Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2801 https://internationalpubls.com Almost GΩα-Closed Functions and Separation Axioms A. Parveen banu1 , N. Selvanayaki2, R.Abinprakash3 1Assistant Professor, 2Assistant Professor (Selection Grade), 3Associate Professor, Department of Mathematics, 1 Sri G.V.G Visalakshi College for Women, Udumalpet, Tamil Nadu, India. 2,3 Akshaya College of Engineering and Technology, Coimbatore, Tamil Nadu, India 1E-mail : banuparveen14@gmail.com,2E-mail : selvanayaki@acetcbe.edu.in 3E-mail : abinprakash6343@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this article explores the concept of almost GΩα -closed sets and almost GΩα -closed functions and their relationship with separation axioms in topology. Almost GΩα -closed functions serve as a generalization of closed functions and play a crucial role in the study of topological properties. We investigate various characterizations and fundamental properties of almost GΩα -closed functions, analyzing their interaction with different separation axioms. AMS Subject Classification: 54C10, 54C08, 54C05 Keywords: almost GΩα-closed sets , almost GΩα -closed functions, Normal Spaces, Weakly Normal Spaces. 1. INTRODUCTION In topological spaces, it is well known that normality is preserved under closed continuous surjections. Many authors have tried to weaken the condition “closed” in this theorem. In 1978, Long and Herrington [5] used almost closedness due to Singal [17]. In 1982, Malghan [7] used g-closedness. In 1986, Greenwood and Reilly [4] used -closedness due to Mashhour et al. [8]. In 1995, Yoshimura et al. [19] used almost g-closedness which is a generalization of both almost closedness and g-closedness. In 1999, Noiri [9] introduced almost ag-closedness using ag-closed sets [35]. Ravi et. al. [14] introduced almost ags-closedness using ags-closed sets [13]. We use GΩα-closed sets to define a new class of functions called almost GΩα -closed functions. The purpose of the present article is to improve preservation theorems of separation axioms, that is, normality, weak normality, mild normality, almost normality, regularity, almost regularity, quasi-regularity and strong s-regularity. Theorem A Normality and weak normality are preserved under almost GΩα -closed continuous surjections. Theorem B Regularity and strong s-regularity are preserved under almost a-open almost GΩα -closed continuous surjections. 2. PRELIMINARIES The family of regular open (resp. regular closed) sets of a space (X, ) is denoted by RO (X, ) (resp. RC (X, ) ) or simply by RO (X) (resp. RC (X)). mailto:selvanayaki@acetcbe.edu.in mailto:abinprakash6343@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2802 https://internationalpubls.com Definition 2.1 A subset A of a space (X, ) is called a rg-closed [9] if αcl(A)  U whenever A  U and U is regular open in (X, ). The complement of rαg-closed set is called rαg-open set. Definition 2.2 A function f : (X, ) → (Y, ) is said to be (i) -closed [40] (resp. ĝ-closed [14], gs-closed [66], g  -closed) if for each closed set F of X, f(F) is -closed (resp. ĝ-closed, gs-closed, g  -closed); (ii) almost -closed [9] (resp. almost ĝ-closed [14], almost gs-closed [14], almost g  -closed, almost g  -closed) if for each F RC(X, ), f(F) is -closed (resp. ĝ-closed, gs-closed, g  - closed, g  -closed). Definition 2.3 A space X is said to be (i) weakly normal [20] if for each decreasing sequence {Fn} of closed sets of X such that  {Fn : n  N} =  and each closed set H of X with H  F1 = , there exist n  N and an open set U of X such that Fn  U and cl(U)  H = ; (ii) mildly normal [18] if for any disjoint regular closed sets A and B, there exist disjoint open sets U and V such that A  U and B  V; (iii) almost normal [15] if for every pair of disjoint sets A and B, one of which is closed and the other is regular closed, there exist disjoint open sets U and V such that A  U and B  V. Lemma 2.4 [9] If A is an -open set of a space X, then the following hold: cl(A) = cl(A) = cl(int(A)). Lemma 2.5 [10] A space X is weakly normal if and only if for each decreasing sequence {Fn} of closed sets of X such that  {Fn : n  N} =  and each open set U of X such that F1  U, there exist n  N and an open set G of X such that Fn  G  cl(G)  U. Definition 2.6 A function f : X → Y is said to be (i) R-map [1] (resp. almost continuous [79]) if f-1(V) is regular open (resp. open) in X for every V  RO (Y); (ii) almost open [17] (resp. almost -open [49]) if f(U) is open (resp. -open) in Y for every regular open set U of X; (iii) -open [8] if f(U) is -open in Y for every open set U of X; (iv) almost g-closed [9] if f(U) is g-closed in Y for every regular closed set U of X. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2803 https://internationalpubls.com Lemma 2.7 [9] If a function f : X → Y is almost continuous almost -open and V is regular open in Y, then f-1(V) is regular open in X. Lemma 2.8 (i) A subset A of a space X is rg-open if and only if F  int(A) whenever F  RC (X) and F  A [9]. (ii) Every gs-closed set is g-closed but not conversely [14]. (iii) Every g-closed set is rg-closed but not conversely [9]. Definition 2.9 A space X is said to be (i) almost regular [16] if for each F  RC (X) and each x  X − F, there exist disjoint open sets U and V of X such that x  U and F  V; (ii) quasi-regular [12] if for every nonempty open set V of X, there exists a nonempty open set U in X such that cl(U)  V; (iii) strongly s-regular [3] if for any closed set A of X and any point x  X − A there exists an F  RC (X) such that x  F and F  A = . Definition 2.10 A function f : X → Y is said to be (i) feebly continuous [2] if int(f-1(V))   for every nonempty open set V of Y; (ii) feebly open [2] if int(f(U))   for every nonempty open set U of X; (iii) almost feebly open [9] if int(f(U))   for every nonempty U  RO (X). Theorem 2.11[9] The following are equivalent for a space (X, ): (i) (X, ) is regular (resp. almost regular); (ii) for each closed (resp. regular closed) set F and each x  X − F, there exist disjoint U, V   such that x  U and F  V; (iii)for each open (resp. regular open) set V and x  V, there exists U   such that x  U  cl(U)  V. 3. ALMOST GΩα -CLOSED FUNCTIONS Definition 3.1 A function f : (X, ) → (Y, ) is said to be almost GΩα- closed if for each F RC(X, ), f(F) is GΩα-closed. Theorem: 3.2 Every almost closed function is almost GΩ closed function. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2804 https://internationalpubls.com Proof Let F be regular closed space and a function f : (X, ) → (Y, ) be almost closed function . Then if for each regular closed set F, f(F) is closed and every closed set is GΩ closed set. Hence f is almost GΩ closed function. The following example shows that the converse of the above theorem is not true. Example: 3.3 Let X = Y = {a, b, c},  = {, {a}, {b}, {a, b}, X} and  = {, {a, b}, Y}. Then RC(X, ) = {, {a, c}, {b, c}, X} and GΩ closed in Y = {, {c}, {b,c}, {a, c}, Y} Then the identity function f : (X, ) → (Y, ) is almost GΩ -closed. However, it is not almost closed since there exists {b, c }  RC (X, ) such that f({b, c}) = {b, c} is not closed in (Y, ). Theorem: 3.4 Every almost α-closed function is almost GΩα closed function. Proof Let F be regular closed space and a function f : (X, ) → (Y, ) be almost α-closed function . Then if for each regular closed set F, f(F) is α-closed and every α-closed set is GΩ closed set. Hence f is almost GΩα closed function. The following example shows that the converse of the above theorem is not true. Example: 3.5 Let X = Y = {a, b, c},  = {, {a}, {b}, {a, b}, {a,c}X} and  = {, {a, b}, Y}. Then RC(X, ) = {, {a, c}, {b, c}, X} , α- closed in Y = {, {c}, Y} and GΩα closed in Y = {, {c},{a, b}, Y} Then the function f : (X, ) → (Y, ) defined as f(a)=c, f(b)=b, f(c)=c is almost GΩα -closed. However, it is not almost α - closed since there exists {b, c }  RC (X, ) such that f({b, c}) = {a, b} is α- closed in (Y, ). Theorem: 3.6 Every GΩ closed function is almost GΩ closed function but not conversely Proof Let F be regular closed space. Then every regular closed set is closed set and Let a function f : (X, ) → (Y, ) be GΩ closed function . Then if for each closed set F, f(F) is GΩ-closed. Hence f is almost GΩ closed function. The following example shows that the converse of the above theorem is not true. Example 3.7 Let X = Y = {a, b, c},  = {, {a}, {b}, {a, b}, {a,c}X} and  = {, {a, b}, Y}. Then RC(X, ) = {, {a, c}, {b, c}, X} and GΩ closed in Y = {, {c}, {b,c}, {a, c}, Y} Then the function f : (X, ) → (Y, ) defined as f(a)=c, f(b)=b, f(c)=c is almost GΩ -closed. However, it is not GΩ - closed since there exists {b }   c such that f({b}) = {b} is not GΩ - closed in (Y, ). Theorem: 3.8 Every GΩα closed function is almost GΩα closed function. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2805 https://internationalpubls.com Proof Let F be regular closed space. Then every regular closed set is closed set and Let a function f : (X, ) → (Y, ) be GΩα closed function . Then if for each closed set F, f(F) is GΩα-closed. Hence f is almost GΩα closed function. The following example shows that the converse of the above theorem is not true. Example 3.9 Let X = Y = {a, b, c},  = {, {a}, {b}, {a, b}, {a,c}X} and  = {, {a, b}, Y}. Then RC(X, ) = {, {a, c}, {b, c}, X} and GΩα closed in Y = {, {c}, {a, b}, Y} Then the function f : (X, ) → (Y, ) defined as f(a)=c, f(b)=b, f(c)=c is almost GΩα -closed. However, it is not GΩ - closed since there exists {b }   c such that f({b}) = {b} is not GΩα - closed in (Y, ). Theorem: 3.10 Every almost GΩ closed function is almost GΩα closed function. Proof Let F be regular closed space and a function f : (X, ) → (Y, ) be GΩ closed function . Then if for each regular closed set F, f(F) is GΩ-closed and every GΩ closed set is GΩα closed set. Hence f is almost GΩα closed function. Example 3.11 Let X = Y = {a, b, c},  = {, {a}, {b}, {a, b}, {a,c}X} and  = {, {a, b}, Y}. Then RC(X, ) = {, {a, c}, {b, c}, X} and GΩα closed in Y = {, {c}, {a, b}, Y} Then the function f : (X, ) → (Y, ) defined as f(a)=c, f(b)=b, f(c)=c is almost GΩα -closed. However, it is not almost GΩ - closed since there exists {b, c }  RC (X, ) such that f({b, c}) = {a, b} is not GΩ- closed in (Y, ). Remark 3.12 We have the following diagram for properties of functions: GΩ-closed almost closed almost GΩ-closed GΩα -closed almost -closed almost GΩα -closed Theorem 3.13 A surjection f : X → Y is almost GΩα -closed if and only if for each subset S of Y and each U  RO (X) containing f-1(S) there exists an GΩα -open set V of Y such that S  V and f-1(V)  U. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2806 https://internationalpubls.com Proof Necessity. Suppose that f is almost GΩα -closed. Let S be a subset of Y and U  RO (X) containing f-1(S). Put V = Y − f (X − U), then V is an GΩα -open set of Y such that S  V and f-1(V)  U. Sufficiency. Let F be any regular closed set of X. Then f-1(Y − f (F))  X − F and X − F  RO (X). There exists an GΩα-open set V of Y such that Y − f (F)  V and f-1(V)  X − F. Therefore, we have f (F)  Y − V and F  f-1(Y − V). Hence, we obtain f (F) = Y − V and f (F) is GΩα -closed in Y. This shows that f is almost GΩα -closed. Corollary 3.14 If f : X → Y is an almost GΩα-closed surjection, then for each sg-closed set F of Y and each U  RO (X) containing f-1(F) there exists an -open set V of Y such that F  V and f-1(V)  U. Proof Let F be a sg-closed set of Y and U  RO (X) containing f-1(F). By Theorem 1.3.13, there exists an GΩα -open set W of Y such that F  W and f-1(W)  U. Since W is GΩα -open, we have F  int(W). Put V = int(W), then V is -open in Y and f-1(V)  U. 4. NORMAL SPACES In this section, we make use of GΩα-closed sets to obtain further characterizations and preservation theorems of normal spaces. Theorem 4.1 The following are equivalent for a space X: (i) X is normal; (ii) For any disjoint closed sets A and B, there exist disjoint GΩα-open sets U, V such that A  U and B  V; (iii) For any closed set A and any open set V containing A, there exists an GΩα -open set U of X such that A  U  cl(U)  V. Proof (i)  (ii). This is obvious since every open set is GΩα -open. (ii)  (iii). Let A be a closed set and V an open set containing A. Then A and X − V are disjoint closed sets. There exist disjoint GΩα -open sets U and W such that A  U and X − V  W. Since X − V is closed and hence sg-closed, we have X − V  int(W) and U  int(W) = . Therefore, we obtain cl(U)  int(W) =  and hence A  U  cl(U)  X − int(W)  V. (iii)  (i). Let A, B be disjoint closed sets of X. Then A  X − B and X − B is open. There exists an GΩα -open set G of X such that A  G  cl(G)  X − B. Since A is closed, we have A  int(G). Put U = int(cl(int(int(G)))) and V = int(cl(int(X − cl(G)))). Then U and V are disjoint open sets of X such that A  U and B  V. Therefore, X is normal. Theorem 4.2 If f : X → Y is a continuous almost GΩα -closed surjection and X is a normal space, then Y is normal. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2807 https://internationalpubls.com Proof Let A and B be any disjoint closed sets of Y. Then f-1(A) and f-1(B) are disjoint closed sets of X. Since X is normal, there exist disjoint open sets U and V such that f-1(A)  U and f-1(B)  V. Let G = int(cl(U)) and H = int(cl(V)), then G and H are disjoint regular open sets of X such that f-1(A)  G and f-1(B)  H. By Theorem 1.3.13, there exists GΩα -open sets K and L of Y such that A  K, B  L, f-1(K)  G and f-1(L)  H. Since G and H are disjoint, so are K and L. It follows from Theorem 1.4.1 that Y is normal. Theorem 4.3 If f : X → Y is an almost GΩα -closed continuous surjection and X is a weakly normal space, then Y is weakly normal. Proof Let {Fn} be any decreasing sequence of closed sets of Y with no common point and any open set V of Y such that F1  V. Then {f-1(Fn)} is a decreasing sequence of closed sets of X with no common point and f-1(V) is an open set of X such that f-1(F1)  f-1(V). Since X is weakly normal, by Lemma 1.4.7, there exist n  N and an open set U of X such that f-1(Fn)  U  cl(U)  f-1(V). Therefore, f- 1(Fn)  int(cl(U)) and by Corollary 1.3.14, there exists an -open set G of Y such that Fn  G and f- 1(G)  int(cl(U)). Since cl(U) is regular closed and f is almost GΩα -closed, f(cl(U)) is GΩα -closed in Y. Thus, we obtain Fn  G  cl(G)  cl(f(cl(U)))  V. Let H = int(cl(int(G))), then by Lemma 7.2.4 we have Fn  H  cl(H) = cl(G)  V. It follows from Lemma 1.2.5 that Y is weakly normal. Theorem 4.4 The following are equivalent for a space X: (i) X is mildly normal; (ii) for any disjoint H, K  RC (X), there exist disjoint GΩα -open sets U, V such that H  U and K  V; (iii) for any disjoint H, K  RC (X), there exist disjoint gs-open sets U, V such that H  U and K  V; (iv) for any disjoint H, K  RC (X), there exist disjoint rg-open sets U, V such that H  U and K  V; (v) for any H  RC (X) and any V  RO (X) containing H, there exists an rg-open set U of X such that H  U  cl(U)  V; (vi) for any H  RC (X) and any V  RO (X) containing H, there exists an -open set U of X such that H  U  cl(U)  V; (vii) for any disjoint H, K  RC (X), there exist disjoint -open sets U, V such that H  U and K  V. Proof It is obvious that (i)  (ii), (ii)  (iii) and (iii)  (iv). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2808 https://internationalpubls.com (iv)  (v). Let H  RC (X) and V  RO (X) containing H. There exist disjoint rg-open sets U, W such that H  U and X − V  W. By Lemma 7.2.8, we have X − V  int(W) and U  int(W) = . Therefore, we obtain cl(U)  int(W) =  and hence H  U  cl(U)  X − int(W)  V. (v)  (vi). Let H  RC (X) and V  RO (X) containing H. There exists an rg-open set G of X such that H  G  cl(G)  V. Since H  RC (X), by Lemma 1.2.8, we have H  int(G). Put U = int(G), then U is -open in X and H  U  cl(U)  V. (vi)  (vii). Let H and K be any disjoint regular closed sets of X. Then, since H  X − K and X − K  RO (X), there exists an -open set U of X such that H  U  cl(U)  X − K. Put V = X − cl(U), then U and V are disjoint -open sets of X such that H  U and K  V. (vii)  (i). Let H and K be any disjoint regular closed sets of X. Then there exist disjoint -open sets A and B of X such that H  A and K  B. Since A and B are disjoint, we have int(cl(int(A)))  int(cl(int(B))) = . Now, Put U = int(cl(int(A))) and V = int(cl(int(B))), then U and V are disjoint open sets of X such that H  U and K  V. Therefore, X is mildly normal. Theorem 4.5 Let f : X → Y be an R-map and an almost GΩα -closed surjection and X is mildly normal then Y is mildly normal. Proof Let A and B be any disjoint regular closed sets of Y. Then f-1(A) and f-1(B) are disjoint regular closed sets of X. Since X is mildly normal, there exist disjoint open sets U and V of X such that f -1(A)  U and f-1(B)  V . Put G = int(cl(U)) and H = int(cl(V)), then G and H are disjoint regular open sets of X such that f-1(A)  G and f-1(B)  H. By Theorem 3.13 [14], there exist GΩα -open sets K and L of Y such that A  K, B  L, f-1(K)  G and f-1(L)  H. Since G and H are disjoint, so are K and L. It follows from Theorem 1.4.4 that Y is mildly normal. Theorem 4.6 If f : X → Y is an almost -open almost gs-closed continuous surjection and X is an almost normal space, then Y is almost normal. Proof Let B be any closed set of Y and V  RO (Y) containing B. Since f is continuous and almost -open, f-1(B) is closed and f-1(V)  RO(X) by Lemma 1.2.7. Since X is almost normal and f-1(B)  f-1(V), there exists U  RO (X) such that f-1(B)  U  cl(U)  f-1(V) [15, Theorem 2.1]. Since f is almost -open and almost gs-closed, f(U) is -open and f(cl(U)) is gs-closed in Y. Therefore, we obtain B  f(U)  cl(f(U))  cl(f(cl(U))  V. Put G = int(cl(int(f(U)))). Then G is open in Y and cl(f(U)) = cl(int(f(U))) = cl(G) by Lemma 1.2.4. Therefore, we obtain B  f(U)  G  cl(G)  V. It follows from [15, Theorem 2.1] that Y is almost normal. 5. REGULAR SPACES In this section, we improve preservation theorems of regularity almost regularity and quasi- regularity. Theorem 5.1 If f : X → Y is an almost -open almost GΩα -closed continuous surjection and X is a regular space, then Y is regular. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2809 https://internationalpubls.com Proof Let y be any point of Y and V any open neighbourhood of y. There exists a point x  X with f(x) = y. Since X is regular and f is continuous, there exists an open set U of X such that x  U  cl(U)  f-1(V). Therefore, we have y  f(U)  f(int(cl(U)))  f(cl(U))  V and f(int(cl(U)) is -open because int(cl(U))  RO (X) and f is almost -open. Since cl(U)  RC (X) and f is almost GΩα -closed, f(cl(U)) is GΩα -closed and hence y  f(int(cl(U)))  cl(f(int(cl(U))))  cl(f(cl(U)))  V. It follows from Theorem 1.2.11 that Y is regular. Theorem 5.2 If f : X → Y is an almost -open almost gs-closed almost continuous surjection and X is an almost regular space, then Y is almost regular. Proof Let y be any point of Y and V  RO (Y) containing y. Since f is almost -open almost continuous, f-1(V)  RO (Y) by Lemma 1.2.7. Take a point x  f-1(y). Since X is almost regular, there exists U  RO (X) such that x  U  cl(U)  f-1(V) [16, Theorem 2.2]. Hence y  f(U)  f(cl(U))  V. Since f is almost -open almost gs-closed, f(U) is -open in Y and f(cl(U)) is gs-closed in Y and hence we have y  f(U)  cl(f(U))  cl(f(cl(U)))  V. It follows from Theorem 1.2.11 that Y is almost regular. Theorem 5.3 If f : X → Y is an almost feebly open feebly continuous almost GΩα -closed surjection and X is a quasi-regular space, then Y is quasi-regular. Proof Let V be any nonempty open set of Y. Since f is feebly continuous, int(f-1(V))   and by the quasi- regularity of X there exists a nonempty open set U of X such that U  cl(U)  int(f-1(V)). We have f(int(cl(U)))  f(cl(U))  V. Since f is almost feebly open, int(f(int(cl(U))))  . Since f is almost GΩα -closed, f(cl(U)) is GΩα -closed and hence cl(f(cl(U)))  V. Now, put G = int(f(int(cl(U)))), then by Lemma 1.2.4 we obtain   G  cl(G) = cl(G)  cl(f(cl(U)))  V. This shows that Y is quasi- regular. Theorem 5.4 If f : X → Y is an almost -open almost GΩα -closed continuous surjection and X is a strongly s-regular space, then Y is strongly s-regular. Proof Let V be any open set of Y and y any point of V. Since f is continuous, f-1(V) is open in X. For a point x  f-1(y), there exists F  RC(X) such that x  F  f-1(V); hence y = f(x)  f(F)  V. Since f is continuous, we have f(F) = f(cl(int(F)))  cl(f(int(F))). Since f is almost GΩα -closed, f(F) is GΩα -closed and cl(f(F))  V. Moreover, f is almost -open, f(int(F)) is -open in Y and by Lemma 1.2.4 we have cl(f(int(F))) = cl(int(f(int(F)))) = cl(f(int(F)))  cl(f(F)). Therefore, we obtain cl(int(f(int(F))))  RC (Y) and y  f(F)  cl(f(int(F))) = cl(int(f(int(F))))  cl(f(F))  V. It follows from [3, Theorem 1] that Y is strongly s-regular. 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