,τ)), that is, a subset A of X is τ∗− open if and only if it is semi-I-open (resp. quasi-I-open). Theorem 3.3. Let (X ,τ, In) be an ideal topological space, where In is the ideal of the nowhere dense sets of (X ,τ). 1. (X ,τ, In) is semi-I-Hausdorff (resp. quasi-I-Hausdorff) if and only if it is semi-Hausdorff (resp. β−Hausdorff). 2. (X ,τ, In) is semi-Hausdorff and semi-I-complete (resp. β−Hausdorff and quasi-I-complete), then it is Hausdorff. Proof. (1) Since In is the ideal of nowhere dense sets of (X ,τ), we have A∗ = Cl(Int(Cl(A))) and hence by Example 2.10 of [8] Cl∗(A) = A∪Cl(Int(Cl(A))) = αCl(A), where αCl(A) denotes the α− closure of A.For every subset A of X , Cl∗(Int(A)) = Int(A)∪ Cl(Int(Cl(Int(A)))) = Int(A)∪ Cl(Int(A)) = Cl(Int(A)). Therefore, A ∈ SIO(X ,τ) if and only if A ∈ SO(X ,τ).By this fact, it follows that (X ,τ, In) is semi-I-Hausdorff if and only if it is semi-Hausdorff. On the other hand, E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 177 Cl(Int(A∗)) = Cl(Int(Cl(Int(Cl(A))))) = Cl(Int(Cl(A))) for every subset A of X . Therefore, A∈ QIO(X ,τ) if and only if A∈ β(X ,τ). It follows that (X ,τ, In) is quasi-I-Hausdorff if and only if it is β −Hausdor f f . (2) Let (X ,τ, In) be semi-Hausdorff and semi-I-complete. Then A ∈ SIO(X ,τ) if and only if A ∈ τ∗ if and only if A is α − open. By the proof of (1), SO(X ,τ) = SIO(X ,τ) and hence (X ,τ, I) is Hausdorff. The another result is shown similarly. Lemma 3.1. Let I and J be two ideals on a topological space (X ,τ). If I ⊂ J , then the following properties hold: 1. C l∗J (A)⊂ Cl∗I (A) for each subset A of X , 2. SIO(X ,τ, J) ⊂ SIO(X ,τ, I). Proof. (1) If I ⊂ J , then A∗(J) ⊂ A∗(I) and Cl∗J (A) = A∪A∗(J) ⊂ A∪A∗(I) = Cl∗I (A) for each subset A of X . (2) Let A ∈ SIO(X ,τ, J). Then A ⊂ Cl∗J (Int(A)) ⊂ Cl∗I (Int(A)) and hence A ∈ SIO(X ,τ, I). Theorem 3.4. Let I and J be two ideals on a topological space (X ,τ) and I ⊂ J . If (X ,τ, J) is semi-I-Hausdorff, then (X ,τ, I) is semi-I-Hausdorff. Proof. This is an immediate consequence of Lemma 1 A semi-I-open subspace of semi-I-Hausdorff space need not be semi-I-Hausdorff as shown in the following example. Example 3.4. Let X = {a, b, c} , τ = {∅, X , {a} , {b} , {a, b}} and I = {∅} . Then (X ,τ, I) is semi-I-Hausdorff. But, take A = {a, c} ∈ SIO(X ,τ), then (A,τ|A, I|A) is not semi-I-Hausdorff. Lemma 3.2. (Hatir and Noiri [5]) Let (X ,τ, I) be an ideal topological space. If U ∈ τ and V ∈ SIO(X ,τ), then U ∩ V ∈ SIO(U ,τ|U , I|U). E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 178 Theorem 3.5. Let (X ,τ, I) be a semi-I-Hausdorff space and A⊂ X . Then if A is open, then (A,τ|A, I|A) semi-I-Hausdorff. Proof. This follows from Lemma 2 4. Semi-I-irresolute Functions In this section, we investigate some properties of semi-I-irresolute functions. First, we shall recall some definition of functions. Definition 4.1. A function f : (X ,τ, I) −→ (Y,σ, J) is said to be 1. Semi-I-continuous [5] if for every V ∈ σ, f −1(V ) is semi-I-open set, 2. Semi-I-irresolute if for every V ∈ SJO(Y,σ), f −1(V ) ∈ SIO(X ,τ), 3. Irresolute [3] if for every V ∈ SO(Y,σ), f −1(V ) ∈ SO(X ,τ). Remark 4.1. In [6] , the present authors called semi-I-irresolute functions I-irresolute. However, Dontchev [4] defined a function f : (X ,τ, I) −→ (Y,σ, J) to be I-irresolute if f −1(V ) is I-open in (X ,τ, I) for every I-open in (Y,σ, J). Theorem 4.1. For a function f : (X ,τ, I) −→ (Y,σ, J), the following properties are equivalent; 1. f is semi-I-irresolute, 2. The inverse image of each semi-I-closed set in (Y,σ, J) is semi-I-closed in (X ,τ, I), 3. For each x ∈ X and each V ∈ SIO(Y,σ) containing f (x), there exists U ∈ SIO(X ,τ) containing x such that f (U)⊂ V. Proof. The proof is obvious from the fact that the arbitrary union of semi-I-open sets is semi-I-open [6, Theorem 3.4] . E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 179 Remark 4.2. Irresolute functions are not in general semi-I-irresolute as shown by the following example. Example 4.1. Let X = {a, b, c} , τ= {∅, X , {a}} and I = {∅, {a}} and J = {∅} . Then the identity function f : (X ,τ, I) −→ (X ,σ, J) is irresolute, but it is not semi-I-irresolute since {a, c} ∈ SIO(X ,σ, J) and f −1({a, c}) = {a, c} /∈ SIO(X ,τ, I). Theorem 4.2. Let f : (X ,τ, I) −→ (Y,σ, J) be a function, where I and J are ideals on Y, respectively. If I = J = {∅} or In, then semi-I-irresoluteness and irresoluteness are equivalent. Proof. This follows from the proofs of Theorems 2(1) and 3(1). Theorem 4.3. Let f be a semi-I-irresolute injection from a space (X ,τ, I) into a space (Y,σ, J). If Y is semi-I-Hausdorff, then X is also semi-I-Hausdorff. Proof. Let x , y ∈ X and x 6= y. Then f (x) 6= f (y) thus f (x) and f (y) are semi-I- separated in Y by semi-I-open sets U and V , respectively. Since f is semi-I-irresolute, f −1(U) and f −1(V ) are disjoint semi-I-open sets containing x and y, respectively. This shows that X is semi-I-Hausdorff. Theorem 4.4. Let (X ,τ, I) be an ideal topological space with the following property; if x 6= y, where x , y ∈ X , then there exist a Hausdorff space (Y,σ) and a semi- I-continuous function f : (X ,τ, I) −→ (Y,σ) such that f (x) 6= f (y). Then X is semi-I-Hausdorff. Proof. The proof is straightforward. Theorem 4.5. Let f : (X ,τ, I) −→ (Y,σ, J) be a function and V ∈ σ. Then E. Hatir and T. Noiri / Eur. J. Pure Appl. Math, 2 (2009), (172-181) 180 f −1(V ∗)⊂ ( f −1(V ))∗ implies f −1(Cl∗(V )) ⊂ Cl∗( f −1(V )). Proof. f −1(Cl∗(V )) = f −1(V ∪ V ∗) = f −1(V )∪ f −1(V ∗) ⊂ f −1(V )∪ ( f −1(V ))∗ = Cl∗( f −1(V )). Remark 4.3. The converse of Theorem 10 is false as shown by the following example. Example 4.2. Let X = {a, b, c} , τ= {∅, X , {a} , {c} , {a, c}} and σ = {∅, X , {c} , {a, b}} . Let us take I = ℘(X ) and J = {∅, {c}} . The define the identity function f : (X ,τ, I) −→ (X ,σ, J). For the subset {a, b} ∈ σ, we have ({a, b})∗ = {a, b} and ( f −1({a, b}))∗ = ({a, b})∗ =∅ and thus f −1(V ∗) ( f −1(V ))∗ for V = {a, b} ∈ σ. But f −1(Cl∗(V ))⊂ Cl∗( f −1(V )) for every V ∈ σ. Lemma 4.1. (Hatir and Noiri [6]) Let A and B be subsets of an ideal topological space (X ,τ, I). Then the following properties hold: 1. A∈ SIO(X ,τ) if and only if there exists U ∈ τ such that U ⊂ A⊂ Cl∗(U), 2. If A∈ SIO(X ,τ) and A⊂ B ⊂ Cl∗(A), then B ∈ SIO(X ,τ). The following theorem slightly improve the Theorem 4.5 in [6] which states that if f : (X ,τ, I) −→ (Y,σ, J) is semi-I-continuous and f −1(V ∗) ⊂ ( f −1(V ))∗ for each V ∈ σ, then f is semi-I-irresolute. Theorem 4.6. If f : (X ,τ, I) −→ (Y,σ, J) is semi-I-continuous and f −1(Cl∗(V ))⊂ Cl∗( f −1(V )) for each V ∈ σ, then f is semi-I-irresolute. Proof. Let B be any semi-I-open set of (Y,σ, J). By Lemma 3, there exists V ∈ σ such that V ⊂ B ⊂ Cl∗(V ). Therefore, we have f −1(V ) ⊂ f −1(B) ⊂ f −1(Cl∗(V )) ⊂ Cl∗( f −1(V )). Since f is semi-I-continuous and V ∈ σ, f −1(V ) ∈ SIO(X ,τ) and hence by Lemma 3 , f −1(B) is semi-I-open in (X ,τ, I). This shows that f is semi-I-irresolute. REFERENCES 181 References [1] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, β − open sets and β − continuous mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90. [2] M. E. Abd El-Monsef, R. A. Mahmoud and A. A. Nasef, On quasi-I-openness and quasi- I-continuity, Tamkang J. Math., 31(2000), 101-108. [3] S. G. Crossley and S. K. Hildebrand, Semi-topological properties, Fund. Math., 74(1972), 233-254. [4] J. Dontchev, On Hausdorff spaces via topological ideals and I-irresolute functions, In pa- pers on General Topology and Applications, Annals of New York Academy of Sciences, Vol.767(1995), 28-38. [5] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96(4)(2002), 314-349. [6] E. Hatir and T. Noiri, On semi-I-open sets and semi-I-continuous functions, Acta Math. Hungar., 107(4)(2005), 345-353. [7] E. Hayashi, Topologies defined by local properties, Math. Ann., 156(1964), 205-215. [8] D. Janković and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97(1990), 295-310. [9] D. Janković and T. R. Hamlett, Compatible extensions of ideals, Boll. Un. Mat. Ital., (7)(6-B)(1992), 453-465. [10] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70(1963), 36-41. [11] S. N. Maheshwari and R. Prasad, Some new separation axioms, Ann. Soc. Sci. Brux- elles, 89(1975), 395-402. [12] R. A. Mahmoud and M. E. Abd El-Monsef, β− ir resolute and β− topo log ical invari- ant, Proc. Pakistan Acad. Sci., 27(1990), 285-296. [13] A. A. Nasef, On Hausdorff spaces via ideals and quai-I-irresolute functions, Chaos Solitons and Fractals, 14(2002), 619-625.