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g#-SEMI-CLOSED SETS IN TOPOLOGICAL SPACES M.K.R.S.VEERA KUMAR J.K.C.COLLEGE, GUNTUR-522 006, A.P. I N D I A 1991 AMS Classification: 54 A 05, 54 D 10 Key words and Phrases: g-open set, g#-semi-closed sets, #Tb, Tb#,, Tb# # and Tb # spaces. In this paper, we introduce and study a new class of sets, namely g#-semi-closed sets for topological spaces. Applying g#-semi-closed sets, we introduce and study four new classes of spaces, namely #Tb spaces, Tb# spaces, Tb# # spaces and Tb# spaces. The class of Tb# spaces (resp. #Tb spaces) is the dual of the class of #Tb spaces (resp. Tb# spaces) to the class of Tb spaces (resp. T1/2 spaces). We also introduce and study g#scontinuous and g#s-irresolute maps. Levine introduced and studied semi-open sets1[11] and generalized closed sets2[10] in 1963 and 1970 respectively. S.P.Arya and T.Nour3[3] defined generalized semi-closed sets (briefly gs-closed sets) in 1990 for obtaining some characterizations of s-normal spaces. In this paper, we introduce and study g#-semi-closed sets. This new class is properly placed in between the class of semi-closed sets and gs-closed sets. Njåstad4[16] and Abd El-Monsef et. al 5[1] introduced -sets (called as -closed sets) and semi-preopen sets respectively. Semi-preopen sets are also known as -sets6[ 2]. Maki et.al. introduced generalized -closed sets (briefly g-closed sets)7[13] and generalized closed sets (briefly g-closed sets)8[12] in 1993 and 1994 respectively. The class of g#-semi-closed sets is independent from the classes of g-closed sets, gclosed sets, g-closed sets and preclosed sets. Levine2[10] and Devi et. al 9[8] introduced T1/2 spaces and Tb spaces respectively. As applications of g#-semi-closed sets, we introduce and study #Tb spaces, Tb# spaces, Tb# # spaces and Tb# spaces. We proved that the class of Tb# spaces( resp. #Tb spaces) is the dual of the class of #Tb spaces (resp. Tb# spaces) to the class of Tb spaces (resp. T1/2 spaces). Recently Devi et. al. 10[6] introduced Tb spaces. We observe that the class of #Tb spaces properly fits in between the class of T1/2 spaces and the class of Tb spaces. We also introduce and study g#s-continuous maps and g#s-irresolute maps. §2.PREREQUISITES Throughout this paper (X,), (Y, ) and (Z, ) represent non-empty topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space (X, ), cl(A), int(A) and C(A) denote the closure of A, the interior of A and the complement of A in X respectively. We recall the following definitions, which will be used often throughout this paper. DEFINITION 2.01 - A subset A of a space (X, ) is called (1) a preopen set11[14] if A int(cl(A)) and a preclosed set if cl(int(A)) A. (2) a semi-open set1[11] if A cl(int(A)) and a semi-closed set if int(cl(A)) A. (3) an -open set4[16] if A int(cl(int(A))) and a -closed set if cl(int(cl(A))) A. (4) a semi-preopen set6[2] (=-open5[1]) if A cl(int(cl(A))) and a semi-preclosed set 6[2] (= -closed 5[1]) if int(cl(int(A))) A. The semi-closure (resp. -closure) of a subset A of (X, ) is denoted by scl(A) (resp. cl(A) and spcl(A))and is the intersection of all semi-closed (resp. -closed and semipreclosed) sets containing A. DEFINITION 2.02 – A subset A of a space (X, ) is called (1) a generalized closed (briefly g-closed) set2[10] if cl(A) U whenever A U and U is open in (X, ). (2) a generalized semi-closed (briefly gs-closed) set3[3] if scl(A) U whenever A U and U is open in (X, ). (3) a generalized semi-preclosed (briefly gsp-closed) set12[9] if spcl(A) U whenever A U and U is open in (X, ). (4) an -generalized closed (briefly g-closed) set8[12] if cl(A) U whenever A U and U is open in (X, ). (5) a generalized -closed (briefly g-closed) set7[13] if cl(A) U whenever A U and U is -open in (X, ). DEFINITION 2.03 – A function f : (X, ) (Y, ) is called (1) semi-continuous1[11] if f –1(V) is semi-open in (X, ) for every open set V of (Y, ). (2) pre-continuous11[14] if f –1(V) is pre-closed in (X, ) for every closed set V of (Y, ). (3) -continuous12[15] if f –1(V) is -closed in (X, ) for every closed set V of (Y, ). (4) -continuous5[1] if f –1(V) is semi-preopen in (X, ) for every open set V of (Y, ). (5) g-continuous13[4] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ). (6) gs-continuous14[7] if f –1(V) is gs-closed in (X, ) for every closed set V of (Y, ). (7) g-continuous2[10] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ). (8) g-continuous7[13] if f –1(V) is g-closed in (X, ) for every closed set V of (Y, ). (9) gsp-continuous16[9] if f –1(V) is gsp-closed in (X, ) for every closed set V of (Y, ). (10) g-irresolute10[6] if f –1(V) is g-closed in (X, ) for every g-closed set V of (Y, ). (11) pre-semi-open15[5] if f(U) is semi-open in (Y, ) for every semi-open set U in (X, ). DEFINITION 2.04 – A topological space (X, ) is said to be (1) a T1/2 space2[10] if every g-closed set in it is closed. (2) a Tb space8[8] if every gs-closed set in it is closed. (3) an Tb space10[6] if every g-closed set in it is closed. §3. BASIC PROPERTIES OF g#-SEMI-CLOSED SETS We introduce the following definition : DEFINITION 3.01 – A subset A of a space (X, ) is called a g#-semi-closed set(written as g#s-closed) if scl(A) U whenever A U and U is a g-open set of (X, ). THEOREM 3.01 – Every semi-closed (resp.-closed, g#s-closed) set is a g#s-closed (resp.g#s-closed, gs-closed) set. PROOF: Follows from the definitions. The following examples show that a g#s-closed (resp.gs-closed) set need not be a semi-closed (resp. g#s-closed) set. EXAMPLE 3.01 – Let X = {a, b, c}, = {, X, {a, b}} and A = {a, c}. A is not even a semi-closed set of (X, ). However A is a g#s-closed set. EXAMPLE 3.02 – Let X = {a, b, c}, = {, X, {a}, {a, c}} and B = {a, b}. B is not a gs-closed set but not a g#s-closed set of (X, ). Thus the class of g#s-closed sets properly contains the class of closed sets, the class of -closed sets and the class of semi-closed sets. Now we show that the class of g#s-closed sets is properly contained in the class of gs-closed sets and the class of gsp-closed sets. THEOREM 3.02 –Every closed (resp. g#s-closed) set is g#s-closed (resp.gs-closed) set and hence gsp-closed. The converses are not true. PROOF: The first assertion follows from the fact that every closed (resp.gs-closed) set is a semi-closed (resp.gsp-closed) set. The set A in the Example 3.01 is is g#s-closed set but not a closed set. The set B in the Example 3.02 is a gs-closed set and hence a gsp-closed set but not a g#s-closed set. So the class of g#s-closed (resp.gsp-closed) sets properly contains the class of closed (resp.g#s-closed) sets. The class of g#s-closed sets properly contains the class of -closed sets in view of the above Theorem since every -closed set is a semi-closed set. g#s-closedness is independent from preclosedness, semi-preclosedness, gclosedness, g-closedness and g-closedness as we see the following examples. EXAMPLE 3.03 –Let X = {a, b, c}, = {, X, {a}, {b}, {a, b}} and C = {a}. C is a g#s-closed set. But C is not even an g-closed set of (X, ). Moreover C is not a preclosed set. EXAMPLE 3.04 – Let X = {a, b, c}, = {, X, {a}, {b, c}} and D = {b}. D is a preclosed and a g-closed set. Hence D is also a g-closed and hence an g-closed set of (X, ). But D is not a g#s-closed set. THEOREM 3.03 – If A is an g-open and g#s-closed set of (X, ), then A is a semiclosed set. PROOF: Obvious. THEOREM 3.04 – If A is a g#s-closed set of (X, ), then scl(A)-A does not contain any non-empty g-closed set. PROOF: Let F be a g-closed set contained in scl(A)-A. Then A X-F and X-F is a g-open set of (X, ). Since A is a g#s-closed set of (X, ), then scl(A) X-F. Now F X-scl(A). Then F (X-scl(A)) (scl(A)-A)) (X-scl(A)) scl(A) = . Hence F = . THEOREM 3.05 – If A is a g#s-closed set of (X, ) and A B scl(A), then B is also a g#s-closed set. PROOF: Let U be an g-open set of (X, ) such that B U. Then A U. Since A is g#sclosed, then scl(A) U. Since B scl(A), then scl(B) scl(scl(A)). Thus scl(B) U. Therefore B is a g#s-closed set of (X, ). 3.01 – The following diagram shows the relationships of g#s-closed sets with other sets. closed g-closed -closed g-closed g-closed g #s-closed semi-closed gs-closed semi-preclosed gsp-closed , where A (resp.A and preclosed B (resp.A B) represents A implies B but not conversely B are independent). § 4. APPLICATIONS OF g#s-CLOSED SETS We introduced the following definition : DEFINITION 4.01 –A space (X, ) is called an Tb# space if every g#s-closed set in it is semi-closed. THEOREM 4.01 – Every T1/2 space and an Tb # space but not conversely. PROOF: Let (X, ) be a T1/2 space and A be a g#s-closed set of (X, ). By the Theorem 3.01, A is a gs-closed set. Since (X, ) is T1/2, then by the Theorem 6.5 of 9[8], A is a semi-closed set of (X, ). Therefore (X, ) is an Tb # space. The space (X, ) in the example 3.04 is not a T1/2 space since {b} is neither open nor closed. However (X, ) is an Thus the class of Tb # Tb # space. spaces properly contains the class of T1/2 spaces. Next we show that the class of Tb # spaces properly contains the class of Tb spaces and the class of Tb spaces. THEOREM 4.02 –Every Tb (resp. Tb ) space is an Tb # space but not conversely. PROOF: The first assertion follows from the Remark 6.10(i)9[8] and the Theorem 5.3(i)10[6] and the above Theorem 4.01. The space in the example 3.04 is neither Tb nor an Tb since {b} is both gs-closed and an g-closed set but not a closed set. Already we know that (X, ) is an Tb # space. Now we characterize Tb# spaces. THEOREM 4.03 – For a space (X, ), the following conditions are equivalent: (1) (X, ) is an Tb# space. (2) Every singleton of X is either g-closed or semi-open. PROOF: (1) (2) Let x X and suppose that {x} is not g-closed of (X,). Then X-{x} is a g#s-closed set since X is the only g-open set containing X-{x}. By (1), X-{x} is a semiclosed set of (X, ) or equivalently {x} is semi-open. (2) (1) Let A be a g#s-closed set of (X, ). Clearly A scl(A). Let x X. By (2), {x} is either g-closed or semi-open. Case (i) Suppose{x} is g-closed. If x A, then x scl(A)-A contains the g-closed set {x} and A is g#s-closed set. In view of the Theorem 3.04, we arrived to a contradiction. Thus x A. Case(ii) Suppose {x} is semi-open. Since x scl(A), then {x} A . So x A. Thus in any case x A. So scl(A) A. Therefore A = scl(A) or equivalently A is semi-closed. Therefore (X, ) is an Tb # -space. We introduce the following definition: DEFINITION 4.02 – A space (X, ) is called a Tb# space if every g#s-closed set in it is closed. THEOREM 4.04 - Every Tb# space is an Tb # space but not conversely. PROOF: The first assertion follows from the fact that every closed set is a semi-closed set. The space (X, ) in the example 3.01 supports the second assertion. Thus the class of show Tb # spaces properly contains the class of Tb# spaces. Next we that the class of Tb# spaces properly contains the class of Tb spaces. THEOREM 4.05 - Every Tb space is a Tb# space but not conversely. PROOF: Let (X, ) is a Tb space and A be a g#s-closed set. By the Theorem 3.01, A is gsclosed of (X, ). Since (X, ) is a Tb space, then A is closed. Therefore (X, ) is a Tb# space. The space (X, ) in the Example 3.04 is a Tb# space but not a Tb space. THEOREM 4.06 – If (X, ) is a Tb# space, then every singleton of X is either gclosed or open. The converse is not true. PROOF: Let x X and suppose that {x} is not g-closed. Then X-{x} is not g-open. Then X is the only g-open set containing X-{x}. So X-{x} is g#s-closed. Since (X, ) is a Tb# space, then X-{x} is closed or equivalently {x} is open. Consider the space (X, ) in the example 3.03. {a}and {b} are open sets and {c} is an gclosed set. But (X, ) is not a Tb# space since {a} is a g#s-closed set but not a closed set of (X,). REMARK 4.01 - Tb#ness is independent from Tbness and T1/2 ness. PROOF: The space (X, ) in the Example 3.03 is an Tb space and hence a T1/2 space. But (X, ) is not a Tb# space. The space (X, ) in the Example 3.04 is a Tb# space. But (X, ) is not even a T1/2 space. We introduce the following definition : DEFINITION 4.03 – A space (X, ) is called a Tb# # space if every g#s-closed set in it is -closed. THEOREM 4.07 – Every Tb# (Tb) space is a Tb# # space but not conversely. PROOF: Let (X, ) be a Tb# space and A be a g#s-closed set of (X, ). Since (X, ) is Tb#space, then A is closed. Since every closed set is an -closed set, then A is -closed. Therefore (X, ) is a Tb# # space. By the Theorem 4.05, (X, ) is also a Tb# # space if (X, ) is a Tb space. The space in the Example 3.02 is a Tb# # space but it is neither a Tb# space nor a Tb space since {c} is a g#-closed set of (X, ) but it is neither a closed set nor a gs-closed set. Thus the class of Tb# # spaces properly contains the class of Tb# spaces and hence the class of Tb spaces. THEOREM 4.08 – If (X, ) is a Tb# # space, then every singleton of X is either gclosed or -open. The converse is not true. PROOF: Suppose (X, ) is a Tb# # space. Suppose that {x} is not g-closed for some x X. Then X-{x} is not g-open. Then X is the only g-open set containing X-{x}. So X-{x} is g# s-closed of (X, ). Since (X, ) is a Tb# # space, then X-{x} is -closed or equivalently {x}-open. The space in the Example 3.03 supports that the converse is not true. We introduce the following definition: DEFINITION 4.04 - A space (X, ) is called a #Tb space if every gs-closed set in it is gs#-closed. THEOREM 4.09 – Every T1/2 space is a #Tb space but not conversely. PROOF: Let (X, ) be a T1/2 space and A be a gs-closed set of (X, ). Since (X, ) is T1/2, then by the Theorem 6.59[8], A is semi-closed in (X, ). By the Theorem 3.01, A is g#sclosed. Therefore (X, ) is a #Tb space. The space in the example 3.03 is a #Tb space but not a T1/2 space. Therefore the class of #Tb spaces properly contains the class of T1/2 spaces. Now we show that the class of Tb spaces and the class of Tb spaces are properly contained in the class of #Tb spaces. THEOREM 4.10 – Every Tb (Tb) space is a #Tb space but not conversely. PROOF: The first part follows from the above Theorem 4.09 and the Remark 6.109[8] and the Theorem 5.3 10[6]. The space (X, ) in the example 3.01 is not even an Tb space. However (X, ) is a #Tb space. Now we show that the class of Tb# spaces is the dual of the class of # Tb spaces to the class of Tb spaces and the that the class of to # Tb spaces is the dual of the class of Tb# spaces the class of T1/2 spaces. THEOREM 4.11 – A space (X, ) is a Tb space if and only if (X, ) is Tb# and #Tb. PROOF: Necessity- Follows from the Theorems 4.05 and 4.10. Sufficiency- Suppose (X, ) is both Tb# and #Tb. Let A be a gs-closed set of (X, ). Since (X, ) is #Tb, then A is a g#s-closed set of (X, ). Since (X, ) is also a Tb#, then A is a closed set of (X, ). Therefore (X, ) is a Tb space. THEOREM 4.12 – A space (X, ) is T1/2 if and only if it is #Tb and Tb#. PROOF: Necessity- Follows from the Theorems 4.01 and 4.09. Sufficiency- Suppose (X, ) is both #Tb and Tb # . Let A be a gs-closed set of (X, ). Since (X, ) is #Tb, then A is a g#s-closed set of (X, ). Since (X, ) is also an Tb # space, then A is a semi-closed set of (X, ). Then by the Theorem 6.59[8], (X, ) is a T1/2 space. 4.01 – The following diagram shows the inter relationships between the separation axioms discussed in this section. Tb Tb # Tb Tb# T1/2 ## Tb # Tb Tb Tb# + #Tb , where A A always B (resp. A T1/2 # Tb + # Tb B) represents A implies B but B need not imply (resp. A and B are independent). §5. g#s-CONTINUOUS MAPS AND g#s-IRRESOLUTE MAPS We introduce the following definition: DEFINITION 5.01 – A function f : (X, ) (Y, ) is called g#s-continuous if f-1(V) is a g#s-closed set of (X, ) for every closed set V of (Y, ). THEOREM 5.01 – Every continuous (resp. -continuous and semi-continuous) map is g#s-continuous. PROOF: Follows from the Theorem 3.01. The following example supports that the converse of the above Theorem is not true in general. EXAMPLE 5.01 – Let X = Y = {a, b, c}, = {, X, {a, b}} and = {, Y, {a}}. Define f : (X, ) (Y, ) by f(a) = b, f(b) = a and f(c) = c. f is not continuous, in fact, it is not even semi-continuous since {b, c} is a closed set of (Y, ) but f -1({b, c}) = {a, c} is not a semi-closed set of (X, ). However f is g#s-continuous. Thus the class of g#s-continuous maps properly contains the class of continuous maps, the class of -continuous maps and the class of semi-continuous maps. Next we show that the class of g#s-continuous maps is properly contained in the class of gs-continuous maps and hence in the class of gsp-continuous maps. THEOREM 5.02 – Every g#s-continuous map is gs-continuous and hence gspcontinuous. PROOF:: Follows from the Theorem 3.01 and the Theorem 3.02. The function in the following example is gs-continuous and hence gsp-continuous but not g#s-continuous. EXAMPLE 5.02 – Let X = {a, b, c} = Y, = {, X, {a}, {a, c}} and = {, Y, {a}}. Define g : (X, ) (Y,) by g(a) = b, g(b) = c and g(c) = a. g is not a g#s-continuous map Since {b, c} is a closed set of (Y, ) but g -1({b, c}) = {a, b} is not a g#s-closed set of (X, ). However g is a gs-continuous map and hence gsp-continuous. g#s-continuity is independent from pre-continuity, g-continuity, g-continuity and gcontinuity. EXAMPLE 5.03 – Let X = {a, b, c} =Y, = {, X, {a}, {b}, {a, b}} and = {, Y, {a}, {b, c}}. Let h : (X, ) (Y,) be the identity map. {a} is a closed set of (Y, ) but h -1({a}) = {a} is neither a preclosed set nor an g-closed set. Therefore h is neither precontinuous nor g-continuous. Since {a} is not an g-closed set of (X, ), then {a} is neither g-closed nor g-closed. Therefore h is neither g-continuous nor g-continuous. However h is g#s-continuous. EXAMPLE 5.04 – Let X= {a, b, c}, = {, X, {a}, {b, c}}. Define : (X, ) (X, ) by (a) = b, (b) = a and (c) = c. is not g#s-continuous since {a} is a closed set of (X, ) but -1({a}) = {b} is not a g#s-closed set of (X, ). However is pre-continuous, gcontinuous and hence g-continuous and g-continuous. The composition of two g#s-continuous maps need not be g#s-continuous as it can be seen from the following Example. EXAMPLE 5.05 – Let X = Y = Z = {a, b, c}, = {, X, {a}, {b, c}}, = {, Y, {a, b}} and = {, Z, {a}}. Define : (X, ) (Y, ) by (a) = c, (b) = a and (c) = b. Define : (Y, ) (Z, ) by (a) = b, (b) = a and (c) = c. Clearly both and are g#scontinuous maps. {b, c} is a closed set of (Z, ) but ( o ) -1({b, c}) = -1( -1({b, c})) = -1({a, c}) = {a, b} is not a g#s-closed set of (X, ). Therefore o : (X, ) (Z, ) is not a g#s-continuous map. We introduce the following definition : DEFINITION 5.02 – A function f : (X, ) (Y, ) is called g#s-irresolute if f -1(V) is a g#s-closed set of (X, ) for every g#s-closed set V of (Y, ). THEOREM 5.03 – Every g#s-irresolute map is g#s-continuous but not conversely. PROOF: Let f : (X, ) (Y, ) be a g#s-irresolute map. Let V be a closed set of (Y, ). Then by the Theorem 3.02, V is g#s-closed. Since f is g#s-irresolute, then f -1(V) is a g#s-closed set of (X, ). Therefore f is g#s-continuous. The function : (X, ) (Y, ) in the Example 5.05 is not g#s-irresolute since {a, c} is a g#s-closed set of (Y, ) but -1({a, c}) = {a, b}is not a g#s-closed set of (X, ). Already we have seen that is g*s-continuous. Thus the class of g#s-continuous maps properly contains the class of g#s-irresolute maps. THEOREM 5.04 – Let f : (X, ) (Y, ) and g : (Y, ) (Z, ) be any two functions. Then (i) f o g : (X, ) (Z, ) is g#s-continuous if f is g#s-irresolute and g is g#s-continuous. (ii) f o g : (X, ) (Z, ) is g#s-irresolute if both f and g are g#s-irresolute. (iii)f o g : (X, ) (Z, ) is g#s-continuous if f is g#s-continuous and g is continuous. PROOF: Omitted. DEFINITION 5.03 – A function f : (X, ) (Y, ) is called pre-g-closed if f(U) is a g-closed set of (Y, ) for every g-closed set U of (X, ). Note that the above definition is different from the definition 3.1(ii) of14 [7]. THEOREM 5.05 - If the domain of a surjective, pre-g-closed and pre-semi-open map is an Tb# space, then so is the range (=codomain). PROOF: Let f : (X, ) (Y, ) be a surjective, pre-g-closed and pre-semi-open map, where (X, ) is an Tb# space. Let y Y. Since f is a surjection, then y = f(x) for some x X. Since (X, ) is an Tb# space, then by the Theorem 4.03, {x} is either g-closed or semi-open. If {x} is g-closed, then {y} = f({x}) is g-closed since f is a pre-g-closed map. If {x} is semi-open, then {y} = f({x}) is semi-open since f is a pre-semi-open map. Thus {y} is either an g-closed or semi-open set of (Y, ). By the Theorem 4.03 again, (Y, ) is an space. Tb # THEORME 5.06 – Let f : (X, ) (Y, ) be a g#s-continuous map. If (X, ), the domain of f is an Tb # space, then f is semi-continuous. PROOF: Let V be a closed set of (Y, ). Then f -1(V) is a g#s-closed set of (X, ) since f is g#s-continuous. Since (X, ) is an Tb # space, then f -1(V) is a semi-closed set of (X, ). Therefore f is semi-continuous. THEOREM 5.07 - Let f : (X, ) (Y, ) be a g#s-continuous map. If (X, ), the domain of f is a Tb# # space, then f is an -continuous map. PROOF: Similar to as that of the above Theorem 5.06. THEOREM 5.08 - Let f : (X, ) (Y, ) be a g#s-continuous map. If (X, ), the domain of f is a Tb# space, then f is continuous. PROOF: Similar to as that of the Theorem 5.06. THEOREM 5.09 - Let f : (X, ) (Y, ) be a gs-continuous map. If (X, ), the domain of f is a #Tb space, then f is g#s-continuous. PROOF: Similar to as that of the Theorem 5.06. THEOREM 5.10 – Let f : (X, ) (Y, ) be an g-irresolute and a pre-semi-closed map. Then f(A) is a g#s-closed set of (Y, ) for every g#s-closed set A of (X, ). PROOF: Let A be a g#s-closed set of (X, ). Let U be an g-open set of (Y, ) such that f(A) U. Since f is g-irresolute, then f -1(U) is an g-open set of (X, ). Since A f -1(U) and A is a g#s-closed set of (X, ), then scl(A) f -1(A). This implies f(scl(A)) U. Since f is a presemi-closed map, then f(scl(A)) scl(f(scl(A))) = f(scl(A)) U. Thus scl(f(A)) U. Therefore f(A) is a g#s-closed set of (Y, ). REFERENCES: [1] . M.E.Abd El-Monsef, S.N.El-Deeb and R.A.Mahmoud, -open sets and -continuous mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90. [2] .D.Andrijevic, Semi-preopen sets, Mat. Vesnik, 38(1)(1986), 24-32. [3] .S.P.Arya and T.Nour, Characterizations of s-normal spaces, Indian J. Pure. Appl. Math., 12(8)(1990), 717-719. [4] .K.Balachandran, P.Sundaram and H.Maki, On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 12(1991), 5-13. [5] .S.G.Crossley and S.K.Hildebrand, Semi-topological properties, Fund. Math., 74(1972), 233-254. 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