3_726_acikgoz.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 1, 2011, 20-33 ISSN 1307-5543 – www.ejpam.com On β∗g−closed Sets and New Separation Axioms Ahu Açıkgöz Department of Mathematics, Aksaray University, 68100 Aksaray Abstract. In this paper, by using β∗−set [24] we introduce a new class of sets called β∗g−closed sets, which is stronger than g−closed sets and weaker than closed sets. We define two new separation ax- ioms called β∗T1/2 and β∗∗T1/2 spaces as applications of β∗g−closed sets. The notions β∗g−continuity and β∗g−irresoluteness are also introduced. 2000 Mathematics Subject Classifications: 54C08, 54A05 Key Words and Phrases: β∗-set, β∗g−closed set, β∗g−continuous,β∗T1/2 space 1. Introduction and Preliminaries To date, many studies have been made on closed sets and set concepts derived from this set. The concept of g−closed sets was introduced by Levine [17] and was used to obtain a T1/2 space in which the closed sets and g−closed sets coincide. This natural generalization of a closed set concept has made it possible to use the concept in many areas, especially in quantum physics [13] and computer graphics [13-15]. The notion has been studied exten- sively in recent years by many topologists. More importantly several new separations which are between T0 and T1 such as T1/2, Tgs, πgp− T1/2 and T3/4 are suggested. Some of these have been found to be useful in computer science and digital topology (see [7, 12-15], for example). As a brief literature review, related studies of g−closed sets can be summarized as follows. Dontchev and Noiri [8] introduced the notion of r g−closed sets which are weaker than that of g−closed sets. Kumar [16] defined the notion of g∗-closed sets that are gener- alizations of g−closed sets and introduced T ∗ 1/2 and ∗T1/2 spaces as applications of g∗-closed sets. Devi et al. [6] introduced and studied gs−closed and sg−closed sets which are weaker than g−closed sets. Arya and Nour [1] gave some properties of s-normal spaces by using gs−open sets. The notion of s-normal space was studied extensively by Noiri [20]. Zaitsev [25] introduced the notions of π-closed sets and guasi normal spaces. Dontchev and Noiri [8] introduced the notion of πg-closed sets and obtained some theo- rems in quasi normal spaces by using this notion. Of course, these studies of general topology Email address: ahua ikgoz�aksaray.edu.tr http://www.ejpam.com 20 c© 2010 EJPAM All rights reserved. A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 21 are not limited with these [4,9-10].More recently, several topologists have defined new sepa- ration axioms at topological space by giving some convenient definitions of variety. Park [21] has introduced the class of πgp−closed sets which is weaker than gp−closed and stronger than gpr-closed sets. Park and Park [22] further studied the class of πgp−closed sets and defined the concepts πGP-compactness and πGP-connectedness. Aslım et al. [2] have in- troduced the notions of πgs−closed sets which are implied by that of gs−closed sets and πgs− T1/2-spaces. On the other hand, recently Yuksel and Beceren [24] have defined the no- tion of β∗-set and established a decomposition of continuity. At this point, we shall introduce and study the notions of β∗g−closed sets which are situated between the class of closed sets and g−closed sets. Using these sets, we introduce two new separation axioms called β∗T1/2 and β∗∗T1/2. (Both β∗T1/2 and β∗∗T1/2 contain the class of T1/2 spaces.) We show that the class of β∗∗T1/2 spaces is the dual of the class of β∗T1/2 spaces to the class of T1/2 spaces. We also introduce β∗g−continuity and β∗g−irresolute functions for preservation theorems. It should be mentioned that the present work may be found relevant to work of Witten [23]. Throughout this paper, spaces (X ,τ) and (Y,σ) (or simply X and Y ) always mean topo- logical spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X . The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A is said to be locally closed (briefly, LC-set) [3] if A = U ∩ V , where U is open and V is closed. A subset A is said to be regular open (resp. regular closed) if A = Int(Cl(A) (resp. A = Cl(Int(A)). The finite union of regular open sets is said to be π-open. The complement of a π-open set is said to be π-closed. A subset A is said to be semiopen [5] if A ⊂ Cl(Int(A)) and the complement of a semiopen set is called semiclosed. The intersection of all semiclosed sets containing A is called the semiclosure [5] of A and is denoted by sC l(A). Dually the semiinterior [5] of A is defined to be the union of all semiopen sets contained in A and is denoted by sInt(A). A subset A is said to be pre open [19] if A ⊂ Int(Cl(A)) and the complement of a pre open set is called pre closed. The intersection of all preclosed sets containing A is called the preclosure [19] of A and is denoted by pCl(A). Dually the preinterior [19] of A is defined to be the union of all pre open sets contained in A and is denoted by pInt(A). Note that sC l(A) = A∪ Int(Cl(A)), sInt(A) = A∩ Cl(Int(A)), pCl(A) = A∪ Cl(Int(A)) and pInt(A) = A∩ Int(Cl(A)). 2. β∗g−-closed Sets Definition 1. A subset A of a space (X ,τ) is called a generalized closed set (briefly, g−closed) [17] if C l(A) ⊂ U whenever A⊂ U and U is open. The complement of a g−closed set is called a g−open set. (a) a regular generalized closed set (for short, r g−closed) [8] if C l(A) ⊂ U whenever A ⊂ U and U is regular open in X; (b) g∗-closed [16] if C l(A)⊂ U whenever A⊂ U and U is g−open; (c) πg−closed [8] if C l(A)⊂ U whenever A⊂ U and U is π-open in X; (d) gp−closed [18] if pC l(A)⊂ U whenever A⊂ U and U is open in X; A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 22 (e) gs−closed [1] if sC l(A)⊂ U whenever A⊂ U and U is open in X; (f) πgp−closed [21] if pC l(A) ⊂ U whenever A⊂ U and U is π-open inX; (g) πgs−closed [2] if sC l(A)⊂ U whenever A⊂ U and U is π-open in X; (h) πgs−open (resp. g∗-open, πg−open, gp−open, πgp−open, gs−open) if the complement of A is πgs−closed (resp. g∗-closed, πg−closed, gp−closed, πgp−closed, gs−closed). Definition 2. A subset A of a space (X,τ) is called (a) a β∗-set [24] if A= U ∩ V , where U is open and Int(V ) = Cl(Int(V )). (b) β∗g−closed if C l(A)⊂ U whenever A⊂ U and U is a β∗-set. (c) β∗sg−closed if sC l(A)⊂ U whenever A⊂ U and U is a β∗-set. (d) β∗pg−closed if pCl(A)⊂ U whenever A⊂ U and U is a β∗-set. (e) β∗pg−open (resp. β∗g−open, β∗sg−open) if the complement of A is β∗pg−closed (resp. β∗g−closed, β∗sg−closed). The class of all β∗g−closed subsets of (X ,τ) is denoted by β∗GC(X ,τ). Levine [17] and Kumar [16] gave the following diagrams using some of the expressions, respectively. DIAGRAM I. closed set −→ g−closed set −→ r g−closed set DIAGRAM II. closed set −→ g∗−closed set −→ g−closed s Furhermore, Aslım et al. [2] indicated that every gs−closed set is a πgs−closed set and every πg−closed set is a πgs−closed set. They gave the following diagram using these properties. DIAGRAM III. pre-closed −→ gp− closed −→ πgp− closed ↑ ↑ ↑ π− closed −→ closed −→ g − closed −→ πg − closed ↓ ↓ ↓ semi-closed −→ gs− closed −→ πgs− closed Remark 1. A LC-set is independent from a g−closed set as it can be seen from the next two examples. A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 23 Remark 2. Let X = {a, b, c} and τ = {X ,;, {a}}. Then {a} is a LC-set, but it is not a g−closed set. Remark 3. Let X = {a, b, c} and τ = {X ,;, {a}}. Then {a, b} is a g−closed set, but it is not a LC-set. Theorem 1. For a subset A of a topological space (X ,τ), the following are equivalent: (a) A is a LC-set. (b) A= U ∩ Cl(A) for some U open set. Proof. (a) → (b): Since A is a LC-set, then A= U ∩ V , where U is open and V is closed. So, A⊂ U and A⊂ V . Hence, Cl(A) ⊂ Cl(V ). Therefore, A⊂ U ∩ Cl(A)⊂ U ∩ Cl(V ) = U ∩ V = A. Thus, A= U ∩ Cl(A). (b) → (a): It is obvious because Cl(A) is closed. Theorem 2. For a subset A of a topological space (X ,τ), the following are equivalent: (a) A is closed. (b) A is a LC-set and g−closed. Proof. (a) → (b): This is obvious. (b) → (a): Since A is a LC-set, then A= U ∩ Cl(A), where U is an open set in X. So, A⊂ U and since A is g−closed, then Cl(A) ⊂ U . Therefore, Cl(A) ⊂U∩Cl(A) = A. Hence, A is closed. Theorem 3. Let (X ,τ) be a topological space. Then we have (a) Every closed set is a β∗g−closed set. (b) Every β∗g−closed set is a g−closed set. Proof. (a) This is obvious. (b) Let A be a β∗g−closed set of (X ,τ) and A⊂ U where U ∈ τ. Since every open set is a β∗-set, so U is a β∗-set of (X ,τ). Since A is a β∗g−closed set, we obtain that Cl(A)⊂ U , hence A is a g−closed set of (X ,τ). A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 24 Remark 4. The converses of Theorem 3 need not be true as shown in the following examples. Example 1. Let X = {a, b, c, d} and τ = {X ,;, {b}, {c}, {a, b}, {b, c}, {a, b, c}, {a, b, d}}. Then {a, b} is a β∗g−closed set, but it is not a closed set. Example 2. Let X = {a, b, c} and τ = {X ,;, {a}, {c}, {a, b}, {a, c}}. Then {c} is a g−closed set, but it is not a β∗g−closed set. Theorem 4. Let (X ,τ) be a topological space. Then we have (a) Every β∗g−closed set is a β∗pg−closed set. (b) Every β∗g−closed set is a β∗sg−closed set. Proof. This is obvious. Remark 5. The converses of Theorem 4 need not be true as shown in the following examples. Example 3. Let X = {a, b, c} and τ = {X ,;, {a}, {c}, {a, b}, {a, c}}. Then {a, b} is a β∗pg−closed set which is not a β∗g−closed set. Example 4. Let X = {a, b, c} and τ = {X ,;, {a}, {c}, {a, b}, {b, c}}. Then {b, c} is a β∗sg−closed set which is not a β∗g−closed set. It can be expanded to the following diagram using Diagrams I, II and III DIAGRAM IV. Remark 6. By the two examples stated below, we show that β∗g−closed and g∗-closed are independent of each other. Example 5. Let X = {a, b, c, d} and τ = {X ,;, {b}, {c}, {a, b}, {b, c}, {a, b, c}, {a, b, d}}. Then {a, b} is a β∗g−closed set, but it is not a g∗-closed set. Example 6. Let X = {a, b, c} and τ = {X ,;, {a}, {c}, {a, b}, {a, c}}. Then {c} is a g∗-closed set, but it is not a β∗g−closed set. A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 25 Remark 7. A β∗-set is independent from β∗g−closed as it can be seen from the next two exam- ples. Example 7. Let X = {a, b, c} and τ = {X ,;, {a}}. Then {a} is a β∗-set, but it is not a β∗g−closed set. Example 8. Let X = {a, b, c} and τ = {X ,;, {b}, {c}, {a, b}, {b, c}}. Then {a, b} is a β∗g−closed set, but it is not a β∗-set. Theorem 5. If A is both β∗-set and β∗g−closed set of (X ,τ), then A is closed. Proof. Let A be both β∗-set and β∗g−closed set of (X ,τ). Then Cl(A) ⊂ A, whenever A is a β∗-set and A⊂ A. So we obtain that A= Cl(A) and hence A is closed. Proposition 1. If A and B are β∗g−closed sets, then A∪ B is β∗g−closed. Proof. Let A∪ B ⊆ U , where U is a β∗-set. Since A, B are β∗g−closed sets, Cl(A)⊆ U and Cl(B) ⊆ U , whenever A⊆ U , B ⊆ U and U is a β∗-set. Therefore, Cl(A∪ B) = Cl(A)∪ Cl(B) ⊆ U . Hence we obtain that A∪ B is a β∗g−closed set of (X ,τ). Remark 8. The intersection of two β∗g−closed sets are not always a β∗g−closed set. Example 9. Let X = {a, b, c, d} and τ = {X ,;, {b}, {c}, {a, b}, {b, c}, {a, b, c}, {a, b, d}}. Then {a, b} and {b, c} are β∗g−closed sets, but {a, b} ⋂ {b, c} = {b} is not β∗g−closed. Theorem 6. If A is a β∗g−closed set of (X ,τ) such that A ⊂ B ⊂ Cl(A), then B is also a β∗g−closed set of (X ,τ). Proof. Let U be a β∗-set of (X ,τ) such that B ⊂ U . Then A ⊂ U . Since A is β∗g−closed, we have Cl(A) ⊂ U . Now Cl(B) ⊂ Cl(Cl(A)) = Cl(A) ⊂ U . Therefore, B is also a β∗g−closed set of (X ,τ). Theorem 7. For any topological space (X ,τ), every singleton {x} of X is a β∗-set. Proof. Let x ∈ X , If {x} ∈ τ, then {x} is a β∗-set [3]. If {x} /∈ τ, then Int({x}) = ; = Cl(Int({x})), so {x} is a β∗-set. Corollary 1. For every x ∈ X , {x} is a β∗g−closed set of (X ,τ) if and only if {x} is a closed set of X. Proof. Necessity: Let {x} be β∗g−closed. Then, by Theorem 7 {x} is closed. Sufficiency: Let {x} be a closed set. By Theorem 3 {x} is β∗g−closed. Theorem 8. Let A be β∗g−closed in (X ,τ). Then Cl(A)− A does not contain any non-empty complement of a β∗-set. Proof. Let A be a β∗g−closed set. Suppose that F is the complement of a β∗-set and F ⊂ Cl(A)− A. Since F ⊂ Cl(A)− A ⊂ X − A, A ⊂ X − F and X − F is a β∗-set. Therefore, Cl(A)⊂ X − F and F ⊂ X − Cl(A). However, since F ⊂ Cl(A)−A, F = ;. A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 26 3. β∗g−closures In this section, the notion of the β∗g−closure is defined and some of its basic properties are studied. Definition 3. For a subset A of (X ,τ), we define the β∗g−closure of as follows: β∗g − Cl(A) = ⋂ F is β∗g−closed in X , A⊂ F}. Lemma 1. Let A be a subset of (X ,τ) and x ∈ X . Then x ∈ β∗g − Cl(A) if and only if V ∩A 6= ; for every β∗g−open set V containing x. Proof. Suppose that there exists a β∗g−open set V containing x such that V ∩ A = ;. Since A⊂ X − V , β∗g − Cl(A) ⊂ X − V and then x /∈ β∗g − Cl(A). Conversely, suppose that x /∈ β∗g − Cl(A). Then there exists a β∗g−closed set F containing A such that x /∈ F . Since x ∈ X − F and X − F is β∗g−open, (X − F)∩ A= ;. Lemma 2. Let A and B be subsets of (X ,τ). Then we have (a) β∗g − Cl(;) = ; and β∗g − Cl(X ) = X . (b) If A⊂ B, then β∗g − Cl(A)⊂ β∗g − Cl(B). (c) β∗g − Cl(A) = β∗g − Cl(β∗g − Cl(A)). (d) β∗g − Cl(A∪ B) = β∗g − Cl(A)∪ β∗g − Cl(B). (e) β∗g − Cl(A∩ B) ⊂ β∗g − Cl(A)∩ β∗g − Cl(B). Proof. Straightforward. Remark 9. (a) If A is β∗g−closed in (X ,τ), then β∗g − Cl(A) = A. but the converse is not true as seen by the following example: (b) In general, β∗g − Cl(A)∩ β∗g − Cl(B) 6⊂ β∗g − Cl(A∩ B). for example, Example 10. Let X = {a, b, c, d} and τ = {X ,;, {b}, {c}, {a, b}, {b, c}, {a, b, c}, {a, b, d}}. Let A= {b} then β∗g − Cl(A) = β∗g − Cl({b}) = {b} but {b} is not β∗g−closed set. Example 11. Let X = {a, b, c, d} and τ = {X ,;, {b}, {c}, {a, b}, {b, c}, {a, b, c}, {a, b, d}}. Let A= {a, c} and B = {a, b}. Then β∗g − Cl(A)∩β∗g − Cl(B) = {a, d} 6⊂ {a} = β∗g − Cl(A∩ B). Definition 4. For a subset A of (X ,τ), (a) c∗(A) = ⋂ {F: F is g−closed, A⊂ F} : g-closure of A [18]; (b) πg − cl(A) = ⋂ {F : F is πg−closed, A⊂ F} : πg−closure of A [11]. A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 27 Definition 5. For a topological space (X ,τ), (a) cτ∗ = {U ⊂ X : c∗(X − U) = (X − U)} [18]; (b) βτ∗ = {U ⊂ X : β∗g − cl(X − U) = (X − U)}; (c) πgτ∗ = {U ⊂ X : πg − cl(X − U) = (X − U)} [11]; Proposition 2. For a subset A of (X ,τ), the following statements hold: (a) A⊂ πg − cl(A) ⊂ c∗(A)⊂ β∗g − cl(A). (b) τ ⊂ βτ∗ ⊂ cτ∗ ⊂ πgτ∗. Proof. The proof follows from definitions. Definition 6. A topological space (X ,τ) is said to be (a) T1/2 space [17] if every g−closed set is closed. (b) T ∗ 1/2 space [16] if every g∗-closed set is closed. (c) ∗T1/2 space [16] if every g−closed set is g∗-closed. Theorem 9. Let (X ,τ) be a space. Then (a) Every g−closed set is closed (i.e. (X ,τ) is T1/2) if and only if cτ∗ = τ. (b) Every β∗g−closed set is closed (i.e. (X ,τ) is β∗T1/2) if and only if βτ∗ = τ. (c) Every g−closed set is β∗g−closed (i.e. (X ,τ) is β∗∗T1/2) if and only if cτ∗ = βτ∗. Proof. (a) Let A ∈ cτ∗. Then c∗(X − A) = (X − A). By hypothesis, Cl(X − A) = c∗(X − A) = X − A and hence A ∈ τ. Conversely, let A be a g−closed set. Then c∗(A) = A and hence X − A∈ cτ∗ = τ, i.e. A is closed. (b) Let A∈ βτ∗. Then β∗g − cl(X − A) = X − A and by hypothesis, Cl(X − A) = β∗g − cl(X − A) = X − A. Hence A∈ τ. (c) Similar to (a). Definition 7. A topological space (X ,τ) is called a β∗T1/2 space if every β∗g−closed set is closed. Theorem 10. A topological space (X ,τ) is β∗T1/2 if and only if each singleton of X is open or X − {x} is a β∗-set for each x ∈ X . Proof. A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 28 Necessity: Let x be a point of X. Suppose that X − {x} is not a β∗-set. Then X − {x} is β∗g−closed. Since (X ,τ) is β∗T1/2 , X − {x} is closed and thus {x} is open in (X ,τ). Sufficiency: Suppose that A is β∗g−closed. We shall show that Cl(A) ⊂ A. Let x be any point of Cl(A). Then {x} is open in (X ,τ) or X − β∗-set. (i) In case {x} is open: Since x ∈ Cl(A), {x} ∩ A 6= ; and hence x ∈ A. (ii) In case X−{x} is a β∗-set: By Theorem 8, Cl(A)−A does not contain any nonempty complement of a β∗-set. Therefore, x /∈ Cl(A)− A but x ∈ Cl(A). Thus, x ∈ A. By (i) and (ii), we obtain Cl(A)⊂ A and hence A is closed. Theorem 11. Every T1/2 space is a β∗T1/2 space. Proof. Let (X ,τ) be a T1/2 space and A a β∗g−closed set of (X ,τ). By Theorem 3, A is a g−closed set of (X ,τ). Since X is a T1/2 space, A is closed. Therefore, X is a β∗T1/2 space. Definition 8. A topological space (X ,τ) is called a β∗∗T1/2 space if every g−closed set is β∗g−closed. Theorem 12. Every T1/2 space is a β∗∗T1/2 space. Proof. Let (X ,τ) be a T1/2 space and A a g−closed set of (X ,τ). Since X is a T1/2 space, A is closed. By Theorem 3, A is a β∗g−closed set of (X ,τ). Therefore, X is a β∗∗T1/2 space. Theorem 13. A space (X ,τ) is T1/2 space if and only if it is β∗T1/2 and β∗∗T1/2. Proof. Necessity: It follows from the Theorems 11 and 12. Sufficiency: Suppose that X is both β∗T1/2 and β∗∗T1/2. Let A be a g−closed set of X. Since X is β∗∗T1/2, then A is β∗g−closed. Since X is a β∗T1/2 space, then A is a closed set of X. Thus X is a T1/2 space. 4. β∗g−open Sets Theorem 14. Let (X ,τ) be a topological space. A ⊂ X is β∗g−open if and only if F ⊂ Int(A) whenever X − F is a β∗-set and F ⊂ A. Proof. Necessity: Let A be β∗g−open. Let X − F be a β∗-set and F ⊂ A. Then X − A⊂ X − F where X − F is a β∗-set. β∗g−closedness of X − A implies Cl(X − A) ⊂ X − F . So F ⊂ Int(A). A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 29 Sufficiency: Suppose X−F is a β∗-set and F ⊂ A imply F ⊂ Int(A). Let X−A⊂ U where U is a β∗-set. Then X−U ⊂ A and X−(X−U) is a β∗-set. By hypothesis X−U ⊂ Int(A), That is X − Int(A) ⊂ U and Cl(X −A)⊂ U . So, X −A is β∗g−closed and A is β∗g−open. Theorem 15. If A is a β∗g−open set of (X ,τ) such that Int(A) ⊂ B ⊂ A, then B is also a β∗g−open set of (X ,τ). Proof. This is an immediate consequence of Theorem 6. Definition 9. A function f : (X ,τ)→ (Y,σ) is said to be (a) β∗g−open if f (V ) is β∗g−open in Y for every open set V of X . (b) β∗g−closed if f (F) is β∗g−closed in Y for every closed set F of X . (c) β∗g−preserving (resp. contra β∗g−open) if f (F) is β∗g−closed (resp. β∗g−closed) in Y for every β∗g−closed (resp. open) set F of X . 5. β∗g−continuity and β∗g−irresoluteness Definition 10. A function f : (X ,τ)→ (Y,σ) is said to be (a) π-continuous [9] (resp. πg−continuous [8], πgp-continuous [22], πgs-continuous [2]) if f −1(F) is π-closed (resp. πg−closed, πgp−closed, πgs−closed) in (X ,τ) for every closed set F of (Y,σ); (b) LC-continuous [4] if f −1(F) is a LC-set in (X ,τ) for every closed set F of (Y,σ); (c) g∗-continuous [17] if f −1(F) is g∗-closed in (X ,τ) for every closed set F of (Y,σ); (d) g−continuous [18] (resp. gp-continuous [19], gs-continuous [1]) if f −1(F) is g−closed (resp. gp−closed, gs−closed) in (X ,τ) for every closed set F of (Y,σ). Definition 11. A function f : (X ,τ)→ (Y,σ) is said to be β∗g−continuous (resp. β∗gp-continuous, β∗gs-continuous) if f −1(F) is β∗g−closed (resp. β∗gp−closed, β∗gs−closed) in (X ,τ) for every closed set F of (Y,σ). Definition 12. A function f : (X ,τ)→ (Y,σ) is said to be β∗g−irresolute (resp. β∗-irresolute) if f −1(V ) is β∗g−closed (resp. β∗-set) in X for every β∗g−closed (resp. β∗-set) set V of Y . Definition 13. A function f : (X ,τ) → (Y,σ) is said to be perfectly β∗g−continuous (resp. strongly β∗g−continuous) if f −1(V ) is clopen (resp. open) in (X ,τ) for every β∗g−open set V of (Y,σ). Definition 14. A function f : (X ,τ)→ (Y,σ) is said to be almost β∗g−continuous if f −1(V ) is β∗g−open in (X ,τ) for every regular open set V of (Y,σ). A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 30 Definition 15. A function f : (X ,τ)→ (Y,σ) is said to be contra β∗g−continuous if f −1(V ) is β∗g−closed in (X ,τ) for every open set V of (Y,σ). Theorem 16. For a function f : (X ,τ)→ (Y,σ), the following properties are equivalent: (a) f is continuous, (b) f is LC-continuous and g−continuous. Theorem 17. For a function f : (X ,τ)→ (Y,σ), the following properties are hold: (a) If f is continuous, then f is β∗g−continuous. (b) If β∗g−continuous, then f is g−continuous. Theorem 18. Let (X ,τ) be a topological space. Then we have (a) If f is β∗g−continuous, then f is β∗pg−continuous. (b) If f is β∗g−continuous, then f is β∗sg−continuous. Theorem 19. For a function f : (X ,τ)→ (Y,σ), the following properties hold: Figure 1: *DIAGRAM IV (repeated). Proof. Obvious by Diagram 4. Theorem 20. If f : (X ,τ)→ (Y,σ) is a β∗-irresolute and closed function, then f (A) is β∗g−closed in Y for every β∗g−closed set A of X . Proof. Let A be any β∗g−closed set of X and U be any β∗-set of Y containing f (A). Since f is β∗-irresolute, f −1(U) is a β∗-set in X and A⊂ f −1(U). Therefore, we have Cl(A)⊂ f −1(U) and hence f (Cl(A))⊂ U . Since f is closed, Cl( f (A))⊂ f (Cl(A))⊂ U . Hence f (A) is β∗g−closed in Y . The composition of two β∗g−continuous functions need not be β∗g−continuous. For, consider the following example: A. Açıkgöz / Eur. J. Pure Appl. Math, 4 (2011), 20-33 31 Example 12. Let X = {a, b, c, d}, τ = {X ,;, {b}, {c}, {b, c}},σ = {X ,;, {a, b, d}}, η = {X ,;, {a, d}}. Define f : (X ,τ)→ (X ,σ) by f (a) = a, f (b) = c, f (c) = b, f (d) = d and g : (X ,σ) → (X ,η) by g(a) = d, g(b) = c, g(c) = b, g(d) = a. Then f and g are β∗g−continuous. {b, c} is closed in (X ,η). (g ◦ f )−1({b, c}) = f −1(g−1({b, c})) = f −1({b, c}) = {b, c} which is not β∗g−closed in (X ,τ). Hence g ◦ f is not β∗g−continuous. Theorem 21. Let f : (X ,τ)→ (Y,σ) and g : (Y,σ)→ (Z ,η) be any two functions. Then (a) g ◦ f is β∗g−continuous, if g is continuous and f is β∗g−continuous. (b) g ◦ f is β∗g−irresolute, if g is β∗g−irresolute and f is β∗g−irresolute. (c) g ◦ f is β∗g−continuous, if g is β∗g−continuous and f is β∗g−irresolute. (d) g ◦ f is β∗g−continuous, if f is β∗g−continuous and g is β∗g−continuous and Y is a β∗T1/2-space. Proof. (a) Let V be closed in (Z ,η). Then g−1(V ) is closed in (Y,σ), since g is continuous. β∗g−continuity of f implies that f −1(g−1(V )) is β∗g−closed in (X ,τ). Hence g ◦ f is β∗g−continuous. (b) Let V be β∗g−closed in (Z ,η). Then g−1(V ) is β∗g−closed in (Y,σ), since g is β∗g−irresolute. Since f is β∗g−irresolute, f −1(g−1(V )) is β∗g−closed in (X ,τ). Hence g ◦ f is β∗g−irresolute. (c) Let V be closed in (Z ,η). Since g is β∗g−continuous, g−1(V ) is β∗g−closed in (Y,σ). As f is β∗g−irresolute, f −1(g−1(V )) is β∗g−closed in (X ,τ). Hence g◦ f is β∗g−continuous. (d) Let V be closed in (Z ,η). Then g−1(V ) is β∗g−closed in (Y,σ), since g is β∗g−continuous. As (Y,σ) is a β∗T1/2 space, g−1(V ) is closed in (Y,σ). β∗g−continuity of f implies that f −1(g−1(V )) is β∗g−closed in (X ,τ). Hence g ◦ f is β∗g−continuous. Theorem 22. Let f : (X ,τ) → (Y,σ) be a β∗g−continuous function. If (X ,τ) is a β∗-T1/2 space, then f is continuous. Proof. Let f be a β∗g−continuous function. Then f −1(V ) is a β∗g−closed set of X for every closed set V of Y . Since X is a β∗T1/2 space, β∗GC(X ,τ) = C(X ,τ). Hence, for every closed set V of Y , f −1(V ) is a closed set of X and so f is continuous. Theorem 23. 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