Generalized Star Closed Sets In Interior Minimal Spaces

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1 Research Paper Volume 2 Issue 11 July 2015 International Journal of Informative & Futuristic Research ISSN (Online): Generalized Star Closed Sets In Interior Paper ID IJIFR/ V2/ E11/ 044 Page No Subject Area Mathematics Key Words M-g* closed, Maximal M-g* closed, Maximal M-g* open, Minimal M-g* closed, Minimal M-g* open Received On Accepted On Published On B. Uma Devi 1 Dr. S. Somasundaram 2 Associate Professor, Department of Mathematics, S. T. Hindu College, Nagercoil, Tamilnadu, India Professor, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu, India Abstract The aim of this paper is to introduce and investigate Generalized Star Closed set in Interior Minimal spaces. Further we define M-g*T 0, M-g*T 1, M-g*T 2,M-g*T 1/2 and,m-g*t 1/3.Minimal M-g*open, Maximal M-g*closed sets are also discussed. Several properties of these new notions are investigated. 1. Introduction Norman Levine [1] introduced the concepts of generalized closed sets in topological spaces. Closed sets are fundamental objects in a topological space. For example, one can define the topology on a set by using either the axioms for the closed sets or the Kuratowski closure axioms. By definition, a subset A of a topological space X is called generalized closed if cla U whenever A U and U is open. This notion has been studied by many topologists because generalized closed sets are not only natural generalization of closed sets but also suggest several new properties of topological spaces. Nakaoka and Oda[ 2,3,4] have introduced minimal open sets and maximal open sets, which are subclasses of open sets. Later on many authors concentrated in this direction and defined many different types of minimal and maximal open sets. Inspired with these developments we further study a new type of closed and open sets namely Minimal M-g*Open sets, Maximal M-g* Closed Copyright IJIFR

2 sets. In this paper a space X means a Minimal space (X, M). For any subset A of X its M-interior and M-closure are denoted respectively by the symbols M-int(A) and M-cl(A). 2. Preliminaries Definition 1: A is said to be M-g * closed (M-generalized star closed) if M-cl(A) U whenever A U and U is M-gopen. Definition 2: A is said to be M-g*open (M-generalized star open) if F M-Int (A) whenever F is M-gclosed and F A. Theorem 3: Every M-closed set is M-g*closed. Proof: Let A be an M-closed subset of X. Let A U and U be M-gopen. Since A is M-closed, then M cl(a) =A gives M-cl(A) U. Thus A is M-g*closed. Converse is not true Remark: Union of two M-g*closed set need not be a M-g*closed. Theorem 4: Every M-g*closed set is M-closed in a M-door space. Theorem 5: In a M-Door space, every M-g*closed set is M-closed. Proof: Let X be a M-door space and let A be a M-g*closed set of X. Then A is M-open or M- closed. If A is M-closed, then nothing has to prove. If A is M-open and A is M-g*closed, we get M- cl(a) U Hence A is M-closed. Theorem 6: M-semiclosed and M-g*closed are independent. Theorem 7: If A is M-g*closed and A B M-cl(A), then B is M-g*closed. Proof: Since B M-cl(A), M-cl(B) M-cl(A). Let B U and U be M-gopen. Then M-cl(A) U. Since A is M-g*closed and U is M-gopen gives M-cl(A) U. So M-cl(B) U. Hence B is M- g*closed. Theorem 8: Let A be M-g*closed and A O where O is M-open. Then M-fr(O) M-int(A c ). That is every boundary point of O is an M-interior point of A c. Proof: Let A be M-g*closed and A O where O is M-open. Here O is M-open and A is M-g*closed gives M-cl(A) O. Let x M-fr(O). Since M-fr(O)= [M-cl(O)]-[M-int(O)] We have x [ M-cl(O)] and x [M-int(O)].As O is M-open, x O. So x [ M-cl(A)]. Thus x [ M- cl(a)] c. Hence x [ M-Int(A c )]. Theorem 9: Every M-open set is M-g*open. Proof: Let A be a M-open set of X. Then A c is M-closed set. A c is M-g*closed, then A is M-g*open Converse is not true Theorem 10: If A is M-g*open and M-int(A) B A. Then B is M-g*open. Proof: A is M-g*open. Hence A c is M-g*closed. Also M-Int(A) B A gives[m-int(a)] c B c A c. SoA c B c M-cl(A c ). Thus B c is M-g*closed and hence B is M-g*open. Theorem 11: If A is M-g*open, A F, F is M-closed then M-Fr(F) M-Int(A). Proof: A is M-g*open, A F, F is M-closed, then A c is M-g*closed, A c F c and F c is M-open by Theorem 6, M-Fr(F c ) M-int(A) Since M-Fr(F c ) M-Fr(F) gives M-Fr(F) M-Int(A). 4255

3 Theorem 12: A set A is M-g*closed if and only if whenever A does not intersect a M-closed set F, M-cl(A) also does not intersect F. Proof: Let A be a M-g*closed subset of X. Let F be a M-closed set where A F=. A F = gives A F c. Since F c is M-open and A is M-g*closed. M-cl(A) F c gives M-cl(A) F =. Hence M- cl(a) does not intersect F. Conversely, let A U and U be M-open. If A U there A does not intersect U c and U c is M-closed and by our assumption M-cl(A) does not intersect U c. Hence M- cl(a) U. Thus A is M-g*closed. Theorem 13: A set A is M-g*open if and only if whenever A contains a M-closed set F, M-Int(A) also contains F. Proof: Let A be a M-g*open subset of X. Let A F where F is M-closed. Then A c F c, F c is M-open and A c is M-sg*closed gives M-cl(A c ) F c. Hence [M-cl(A c )] c F gives M-Int(A) F. Conversely, Let A c U and U be M open, then A U c, U c is M-closed. By hypothesis M-IntA U c. Thus M- cl(a c ) U. So A c is M-g*closed. Hence A is M-g*open. Theorem 14: Let A be a M-g*closed. Then A is M-closed if and only if [M-cl(A)]\A is M-closed. Proof: Suppose A is M-g*closed which is M-closed. So M-cl(A) A gives M-cl(A)\A = Hence M-cl(A)\A is M-semiclosed. Conversely, Since A is M-g*closed and [M-cl(A)]\A is M- semiclosed. M-cl(A)\A cannot contain a non-empty M-closed set. So [M-cl(A)]\A =.Thus M- cl(a) A. Hence A is M-closed. Theorem 15: A and B are M -g*closed sets in X and M -cl(a B) = M -cl(a) M -cl(b), then A B is also M -g*closed set in X. Proof: Suppose that U is M-gopen and A B U. Then A U and B U. Since A and B are M- g*closed subsets in X, M-cl(A) U and M-cl(B) U. Hence M-cl(A) M-cl(B) U. Given M-cl(A B)=M-cl(A) M-cl(B) U. Therefore A B is also M-g*closed set in X. Definition 16: An Interior Minimal space (X, M) is said to be a M-g*T 0 space if for every pair of points x y in X either there exists M-g*open set U such that x U, y U or y U, x U. Theorem 17: Every M-T 0 space is M-g*T 0 space but not conversely. Proof is obvious since every M-open set is M-g*open. Definition 18: An Interior Minimal space (X, M) is said to be a M-g*T 1 space if for every pair of points x yin X there exists M-g*open sets U and V such that x U, y U and y V, x V. Example 1: Let X = {0, 1, 2} with M = {, {1}, {0, 1}, {0, 2}, X} is not a M-g*T 1 space. Theorem 19: Every M-T 1 space is M-g*T 1 space. Proof is obvious since every M-open set is M-g*open. Definition 20: An Interior Minimal space (X, M) is said to be a M-g*T 2 space if for every pair of distinct points x, y in X there exists disjoint M-g*open sets U and V in X such that U V =. Example 2: Let X = {0, 1, 2} with M = {, {0}, {1}, {0, 1}, {0, 2}, {1, 2}, X}. X is a M-g*T 2 space. Theorem 21: Every M-T 2 space is M-g*T 2 space. Proof is obvious since every M-open set is M-g*open. Theorem 22: Every M-g*T 2 space is M-g*T 1 space but not conversely. Proof is obvious from the definitions. Theorem 23: Any singleton set in an Interior minimal space is either M-semiclosed or M-g*open 4256

4 Proof Let {x} be a singleton set in an Interior minimal space X. If {x} is M-semiclosed, then proof is over. If {x} is not a M-semiclosed set, Then {x} c is not M semiopen. Therefore X is the only M- semiopen set containing {x} c and {x} c is M-g*closed. Hence {x} is M-g*open. Thus {x} is M- semiclosed or M-g*open. Definition 24: An Interior minimal space X is called M-g * T 1/3 space if every M gclosed set in X is M-g*closed in X. Theorem 25: An Interior minimal space X is M-g * T 1/3 if and only if every M gopen set is M- g*open. Proof: Let X be M-g * T 1/3. Let A be M-gopen set. To prove, A is M-g*open. Since A is M gopen then A c is M-gclosed. Thus A c is M-g*closed. [X is M-g * T 1/3 ] gives A is M-g*open. Conversely, Suppose every M-gopen is M-g*open. Then every M-gclosed set is M-g*closed. Hence X is M- g * T 1/3. Definition 26: An Interior minimal space X is called M-g*T 1/2 spaceif every M g*closed set is M- closed. Theorem 27: An Interior minimal space X is M g*t 1/2 if and only if every M g*open set is M- open. Proof: Let X be M-g*T 1/2 and let A be M-g*open. Then A c is M-g*closed and so A c is M-closed. Thus A is M-open. Conversely, Suppose every M-g*open set is M-open. Let A be any M-g*closed set. Then A c is M-g*open gives A c is M-open and A is M-closed. Thus every M-g*closed set is M- closed set. Hence X is M-g * T 1/2. 3. Minimal M-g*open sets and Maximal M-g*closed sets We now introduce Minimal M-g*open sets and Maximal M-g*closed sets in minimal spaces as follows. Definition 1: A proper nonempty M-g*open subset U of X is said to be a Minimal M-g*open set if any M-g*open set contained in U is or U. Remark 1: Minimal M-open set and Minimal M-g*open set are independent to each other. Theorem 2: (i) Proof (i) (ii) Let U be a Minimal M-g*open set and W be a M-g*open set. Then U W = or U W. Let U and V be Minimal M-g*open sets. Then U V = or U = V. Let U be a Minimal M-g*open set and W be a M-g*open set. If U W =, then there is nothing to prove. (ii) If U W. Then U W U. Since U is a Minimal M-g*open set, we have U W = (iii) U. Therefore U W. Let U and V be Minimal M-g*open sets. If U V, then U V and V U by (i). Therefore U = V. Theorem 3: Let U be a Minimal M-g*open set. If x U, then U W for some M-g*open set W containing x. Proof: Let U be a Minimal M-g*open set and x U. Let W be any other M-g*open set. Then by Theorem 2 (i), U W = or U W. If U W x U x W. 4257

5 Since W is arbitrary, there exists a M-g*open set W containing x such that U W. Theorem 4: Let U be a Minimal M-g*open set. Then U = {W: W is a M-g*open set of X containing x} for any element x of U. Proof: U is a Minimal M-g*open set containing x, then U W for some M-g*open set W containing x. We have U {W: W is a M-g*open set of X containing x} U. Thus U = {W: W is a M-g*open set of X containing x} U. Theorem 5: Let V be a nonempty finite M-g*open set. Then there exists at least one (finite) Minimal M-g*open set U such that U V. Proof: Let V be a nonempty finite M-g*open set. If V is a Minimal M-g*open set, we may set U = V. If V is not a Minimal M-g*open set, then there exists (finite) M-g*open set V 1 such that V 1 V. If V 1 is a Minimal M-g*open set, we may set U = V 1. If V 1 is not a Minimal M-g*open set, then there exists (finite)m-g*open set V 2 such that V 2 V 1. Continuing this process, we have a sequence of M-g*open sets V V 1 V 2 V 3... V k... Since V is a finite set, this process repeats only finitely. Then finally we get a Minimal M-g*open set U = V n for some positive integer n. Hence there exists at least one Minimal M-g*open set U such that U V. [An Interior minimal space X is said to be M-locally finite space if each of its elements is contained in a finite M-g*open set.] Corollary 6: Let X be a M-locally finite space and V be a nonempty M-g*open set. Then there exists at least one (finite) Minimal M-g*open set U such that U V. Proof: Let X be a M-locally finite space and V be a nonempty M-g*open set. Let x V. Since X is M-locally finite space, we have a finite M-open set V x such that x V x. Then V V x is a finite M- g*open set. By Theorem 5, there exists at least one (finite) Minimal M-g*open set U such that U V Vx. That is U V V x V. Hence there exists at least one (finite) Minimal M-g*open set U such that U V. Corollary 7: Let V be a finite Minimal M-open set. Then there exists at least one (finite) Minimal M-g*open set U such that U V. Proof: Let V be a finite Minimal M-open set. Then V is a nonempty finite M-g*open set. By Theorem 5, there exists at least one (finite) Minimal M-g*open set U such that U V. Theorem 8: Let U; U λ be Minimal M-g*open sets for any element λ Γ. If U λ Γ U λ, then there exists an element λ Γ such that U = U λ. Proof: Let U λ Γ U λ. Then U ( λ Γ U λ ) = U. That is λ Γ (U U λ ) = U. Also by Theorem 2 (ii), U U λ = or U = U λ for any λ Γ. Then there exists an element λ Γ such that U = U λ. Theorem 9: Let U; U λ be Minimal M-g*open sets for any λ Γ. If U = U λ for any λ Γ, then ( λ Γ U λ ) U =. Proof: Suppose that ( λ Γ U λ ) U for any λ Γ such that U U λ. That is λ Γ (U λ U). Then there exists an element λ Γ such that U U λ. By Theorem 2(ii), we have U = U λ, which contradicts the fact that U U λ for any λ Γ. Hence ( λ Γ U λ ) U =. We now introduce Maximal M-g*closed sets in Interior Minimal spaces as follows. Definition 10: A proper nonempty M-g*closed set F X is said to be Maximal M-g*closed set if any M-g*closed set containing F is either X or F. 4258

6 Theorem 11: A proper nonempty subset F of X is Maximal M-g*closed set if and only if X\F is a Minimal M-g*open set. Proof: Let F be a Maximal M-g*closed set. Suppose X\F is not a Minimal M-g*open set. Then there exists a M-g*open set U X\F such that U X\F. That is F X\U and X\U is a M- g*closed set gives a contradiction to F is a Maximal M-g*closed set. So X\F is a Minimal M-g*open set. Conversely, Let X\F be a Minimal M-g*open set. Suppose F is not a Maximal M-g*closed set. Then there exists a M-g*closed set E F such that F E X. That is X\E X\F and X\E is a M-g*open set gives a contradiction to X\F is a Minimal M- g*open set. Thus F is a Maximal M-g*closed set. Theorem 12: (i) Let F be a Maximal M-g*closed set and W be a M-g*closed set. Then F W = X or W F. (ii) Let F and S be Maximal M-g*closed sets. Then F S = X or F = S. Proof (i) Let F be a Maximal M-g*closed set and W be a M-g*closed set. If F W = X, then there is nothing to prove. Suppose F W X. Then F F W. Therefore F W= F implies W F. (ii) Let F and S be Maximal M-g*closed sets. If F S X, then we have F S and S F by (i). Therefore F = S. Theorem 13: Let F α, F β, F δ be Maximal M-g*closed sets such that F α F β. If F α F β F δ, then either F α = F δ or F β = F δ Proof : Given that F α F β F δ. If F α = F δ then there is nothing to prove. If F α F δ then we have to prove F β = F δ. Now F β F δ = F β (F δ X) = F β (F δ (F α F β ) (by Theorem 12(ii)) = F β ((F δ F α ) (F δ F β )) = (F β F δ F α ) (F β F δ F β ) = (F α F β ) (F δ F β ) (by F α F β F δ ) = (F α F δ ) F β = X F β (Since F α and F δ are Maximal M-g*closed sets by Theorem (ii), F α F δ = X) = F β. That is F β F δ = F β implies F β F δ. Since F β and F δ are Maximal M-g*closed sets, we have F β = F δ Therefore F β = F δ Theorem 14: Let F α, F β and F δ be different Maximal M-g*closed sets to each other. Then (F α F β ) (F α F δ ). Proof: Let (F α F β ) (F α F δ ) (F α F β ) (F δ F β ) (F α F δ ) (F δ F β ) Gives (F α F δ ) F β F δ (F α F β ). Since by Theorem 12(ii), F α F δ =X and F α F β =X gives X F β F δ X. That is F β F δ. From the definition of Maximal M-g*closed set, we get F β = F δ which gives a contradiction to the fact that F α, F β and F δ are different to each other. Thus (F α F β ) (F α F δ ). Theorem 15: Let F be a Maximal M-g*closed set and x be an element of F. Then F= {S:S is a M- g*closed set containing x such that F S X}. Proof: By Theorem 13, and fact that F is a M-g*closed set containing x, we have F {S: S is a M-g*closed set containing x such that F S X} F. So we have F= {S: S is a M-g*closed set containing x such that F S X}. 4259

7 Theorem 16: Let F be a Maximal M-g*closed set. If x is an element of X\F. Then X\F E for any M-g*closed set E containing x. Proof: Let F be a Maximal M-g*closed set and x is in X\F. E F for any M-g*closed set E containing x. Then E F = X (by Theorem 12 (ii)). Thus X\F E. 4. Minimal M-g*closed set and Maximal M-g*open set We now introduce Minimal M-g*closed sets and Maximal M-g*open sets in Interior Minimal spaces as follows. Definition 17: A proper nonempty M-g*closed subset F of X is said to be a Minimal M-g*closed set if any M-g*closed set contained in F is or F. In Example 1, {2} is both Minimal M-closed set and Minimal M-g*closed set. Remark: Every Minimal M-closed sets are Minimal M-g*closed sets. Definition 18: A proper nonempty M-g*open U X is said to be a Maximal M-g*open set if any M-g*open set containing U is either X or U. In Example 1, {0, 2} is both Maximal M-g*open and Maximal M-open Remark: Every Maximal M-open sets are Maximal M-g*open sets. Theorem 19: A proper nonempty subset U of X is Maximal M-g*open set if and only if X\U is a Minimal M-g*closed set. Proof: Let U be a Maximal M-g*open set. Suppose X\U is not a Minimal M-g*closed set. Then there exists M-g*closed set V X\U such that V X\U. That is U X\V and X\V is a M- g*open set gives a contradiction to U is a Minimal M-g*closed set. Conversely, Let X\U be a Minimal M-g*closed set. Suppose U is not a Maximal M-g*open set. Then there exists M-g*open set E U such that U E X. That is X\E X\U and X\E is a M-g*closed set which is a contradiction for X\U is a Minimal M-g*closed set. Therefore U is a Maximal M-g*closed set. Theorem 20: (i) Let U be a Minimal M-g*closed set and W be a M-g*closed set. Then U W = or U W. (ii) Let U and V be Minimal M-g*closed sets. Then U V = or U = V. Proof (i) Let U be a Minimal M-g*closed set and W be a M-g*closed set. If U W =, then there is nothing to prove. If U W, then U W U. Since U is a Minimal M-g*closed set, we have U W = U. Therefore U W. (ii) Let U and V be Minimal M-g*closed sets. If U V, then U V and V U by (i). Therefore U = V. Theorem 21: Let V be a nonempty finite M-g*closed set. Then there exists at least one (finite) Minimal M-g*closed set U such that U V. Proof: Let V be a nonempty finite M-g*closed set. If V is a minimal M-g*closed set, we may set U = V. If V is not a Minimal M-g*closed set, then there exists (finite) M-g*closed set V 1 such that V 1 V. If V 1 is a Minimal M-g*closed set, we may set U = V 1. If V 1 is not a minimal M-g*closed set, then there exists (finite) M-g*closed set V 2 such that V 2 V 1. Continuing this process, we have a sequence of M-g*closed sets V V 1 V 2 V 3... V k... Since V is a finite set, this process repeats only finitely. Then finally we get a Minimal M-g*closed set U = V n for some positive integer n. 4260

8 Corollary 22: Let X be a M-locally finite space and V be a nonempty M-g*closed set. Then there exists at least one (finite) Minimal M-g*closed set U such that U V. Proof: Let X be a M-locally finite space and V be a nonempty M-g*closed set. Let x V. Since X is M-locally finite space, we have a finite M-open set V x such that x V x. Then V V x is a finite M- g*closed set. By Theorem 21, there exists at least one (finite) Minimal M-g*closed set U such that U V V x. That is U V V x V. Hence there exists at least one (finite) Minimal M-g*closed set U such that U V. Corollary 23: Let V be a finite Minimal M-open set. Then there exists at least one (finite) Minimal M-g*closed set U such that U V. Proof: Let V be a finite Minimal M-open set. Then V is a nonempty finite M-g*closed set. By Theorem 21, there exists at least one (finite) Minimal M-g*closed set U such that U V. Theorem 24: Let U; U λ be Minimal M-g*closed sets for any element λ Γ. If U λ Γ U λ, then there exists an element λ Γ such that U = U λ. Proof: Let U λ Γ U λ. Then U ( λ Γ U λ ) = U. That is λ Γ (U U λ ) = U. Also by Theorem 20 (ii), U U λ = or U = U λ for any λ Γ. This implies that there exists an element λ Γ such that U = U λ. Theorem 25: Let U; U λ be Minimal M-g*closed sets for any λ Γ. If U = U λ for any λ Γ, then ( λ Γ U λ ) U =. Proof: Suppose that ( λ Γ U λ ) U. That is λ Γ (U λ U). Then there exists an element λ Γ such that U U λ. By Theorem 20(ii), we have U = U λ, which contradicts the fact that U U λ for any λ Γ. Hence ( λ Γ U λ ) U =. Theorem 26: A proper nonempty subset F of X is Maximal M--g*open set if and only if X\F is a Minimal M-g*closed set. Proof: Let F be a Maximal M-g*open set. Suppose X\F is not a Minimal M-g*open set. Then there exists M-g*open set U X\F such that U X\F. That is F X\U and X\U is a M-g*open set which is a contradiction for F is a Minimal M-g*closed set. Conversely, Let X\F be a Minimal M- g*closed set. Suppose F is not a Maximal M-g*open set. Then there exists M-g*open set E F such that F E X. That is X\E X\F and X\E is a M-g*open set which is a contradiction for X\F is a Minimal M-g*closed set. Thus F is a Maximal M-g*open set. Theorem 27: (i) Let F be a Maximal M-g*open set and W be a M-g*open set. Then F W = X or W F. (ii) Let F and S be Maximal M-g*open sets. Then F S=X or F = S. Proof: (i) Let F be a Maximal M-g*open set and W be a M-g*open set. If F W = X, then there is nothing to prove. Suppose F W X.Then F F W. Therefore F W = F gives W F. (ii) Let F and S be Maximal M-g*open sets. If F S X, then we have F S and S F by (i). Therefore F = S. Theorem 28: Let F α, F β, F δ be Maximal M-g*open sets such that F α F β. If F α F β F δ, then either F α = F δ or F β = F δ Proof: Given that F α F β F δ. If F α = F δ then there is nothing to prove. If F α F δ then we have to prove F β = F δ. Now F β F δ = F β (F δ X) = F β (F δ (F α F β )(by Theorem 27 (ii)) = F β ((F δ F α ) (F δ F β )) = (F β F δ F α ) (F β F δ F β ) = (F α F β ) (F δ F β ) (by F α F β F δ ) 4261

9 = (F α F δ ) F β = X F β. (Since F α and F δ are Maximal M-g*open sets by Theorem 27 (ii), F α F δ = X) = F β. That is F β F δ = F β. Thus F β F δ. Since F β and F δ are Maximal M-g*open sets, we have F β = F δ Therefore F β = F δ. Theorem 29: Let F α, F β and F δ be different Maximal M-g*open sets to each other. Then (F α F β ) (F α F δ ). Proof: Let (F α F β ) (F α F δ ) (F α F β ) (F δ F β ) (F α F δ ) (F δ F β ) (F α F δ ) F β F δ (F α F β ). (Since by Theorem 27(ii), F α F δ = X and F α F β = X) X F β F δ X F β F δ. From the definition of Maximal M-g*open set it follows that F β = F δ, which is a contradiction to the fact that F α, F β and F δ are different to each other. Therefore (F α F β ) (F α F δ ). Theorem 30: Let F be a Maximal M-g*open set. If x is an element of X\F. Then X\F E for any M-g*open set E containing x. Proof: Let F be a Maximal M-g*open set and x X\F. E F for any M-g*open set E containing x. Then E F = X by Theorem 27(ii). Therefore X\F E. 5. Conclusion The above study does not stand just as generalizations of the corresponding results in topological spaces. It also works in non-topological spaces. Although, most of the results are carried out in a routine manner to Interior minimal spaces, there are some results which do not carry over to Interior minimal spaces. For instance, finite unions and finite intersections of generalized star closed sets in all the above mentioned generalizations turn out are the corresponding generalized star closed sets. This does not hold in Interior minimal spaces. References [1] [1] Norman Levine, Generalized closed sets in topology, Tend Circ., Mat. Palermo (2) 19 (1970), [2] [2] F. Nakaoka and N. Oda, Some Properties of Maximal Open Sets, Int. J. Math. Sci. 21,(2003) [3] [3] F. Nakaoka and N. Oda, Some Applications of Minimal Open Sets, Int. J. Math. Sci. 27-8, (2001), [4] [4] F. Nakaoka and N. Oda, On Minimal Closed Sets, Proceeding of Topological spaces Theory and its Applications, 5(2003),