BETWEEN SEMI-CLOSED SETS AND
SEMI-PRE-CLOSED SETS
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: semi-closure, sg-open sets, semi-T1/2 spaces, semi- T1/3 spaces.
Summary – In this paper a new class of sets, namelyy–closed sets is introduced for topological spaces. This class falls strictly in between the class of semi-clsoed sets and the class of semi-preclosed sets. This class also sits strictly in between the class of semi-closed sets and the class of semi-generalized closed sets.We also introduce and study a new class of spaces, namely semi-T1/3 spaces. Further we introduce and studyy–continuous maps and y–irresolute maps.
§ 1. INTRODUCTION
....................N. Levine [21] and M. E.Abd El-Monsef et.al [1] introduced semi-open sets and b-sets respectively. b-sets are also called as semi-preopen sets by Andrijevic [2]. Levine[20] generalized the concept of closed sets to generalized closed sets. Bhattacharya and Lahiri[5] generalized the concept of closed sets to semi-generalized closed sets via semi-open sets. The complement of a semi-open (resp. semi-generalized closed) set is called a semi-closed [6] (resp. semi-generalized open [5]) set. A lot of work was done in the field of generalized closed sets. In this paper we employ a new technique to obtain a new class of sets, called y–closed sets. This class is obtained by generalizing semi-closed sets via semi-generalized open sets. It is shown that the class of y-closed sets properly contains the class of semi-closed sets and is properly contained in the class of semi-preclosed sets. Further it is observed that the class of y -closed sets is independent from the class of preclosed sets, the class of g-closed sets, the class of ga-closed sets and the class of ag-closed sets. Moreover this class sits properly in between the class of semi-closed sets and the class of semi-generalized closed sets.
..........Bhattacharya and Lahiri [5], Jankovic and Reilly[19] and Maki et. al.[26] introduced semi-T1/2 spaces, semi-TD spaces and aT1/2 spaces respectively. Later Dontchev [13,14] proved that aT1/2, semi-TD and semi-T1/2 separation axioms are equivalent. R. Devi, K.Balachandran and H.Maki[8] and R. Devi, H.Maki and K.Balachandran[10] introduced aTb
spaces and Tb spaces respectively. As an application of y -closed sets, we introduced a new class of spaces, namely semi-T1/3 spaces. We also characterize semi-T1/3 spaces and show that the class of semi-T1/3 spaces properly contains the class of semi-T1/2 spaces, the class of aTb spaces and the class of semi-T1/3 spaces.
..........We also introduce and study two classes of maps, namely, y -continuity and y-irresoluteness. y-continuity falls strictly in between semi-continuity [21] and b–continuity [1]. y-continuity also falls strictly in between semi-continuity [21] and sg-continuity[30].
§ 2.PRELIMINARIES
..........Throughout this paper (X, t ), (Y, s ) and (Z, h ) represent non-empty topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space (X, t ), cl(A), int(A) and C(A) denote the closure of A, the interior of A and the complement of A in X respectively.
Let us recall the following definitions, which are useful in the sequel.
..........DEFINITION 2.01 - A subset A of a space (X, t ) is called
(1) a semi-open set[21] if A Í cl(int(A)) and a semi-closed set if int(cl(A)) Í A.
(2) a preopen set[27] if A Í int(cl(A)) and a preclosed set if cl(int(A)) Í A.
(3) an a-open set[29] if A Í int(cl(int(A))) and a a-closed set if cl(int(cl(A))) Í A.
(4) a semi-preopen set[2] (=b-open[1]) if A Í cl(int(cl(A))) and a semi-preclosed set[2] (=b-closed [1]) if int(cl(int(A))) Í A.
(5) a regular-open set if A = int(cl(A)) and a regular-closed set if cl(int(A)) = A.
(6) a semi-regular set[11] if it both semi-open and semi-closed in (X, t ).
(7) a d-closed set[31] if A = cld (A), where cld (A) = {x Î X / int(cl(U)) Ç A ¹ f , x Î U and U Î t }.
..........The semi-closure (resp. a-closure, semi-preclosure) of a subset A of (X, t ) is the intersection of all semi-closed (resp. a–closed, semi-preclosed) sets that contains A and is denoted by scl(A) (resp. a cl(A), spcl(A)). The union of all semi-open subsets of X is called the semi-interior of A and is denoted by sint(A).
..........The following definitions are useful in the sequel.
..........DEFINITION 2.02 – A subset A of a space (X, t ) is called
(1) a generalized closed (briefly g-closed) set [20] if cl(A) Í U whenever A Í U and U is.open in (X, t ).
(2) a semi-generalized closed set (briefly sg-closed) [5] if scl(A) Í U whenever A Í U and .U is semi-open in (X, t ). The ......complement of a sg-closed set is called a sg-open set.
(3) a generalized semi-closed set (briefly gs-closed) [3] if scl(A) Í U whenever A Í U and U is open in (X, t ).
(4) an a-generalized closed set (briefly ag-closed) [25] if a cl(A) Í U whenever A Í U and .U is open in (X, t ).
(5) a generalized a-closed set (briefly ga -closed) [26] if a cl(A) Í U whenever A Í U and .U is a-open in (X, t ).
(6) a ga**-closed set [26] if cl(A) Í int(cl(U)) whenever A Í U and U is a-open in (X, t ).
(7) a generalized semi-preclosed (briefly gsp-closed) set[12] if spcl(A) Í U whenever A Í U .and U is open in (X, t ).
(8) a d-generalized closed (briefly d g-closed) set [15] if cld (A) Í U whenever A Í U and U is open in (X, t ).
(9) a Q-set[22] if int(cl(A)) = cl(int(A)).
..........DEFINITION 2.03 – A function f : (X, t ) ® (Y, s ) is said to be
(1) semi-continuous[21] if f –1(V) is semi-open in (X, t ) for every open set V of (Y, s ).
(2) pre-continuous[27] if f –1(V) is pre-closed in (X, t ) for every closed set V of (Y, s ).
(3) a -continuous[28] if f –1(V) is a-closed in (X, t ) for every closed set V of (Y, s ).
(4) b -continuous[1] if f –1(V) is semi-preopen in (X, t ) for every open set V of (Y, s ).
(5) g-continuous[4] if f –1(V) is g-closed in (X, t ) for every closed set V of (Y, s ).
(6) sg-continuous[30] if f –1(V) is sg-closed in (X, t ) for every closed set V of (Y, s ).
(7) gs-continuous[9] if f –1(V) is gs-closed in (X, t ) for every closed set V of (Y, s ).
(8) ga -continuous[26] if f –1(V) is ga-closed in (X, t ) for every closed set V of (Y, s ).
(9) a g-continuous[18] if f –1(V) is ag-closed in (X, t ) for every closed set V of (Y, s ).
(10) gsp-continuous[12] if f –1(V) is gsp-closed in (X, t ) for every closed set V of (Y, s ).
(11) irresolute[7] if f –1(V) is semi-open in (X, t ) for every semi-open set V of (Y, s ).
(12) sg-irresolute[30] if f –1(V) is sg-closed in (X, t ) for every sg-closed set V of (Y, s ).
(13) pre-semi-open[7] if f(U) is semi-open in (Y, s ) for every semi-open set U in (X, t ).
(14) pre-semi-closed[7] if f(U) is semi-closed in (Y, s ) for every semi-closed set U in (X, t ).
..........DEFINITION 2.04 – A space (X, t ) is called a
(1) T1/2 space[20] if every g-closed set is closed.
(2) semi-T1/2 space[5] if every sg-closed set is semi-closed.
(3) semi-TD space[19] if every singleton is either open or nowhere dense.
(4) a Ti space[26] if a space (X, t a ) is Ti, where i = ½, 1.
(5) a T1/2* space[26] if every ga **-closed set is a -closed.
(6) a Tm space[26] if every ga**-closed set is closed.
(7) Tb space[10] if every gs-closed set is closed.
(8) a Tb space[8] if every ag-closed set is closed.
(9) semi-T1 space [23] if for any x, y Î X such that x ¹ y, there exist two semi-open sets G
and H such that x Î G, y Î H but x Ï H and y Ï G.
(10) feebly-T1 space [19,24] if every singleton is either nowhere dense or clopen.
(11) T3/4 space [15] if every d-g-closed set is d-closed.
§ 3. BASIC PROPERTIES OF –-CLOSED SETS
......We introduce the following definition :
..........DEFINITION 3.01 – A subset A of (X, t ) is called a y-closed set if scl(A) Í U whenever A Í U and U is a ......sg-open set of (X, t ).
..........REMARK 3.02 – If A is y -closed and U is sg-open with A Í U, then scl(A) Í sint(U). This follows from the ......Theorem 6 of [5].
..........THEOREM 3.03 – 1) Every semi-closed set, and thus every closed set and every a-closed set is y -closed.
2) Every y-closed set is sg-closed, and thus semi-preclosed (by Theorem2.4(i) in [14]) and also gs-closed.
PROOF: Follows immediately from the definitions.
..........The following examples show that these implications are not reversible.
..........EXAMPLE 3.04 – Let X = {a, b, c}, t = {f , X, {a, b}}. Then A = {a, c}. A is y-closed. B is not a semi-closed set.
..........EXAMPLE 3.05 – Let X = {a, b, c}, t = {f , X, {a}, {b, c}}. Then B = {b} is sg-open and sg-closed. Since scl(B) = {b, c}, B is not y-closed.
..........Thus the class of y-closed sets properly contains the class of semi-closed sets, and thus properly contains the class of a-closed sets and also properly contains the class of closed sets. Also the class of y-closed sets is properly contained in the class of sg-closed sets, and hence it is properly contained in the class of semi-preclosed sets and contained in the class of gs-closed sets.
..........THEOREM 3.06 – 1) y-closedness and g-closedness are independent notions.
2) y-closedness is independent from ga-closedness, ag-closedness and preclosedness.
PROOF: It can be seen by the following examples.
..........EXAMPLE 3.07 –Let X = {a, b, c}, t = {f , X, {a}, {a, c}} and C = {c} and D = {a, b}. C is a y-closed set but not even a g-closed set of (X, t ). D is a g-closed set but not a y-closed set of (X, t ).
..........The following two examples show that y-closedness is independent from ga -closedness, ag-clsoedness and preclosedness.
..........EXAMPLE 3.08 – Let X = {a, b, c}, t = {f , X, {a}, {b}, {a, b}} and E = {a}. E is y-closed but it is neither a ga-closed set nor an ag-closed set. Also E is not a preclosed set.
..........EXAMPLE 3.09 – Let X, t and B be as in the example 3.05. B is not a y-closed set of (X, t ). However B is a ga-closed set and hence it is an ag-closed set. Moreover B is also a preclosed set of (X, t ).
..........The following Theorem characterizes y-closed sets.
..........THEOREM 3.10 – Let A be a subset of (X, t ). Then
1) A is y-closed if and only if scl(A)-A does not contain any non-empty sg-closed set.
2) If A is y-closed and A Í B Í scl(A), then B is y-closed.
PROOF: 1) Necessity: Suppose that A is y-closed and let F be a nonempty sg-closed set with F Í scl(A)-A. Then A Í X-F and so scl(A) Í X-F. Hence F Í X-scl(A), a contradiction.Sufficiency: Suppose that for A Í X, scl(A)-A does not contain a nonempty sg-closed set. Let U be a sg-open set such that A Í U. If scl(A) Ë U, then scl(A) Ç C(U) ¹ f . It follows from Theorem 2.3 in [16] that scl(A) Ç C(U) is sg-closed, a contradiction.
2) Follows from the fact that scl(A) = scl(B).
..........THEOREM 3.11 – For a subset A of (X, t ), the following conditions are equivalent:
1) A is sg-open and y-closed.
2) A is semi-regular.
..........COROLLARY 3.12 - For a subset A of a space (X, t ), the following conditions are
equivalent:
(1) A is pre-open, sg-open and y-closed.
(2) A is regular open.
(3) A is pre-open, sg-open and semi-closed.
..........The following example shows that a subset G of a space (X, t ) need not be a closed set even though G is pre-open, sg-open and a Q-set.
..........EXAMPLE 3.13 – Let X = {a, b, c} and t = {f , X, {a}} and G = {a}. Clearly G is pre-open, sg-open and a Q-set but not a closed set.
..........THEOREM 3.14 – For a subset A of a space (X, t ), the following conditions are equivalent.
(1) A is clopen.
(2)A is preopen, sg-open, Q-set and y-closed.
PROOF: (1) Þ (2) is obvious.
(2) Þ (1) Since A is preopen, sg-open and a y-closed set of (X, t ), then by the Theorem 3.12, A is a regular open set. This implies A is open. On the other side, A = int(cl(A)) = cl(int(A)) Í cl(A) since A is a Q-set. So A is closed. Therefore A is a clopen set of (X, t ).
..........REMARK 3.15 – Union of two y-closed sets need not be y-closed. Let X = {a, b, c}, t = {f , X, {a}, {b}, {a, b}}. A ={a} and B = {b}. Both A and B are y-closed but A È B, their union is not a y-closed set of (X, t ).
..........3.16 - The following diagram shows the relationships established between y -closed sets and some other sets.
[see my original paper]
§ 4. SEMI-T1/3 SPACES
..........We introduce the following definition:
..........DEFINITION 4.01 –A space (X, t ) is said to be a semi-T1/3 space if every y-closed set in it is semi-closed.
..........THEOREM 4.02 – Every semi-T1/2 space and a semi-T1/3 space.
......The converse of the above theorem is not true as it can be seen from the following example.
..........EXAMPLE 4.03 – Let X = {a, b, c} and t = {f , X, {a}, {b, c}}. (X, t ) is not a semi-T1/2 space since {b} is a sg-closed set but not a semi-closed set of (X, t ). However (X, t ) is a semi-T1/3 space.
..........We characterize semi-T1/3 spaces in the following Theorem.
..........THEOREM 4.04 – For a space (X, t ), the following conditions are equivalent:
(1) (X, t ) is a semi-T1/3 space.
(2) Every singleton of X is either sg-closed or semi-open.
(3) Every singleton of X is either sg-closed or open.
PROOF: (1) Þ (2) Let x Î X and suppose that {x} is not a sg-closed of (X,t ). Then X-{x} is a sg-open set of (X, t ). So X is the only sg-open set containing X-{x}. Hence X-{x} is a y-closed set of (X, t ). Since (X, t ) is a semi-T1/3 space, then X-{x} is a semi-closed set of (X, t ) or equivalently {x} is semi-open set of (X, t ).
(2) Þ (1) Let A be a y -closed set of (X, t ). Clearly A Í scl(A). Let x Î X. By assumption, {x} is either sg-closed or semi-open.
Case (i) Suppose{x} is sg-closed. By the Theorem 3.10, scl(A)-A does not contain any non-empty sg-closed set. Since x Î scl(A), then x Î A.
Case(ii) Suppose {x} is a semi-open set. Since x Î scl(A), then {x} Ç A ¹ f . So x Î A.Thus in any case, scl(A) Í A.
Therefore A = scl(A) or equivalently A is a semi-closed set of (X, t ).Hence (X, t ) is an semi-T1/3 space.
(2) Û (3). Follows from the fact that a singleton is semi-open if and only if it is open.
..........THEOREM 4.05 – Every T1 space (resp. T3/4 space, T1/2 space, aT*1/2 space, aTm space, aT1 space, aT1/2 space) is a semi-T1/3 space but not conversely.
PROOF: Since every T1 space (resp. T3/4 space, T1/2 space, aT1 space, a Tm space, a T*1/2 space, aT1/2 space) is a T3/4 space[15] (resp. T1/2 space[15], semi-T1/2 space[5], aT1/2 space[26], aT*1/2 space[26], aT1/2 space[26], semi-T1/2 space[14]), the first assertion is true. The space (X, t ) in the example 4.03 is a semi-T1/3 space but not even a semi-T1/2 space.
..........REMARK 4.06 – Dontchev [13, 14] showed that aT1/2, semi-TD, semi-T1/2 separation axioms are equivalent and also that aT1 ness and feebly-T1 ness are equivalent. Dontchev and Ganster [15] proved that every T3/4 space is a semi-T1 space but not conversely.
..........THEOREM 4.07 - Every Tb space is a semi-T1/3 space and an aTb space but the respective converses are not true.
PROOF: First we observe that every Tb space is an aTb space since every ag-closed set is a gs-closed set. The fact that every Tb space is a semi-T1/3 space follows from the Remark 6.10 of [9] since every Tb space is a T1/2 space. The space in the example 3.08 is an aTb space but not a Tb space. The space in the example 3.05 is a semi-T1/3 space but not a Tb space.
..........THEOREM 4.08 - Every aTb space is a semi-T1/3 space but not conversely.
PROOF: The first assertion follows from the Theorem 5.3 [8] and the Theorem 4.02 since every T1/2 space is a semi-T1/2 space. The space in the example 3.05 is a semi-T1/3 space but not an aTb space.
..........DEFINITION 4.09 – A function f : (X, t ) ® (Y, s ) is called a pre-sg-closed if f(U) is sg-closed in (Y, s ) for every sg-closed set of (X, t ).
..........THEOREM 4.10 – If the domain of a bijective, pre-sg-closed and pre-semi-open map is a semi-T1/3 space, then so is the codomain(=range).
PROOF: Let f : (X, t ) ® (Y, s ) be a bijective, pre-sg-closed and pre-semi-open map. Suppose (X, t ) is a semi-T1/3 space. Let y Î Y. Since f is a bijection, then y = f(x) for some x Î X. Since (X, t ) is a semi-T1/3 space, then by the Theorem 4.04, {x} is either sg-closed or semi-open. If {x} is sg-closed, then {y} = f({x}) is sg-closed since f is a pre-sg-closed map. If {x} is semi-open, then {y} = f({x}) is semi-open since f is a pre-semi-open map. Thus every singleton of Y is either sg-closed or semi-open in (Y, s ). By the Theorem 4.04 again, (Y, s ) is also a semi-T1/3 space.
..........4.11 – The following diagram shows the relationships among the separation axioms considered in this paper.
§ 5. –-CONTINUOUS AND y–-IRRESOLUTE MAPS
..........We introduce the following definition:
..........DEFINITION 5.01 – A function f : (X, t ) ® (Y, s ) is called y-continuous if f-1(V)
is a y-closed set of (X, t ) for every closed set V of (Y, s ).
..........THEOREM 5.02 – 1) Every semi-continuous map and thus every continuous map and every a–continuous map is y–continuous.
2) Every y–continuous map is sg-continuous and thus b–continuous, gs-continuous and gsp-continuous.
PROOF: (1). Let f : (X, t ) ® (Y, s ) be a semi-continuous map. Let V be a closed set of (Y, s ). Since f is semi-continuous, then f –1(V) is a semi-closed set of (X, t ). By the Theorem 3.03, f –1(V) is also a y–closed set of (X, t ). Therefore f is a y–continuous map.
(2) Let f : (X, t ) ® (Y, s ) be a y–continuous map. Let V be a closed set of (Y, s ). Since f is y-continuous, then f –1(V) is a y–closed set of (Y, s ). By the theorem 3.03, f –1(V) is a sg-closed and thus b –closed, gs-closed and gsp-closed set of (Y, s ). Therefore f is a sg-continuous map and thus b–continuous, gs-continuous and gsp-continuous.
..........The converses in the above Theorem are not true as it can be seen from the following examples.
..........EXAMPLE 5.03 – Let X = {a, b, c} = Y, t = {f , X, {a, b}} and s = {f , Y, {a}, {b}, {a, b}}. Let f be the identity map from (X, t ) into (Y, s ). f is not even semi-continuous since {a, c} is a closed set of (Y, s ) but f -1({a, c}) = {a, c} is not a semi-closed set of (X, t ). However f is a y-continuous map.
..........EXAMPLE 5.04 – Let X = {a, b, c} = Y, t = {f , X, {a}, {b, c}} = s . Define g : (X, t ) ® (Y,s ) by g(a) = c, g(b) = a and g(c) = c. g is not a y-continuous map since {a} is a closed set of (Y, s ) but g -1({a}) = { b} is not a y-closed set of (X, t ). However g is a sg-continuous map.
..........Thus the class of y-continuous maps properly contains the class of semi-continuous maps and thus it contains the class of continuous maps and the class of a–continuous maps. Also the class of y -continuous maps is properly contained in the class of sg-continuous maps and hence it is contained in the classes of b–continuous maps, gs-continuous maps and gsp-continuous maps.
..........THEOREM 5.05 – 1) y-continuity and g-continuity are independent of each other.
2) y-continuity is independent from ag-continuity, ga-continuity and precontinuity.
PROOF: 1) Let X = {a, b, c} = Y, t = {f , X, {a}, {a, c}} = s . Define h : (X, t ) ® (Y,s ) by h(a) = a, h(b) = c and h(c) = b. h is not g-continuous since {b} is a closed set of (Y, s ) but h -1({b}) = {c} is not a g-closed set of (X, t ). However h is a y-continuous map. Define q : (X, t ) ® (Y,s ) by q (a) = c, q (b) = b and q (c) = a. q is not y–continuous since {b, c} is a closed set of (Y, s ) but q -1({b, c}) = {a, b} is not a y–closed set of (X, t ). However q is a g-continuous map.
2) Let X = {a, b, c} = Y, t = {f , X, {a}, {b}, {a, b}} and s = {f , Y, {a}, {a, c}}. Define j : (X, t ) ® (Y, s ) by j (a) = a, j (b) = b and j (c) = c. j is a y-continuous map. j is neither a a pre-continuous nor an ag-continuous map. Moreover j is not a ga-continuous map. The function g in the example 5.04 is not y-continuous. However g is pre-continuous, ag-
continuous and ga-continuous.
..........The composition of two y -continuous maps need not be y -continuous as it can be seen from the following Example.
..........EXAMPLE 5.06 – Let X, Y, t , s and j be as in the above result. Let Z = X and h = {f , Z, {a}, {b}, {a, b}, {a, c}}. Define f : (Z, h ) ® (X, t ) by f(a) = b, f(b) = a and f(c) = c. Clearly both f and j are y-continuous maps. But j o f : (Z, h ) ® (Y, s ) is not y-continuous since {b} is a closed set of (Y,s ) but (j of) -1({b}) = f -1(j -1({b})) = f -1({b}) = {a} is not a y-closed set of (Z, h ).
..........We introduce the following definition:
..........DEFINITION 5.07 – A function f : (X, t ) ® (Y, s ) is called y-irresolute if f -1(V) is a y-closed set of (X, t ) for every y-closed set V of (Y, s ).
..........Clearly every y-irresolute map is y-continuous. The converse, however is not true as it can be seen by the following example.
..........EXAMPLE 5.08 – Let X, Y, t , s and f be as in the example 5.03. f is not a y–irresolute since {a} is a y–closed set of (Y, s ) but f –1({a}) = {a} is not a y–closed set of (X, t ). However f is a y–continuous map.
..........THEOREM 5.09 – Let f : (X, t ) ® (Y, s ) and g : (Y, s ) ® (Z, h ) be any two functions. Then
(i) g o f : (X, t ) ® (Z, h ) is y-continuous if g is continuous and f is y-continuous.
(ii) g o f : (X, t ) ® (Z, h ) is y-irresolute if g is y -irresolute and f is y-irresolute.
(iii)g o f : (X, t ) ® (Z, h ) is y-continuous if g is y -continuous and f is y-irresolute.
PROOF: Omitted.
..........THEOREM 5.10 – Let f : (X, t ) ® (Y, s ) be a bijective y-irresolute map. If (X, t ) is a semi-T1/3 space, then f is an irresolute map.
PROOF: Let V be a semi-open set of (Y, s ). Then C(V) is a semi-closed set of (Y, s ). By the Theorem 3.03, C(V) is a y-closed set of (Y, s ). Since f is a y-irresolute map, then f -1(C(V)) is a y-closed set of (X, t ). Since (X, t ) is a semi-T1/3 space, then f –1(C(V)) is a semi-closed set of (X, t ). Since f is a bijection, f -1(V) = C(f -1(C(V))). Thus f -1(V) is a semi-open set of (X, t ). Therefore f is an irresolute map.
..........THEOREM 5.11 – Let f : (X, t ) ® (Y, s ) be a surjective sg-irresolute and a pre-semi-closed map. Then for every y-closed set A of (X, t ), f(A) is a y-closed set of (Y, s ).
PROOF: Let A be a y-closed set of (X, t ). Let U be a sg-open set of (Y, s ) such that f(A) Í U. Since f is a surjective, sg-irresolute map, then f -1(U) is a sg-open set of (X, t ). Then scl(A) Í f -1(A) since A is a y-closed set and A Í f -1(U). This implies f(scl(A)) Í U. Since f is a pre-semi-closed, then f(scl(A)) Í scl(f(scl(A))). Now scl(f(A)) Í scl(f(scl(A))) = f(scl(A)) Í U. Therefore f(A) is a y-closed set of (Y, s ).
..........THEOREM 5.12 – Let f : (X, t ) ® (Y, s ) be a surjective, y-irresolute and a pre-semi-closed map. If (X, t ) is a semi-T1/3 space, then (Y, s ) is also a semi-T1/3 space.PROOF: Let A be a y-closed set of (Y, s ). Since f is a y-irresolute map, then f -1(A) is a y-closed set of (X, t ). Since (X, t ) is a semi-T1/3 space, then f -1(A) is semi-closed in (X, t ). Then f(f -1(A)) is semi-closed in (Y, s ) since f is a pre-semi-closed map. Since f is a surjection, then A = f(f -1(A)). Thus A is a semi-closed set of (Y, s ). Therefore (Y, s ) is a semi-T1/3 space.
REFERENCES:
[1]. M.E.Abd El-Monsef, S.N.El-Deeb and R.A.Mahmoud, b-open sets and b-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., 21(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]. P.Bhattacharya and B.K.Lahiri, Semi-generalized closed sets in topology, Indian J. Math., 29(3)(1987),375-382.
[6]. N.Biswas, On characterizations of semi-continuous functions, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., .......(8)48(1970), 399-402.
[7]. S.G.Crossley and S.K.Hildebrand, Semi-topological properties, Fund. Math., 74(1972), 233-254.
[8]. R.Devi, K.Balachandran and H.Maki, Generalized a-closed maps and a -generalized closed maps, Indian J. Pure. .......Appl. Math., 29(1)(1998), 37-49.
[9]. R.Devi, K.Balachandran and H.Maki, Semi-generalized homeomorphisms and generalized semi-homeomorphisms in .......topological spaces, Indian J. Pure. Appl. Math., 26(3)(1995), 271-284.
[10]. R.Devi, H.Maki and K.Balachandran, Semi-generalized closed maps and generalized semi-closed maps, Mem. Fac. .......Sci. Kochi Univ. Ser.A., Math., 14(1993), 41-54.
[11]. G. Di Maio and T.Noiri, On s-closed spaces, Indian J. Pure. Appl. Math., 18(3)(1987), 226-233.
[12]. J.Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 16(1995), 35-48.
[13]. J.Dontchev, On point generated spaces, Questions Answers Gen. Topology, 13(1995), 63-69.
[14]. J.Dontchev, On some separation axioms associated with the a-topology, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., .......18(1997), 31-35.
[15]. J.Dontchev and M.Ganster, On d-generalized closed sets and T3/4 -spaces, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., .......17(1996), 15-31.
[16]. J.Dontchev and H.Maki, On sg-closed sets and semi-l-closed sets, Questions Answers. Gen. Topology, 15(1997), .......259-266.
[17]. W. Dunham, T1/2-spaces, Kyungpook Math. J., 17(1977), 161-169.
[18]. Y.Gnanambal, On generalized preregular closed sets in topological spaces, Indian J. Pure. Appl. Math., 28(3)(1997), .......351-360.
[19]. D.S.Jankovic and I.L.Reilly, On semi separation properties, Indian J. Pure. Appl. Math., 16(9)(1985), 957-964.
[20]. N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19(2)(1970), 89-96.
[21]. N.Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70(1963), 36-41.
[22]. N.Levine, On the commutativity of the closure and interior operator in topological spaces, Amer. Math. Monthly, .........68(1961), 474-477.
[23]. S.N.Maheswari and R.Prasad, Some new separation axioms, Ann. Soc. Sco. Bruxelles Ser.I., 89(1975), 395-402.
[24]. S.N.Maheswari and U.Tapi, Feebly T1-spaces, An. Univ. Timisoara Ser. Stiint. Mat., 16(2)(1978), 173-177.
[25]. H.Maki, R.Devi and K.Balachandran, Associated topologies of generalized a-closed sets and a -generalized closed .......sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 15(1994), 51-63.
[26]. H.Maki, R.Devi and K.Balachandran, Generalized a-closed sets in topology, Bull. Fukuoka Univ. Ed. Part III, .......42(1993), 13- 21.
[27]. A.S.Mashhour, M.E.Abd El-Monsef and S.N.El-Deeb, On pre-continuous and weak .pre-continuous mappings, Proc. ....... Math. and Phys. Soc. Egypt, 53(1982), 47-53.
[28]. A.S.Mashhour, I.A.Hasanein and S.N.El-Deeb, a-continuous and a-open mappings, Acta Math. Hung., . .........41(3-4)(1983), 213-218.
[29]. O.Njåstad, On some classes of nearly open sets, Pacific J. Math., 15(1965), 961-970.
[30]. P.Sundaram, H.Maki and K.Balachandran, Semi-generalized continuous maps and semi-T1/2 spaces, Bull. Fukuoka ........Univ. Ed. Part III, 40(1991), 33-40.
[31]. N.V.Velicko, H-closed topological spaces, Amer. Math. Soc. Transl., 78(1968), 103-118.
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