- Closed Sets in Strong Generalized Topological Spaces

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1 Volume-6, Issue-3, May-June 2016 International Journal of Enineerin and Manaement Research Pae Number: A New lass of - losed Sets in Stron Generalized Topoloical Spaces G Selvi 1, R Usha Devi 2, S Muruesan 3 1, 2 Assistant Professor, Department of Mathematics, Panimalar Institute of Technoloy, hennai, INDIA 3 Associate Professor, PG and Research Department of Mathematics, Sri S Ramasamy Naidu Memorial ollee, Sattur, Virudhunaar District, Tamil Nadu, INDIA ABSTRAT This paper aims at providin the necessary the basic concepts in Stron Generalized We roduce in this paper eneralized osed sets in Stron Generalized Topoloical spaces Knowlede of fundamentals will reatly help in understandin of topics such as eneralized continuous and eneralized irresolute in Stron Generalized Topoloical spaces Keywords - osed sets, - osed sets, - continuous, - irresolute, - continuous maps, - irresolute maps I INTRODUTION In the year 1963, Levine of semi open sets and semi continuity in Topoloical spaces tried to Generalize Topoloy by replacin open sets with semi open sets In 1970 Levine [13] first roduced the concepts of Generalized osed sets in Topoloical spaces After him saszar roduced and studied the notions of - - open sets, semi open sets preopen sets, - open sets in Generalized Topoloical spaces This concept of Generalized Topoloy was devised by him in 2002 His fundamental concepts have been studied by many topoloists in the recent years After that Min has roduced and studied various types of continuous function and almost continuous functions in Generalized Topoloical spaces In this paper, we roduced - osed sets, - osed sets, - continuous maps, - irresolute maps, - osed sets, - continuous maps in Stronly Generalized Topoloical spaces 697 opyriht 2016 Vandana Publications All Rihts Reserved II PRELIMINARIES Throuhout this paper (X, ), (Y, σ) and (Z, ) represent non-empty Generalized Topoloical spaces on which no separation axioms are assumed unless otherwise mentioned For a subset A of a space (X, ),, represent the osure of A and the erior of A respectively Definition 21: A subset A of a Stron Generalized Topoloical spaces (X, ) is called 1 a pre - open set(2) if A ( ( A))) and pre - - osed set if ( ( ( A))) A 2 a semi- -open set [3] if A ( ( ( A))) and semi- -osed set if A ( ( A))) A 3 a semi--pre-open set [1] if A ( ( ( A))) and a semi pre- -osed set [1] if ( ( ( A))) A 4 an α- -open set [3] if A ( ( ( A))) and an α--osed set [19] if ( ( ( A))) A Definition 22: A subset A of a Stron Generalized Topoloical spaces (X, ) is called 1 a eneralized- - osed set (briefly -osed) [9] if (A) U Whenever U and U is -open in (X, ) 2 a semi eneralized- osed set (briefly - osed)[10] if s (A) Uwhenever U and U is semi- -open in (X, ) 3 a eneralized semi - -osed set (briefly -osed) [9] if s (A) Uwhenever U and U is - open in (X, )

2 4 an α-eneralized - -osed set (briefly -osed) [15] if (A) Uwhenever U and U is - open in (X, ) 5 a eneralized α- -osed set (briefly -osed) [18] if (A) Uwhenever U and U is α- -open in (X, ) 6 a eneralized--semi pre osed set (briefly - osed)[10] if sp (A) Uwhenever U and U is -open in (X, ) 7 a reular eneralized--osed set(briefly - osed)[19]if (A) Uwhenever U and U is reular - open in(x, ) 8 a eneralized- - pre osed set (briefly -osed) [2] if p (A) U whenever U and U is - open in (X, ) 9 a eneralized--pre reular osed set (briefly - osed)[6] if p (A) U whenever U and U is reular open in (X, ) 10 a (A) {F : Fis osed and F} Definition 23: A function f :(X, ) (Y, σ) is called 1 a continuous [1] if f (v) 1 is a -osed set of (X, ) for every - osed set V of (Y, σ) 2 an -continuous [4] if f (v) 1 is an -osed set of (X, ) for every - osed set V of (Y, σ) 3 a s-continuous [3] if f (v) 1 is a s-osed set of (X, ) for every -osed set V of (Y, σ) 4 a -continuous [1] if f (v) 1 is a -osed set of (X, ) for every - osed set V of (Y, σ) 5 a r -continuous [4] if f (v) 1 is a r -osed set of (X, ) for every -osed set V of (Y, σ) 6 a p-continuous [3] if f (v) 1 is a p-osed set of (X, ) for every -osed set V of (Y, σ) Definition 24: A topoloical space (X, τ) is said to be 1 a - osed set [18] if α(a) ( U) U whenever and U is α-open in (X, τ) 2 a -osed set [20] if Uwhenever 3 a 4 a U and U is -open in (X, τ) - eneralized osed set (briefly osed) [21] if A U ) U - α whenever and U is open in (X, τ) -continuous [20] if f 1 (v) is a ((X, τ) for every osed set V of (Y, σ) --osed set of 5 a -irresolute [20] if (X, τ) for every f 1 (v) is a --osed set V of (Y, σ) --osed set of 3(a) Stronly losed Sets In this chapter we have roduce the concept of a new ass of stronly - losed Sets in Stronly Generalized Topoloical spaces Definition 3(a)1A subset of a Stronly Generalized topoloical space (X, ) is said to be stronly -osed set If -open ( ( ( A))) U whenever Uand U is Theorem 3(a)1: Every osed set is stronly - osed set Proof: If U, W h e n e v e r U where U is called - open Example 3(a)1: The converse of the above theorem need not be true from the followin example Let X = {a, b, c} = {, X, {a}, {b}, {a, b}, {a, c}} Let A = {a} A is a stronly - osed set but not a osed set of (X, ) Theorem 3(a)2: If a subset A of a Stron Generalized topoloical space(x, ) is - osed then it is stronly - osed in (X, ) but not conversely Proof: Suppose A is -osed in (X, )Let U be an open set containin A in (X, ) Then U contains A U A A U Now Thus A is stronly - osed in (X, ) Example 3(a)2: The converse of the above theorem need not be true as seen from the followin example Let X = {a, b,c} with topoloy = {X,, {a},{c},{b, c},{a, b}} In this Stronly Generalized topoloical space (X, ) the subset {a, c} is stronly -osed but not -osed set Theorem 3(a)3: If A is a subset of a Stronly Generalized topoloical space(x, ) is open and stronly -osed then it is osed Proof: Suppose a subset A of X is both open and stronly -osed A Now A A 698 opyriht 2016 Vandana Publications All Rihts Reserved

3 Therefore A A Since A A have A A Thus A is osed in (X, ), We orollary 3(a)1: If A is both open and stronly -osed in (X, ) then it is both reular open and reular osed in (X, ) A = A Proof: As A is open A =, since A is osed Thus A is reular open Aain A is open in X, A = A osed A As A is =A Thus A is reular osed orollary 3(a)2: If A is both open and stronly osed then it is r osed - Theorem 3(a)4: If a subset A of a stronly Generalized topoloical space (X, ) is both stronly -osed and semi open then it is -osed Proof: Suppose A is both stronly osed and semi open in (X, ), Let U be an open set containin A As A is stronly A - osed, A U U since A is semi open Thus A is Now - osed in (X, ) orollary 3(a)3: If a subset A of a stronly Generalized topoloical space (X, ) is both stronly osed and open then it is osed set Proof: Suppose A is both stronly -osed and semi open in ( X, ), Let U be an open set containin A As A is stronly Now A U - osed, A U since A is semi open Thus A is osed in (X, ) Theorem 3(a)5: A set A is stronly - osed iff A A contains no non-empty osed set Proof: Necessary: Suppose that F is non-empty osed subset of A Now F A A implies F A A, since A A = A A Thus F A Now F A i mplies A F Here F is -open and A is stronly osed, we have A F Thus F A Hence F A A =υ Therefore F = υ A A contains no non- empty osed sets Sufficient: Let A U, U is -open Suppose that A is not contained in G then ( ( (A))) c is a nonempty osed set of contradiction Therefore is stronly -osed A A U which is a and hence A orollary 3(a)4: A stronly -osed set A is reular osed iff A A is osed A and A Proof Assume that A is reular osed Since A = A, A A = is reular osed and hence osed onversely assume that A A is osed By the above theorem A A contains no nonempty osed set Therefore A A = Thus A is reular osed Theorem 3(a)6: Suppose thatb A X, B is stronly and stronly -osed set relative to a and that both open -osed subset of X then B is stronly -osed set relative to(x, ) Proof: Let B U and U is an open set in X But iven that B A X, therefore B A and B Uthis impliesb A U Since B is stronly - osed B B relative to A, (ie) A A B U B Thus( A B implies A B U stronly osed in X, A U A U This implies B B G We have A U B B A B A, since A is Also B A U B Therefore B is stronly osed set relative to (X, ) Thus 699 opyriht 2016 Vandana Publications All Rihts Reserved

4 orollary 3(a)5: Let A be stronly suppose that F is osed then set Proof: To show that to show A F A F is stronly osed and osed A F is stronly osed, we have U whenever A F U and U is -open A F is osed in A and so stronly osed in B By the above theorem stronly A F is osed in ( X, ), Since A F A X Theorem 3(a)7: If A is stronly A B osed and A then B is stronly osed Proof: Given that B A B A, B A A Since A B As A is stronly osed by the above theorem A contains no non-empty c l o s e d set, B, then B A B contains no empty osed set Aain by theorem 313, B is stronly osed set Theorem 3(a)8: Let A is stronly - osed relative to ( Y, ) A Y X and suppose that osed in ( X, ) then A is stronly Proof: Given that A Y X and A is stronly osed in ( X, ) To show that A is stronly osed relative to ( Y, ), let A Y U, where U is osed in (X, ), A UImplies A -open in ( X, ) Since A is stronly U (ie) Y A Y U,where Y A is osure of erior of A in ( Y, ) Thus A is stronly osed relative to (Y, ) Theorem 3(a)9: If a subset A of a Stronly Generalized topoloical space (X, ) is -s p-osed then it is stronly Proof: Suppose that A is -osed but not conversely -s p- osed set in ( X, ) let U be open set containin A then A U A U A U A A as U is open sp, which implies U (i e) A A U is stronly (X, ) osed set in Example 3(a)3: The converse of the above theorem need not be true from the followin example Let X={a,b,c} with topoloy,={φ,x,,{a},{b}, {a, b},{b, c},{a, c}} and B={a,b} B is stronly osed not a -s p-osed set of (X, ) Theorem 3(a)10: Every δ -osed set is a stronly osed set Proof: The Proof of the theorem is immediate from the definition Example 3(a)4: The converse of the above theorem need not be true from the followin example Let X = {a, b, c} = {Φ, X,{a},{b},{c},{a,c},{a, b}}, D = {b, c} D is not a δ- -osed set and also not even osed set Hence D is stronly osed set Theorem 3(a)11: Every θ -osed set is a stronly osed set Proof: The Proof of the theorem is immediate from the definition Example 3(a)5: The c onverse of the above theorem need not be true from the followin example Let X = {a, b, c}, = {Φ, X, {b}, {c}, {a, b}} and E ={a, c} learly E is osed and hence stronly osed E is not θ- osed set of (X, ) 3(b) - osed sets In this chapter we have roduce the concepts of a new ass of osed sets in Stronly Generalized Topoloical spaces III DEFINITION 3(B)1 Let (X, ) be a Stronly Generalized topoloical space (X, ) A subset A of(x, ) is said to be - open set if U A U The set of all -open set is denoted by G O X A subset A of a Stronly Generalized topoloical space (X, ) is said to be osed set if X - A is - open 700 opyriht 2016 Vandana Publications All Rihts Reserved

5 set The set of all by G O X Theorem 3(b)1: Every osed set is -osed sets is denoted osed set of a Stronly Generalized topoloical space (X, )But the converse part is not true Proof: Let A U and u be osed then A Uand where U is Therefore a subset A of (X, ) is said to be a set But the converse part is not true (i e) -open in (X, ) osed osed set of a Stronly Generalized topoloical space (X, ) need not be osed in eneral Example 3(b)1: Let X = {a, b, c} = {, x, {a}, {b}, {a, b}, {a,c}} Let A = {a} A is osed set but not a osed set of (X, ) Theorem 3(b)2: Every osed set of stronly Generalized topoloical space (X, ) is Generalized topoloical space (X, ) Proof: Let osed, and then X - A is osed of a A U and A U therefore A is U A U where A is open set osed set of a Stronly Generalized topoloical space (X, ) But the converse part is not true Example 3(b)2: Let X ={a, b, c}, = {, x,{a},{b},{c},{a,b},{b,c}} Let A = {{a},{b},{a,b}} A is a osed set of (X, ) but not in osed set of (X, ) Theorem 3(b)3: Every osed set of (X, ) is r osed of (X, ) Proof: Let U then A is where A is A and A U r osed of (X,) and open set then X-A is,u be reular open U A U osed set of (X, ) But the converse part is not true Example 3(b) 3: Let X ={a, b, c }, = {, x,{b},{c},{b, c},{a, c}} Let A = {b, c}a is a r osed set of (X, ) but not in osed set of (X, ) Theorem 3(b) 4: Every osed set of (X, ) is osed of space(x, ) Proof: Let A Uand p A U and U be reular open Then A is pr osed, and U A U where A is open set Then X - A is osed set of (X, ) But the converse part is not true Example 3(b)4: Let X ={a,b,c, d }, = {, x,{a},{a, b},{a, c},{a, b, c}} Let A = {{a, c}, {a, d}} A is a osed set but not in osed set of (X, ) Theorem 3(b)5: Every osed set of a Stronly Generalized topoloical space (X,) is w -osed of (X, ) Proof: Let A U and A U, U be open then A is osed of (X, ) and U A U is open set then X-A is the converse part is not true where A osed set of (X, ) But Example 3(b) 5: Let X = {a, b, c}, = {, x, {a},{b},{a, b},{a,c}} Let A = {{b}, {a, b}, {a, c}}a is a w osed set of (X,) but not in osed set of (X, ) Theorem 3(b) 6: Every osed set of (X, ) is osed set of (X, ) Proof: Let A Uand A U, U be open then A is osed, and U A U where A is open set of (X, ) Then X -A is osed set of (X, ) But the converse part is not true Example 3(b)6: Let X ={a, b, c }, = {, x, {b},{c},{a, b},{b, c}}let A = {{a}, {b, c}} A is a osed set of (X,) but not in Theorem 3(b)7: Every osed set of (X, ) osed set of (X, ) is s osed, sp osed and p osed set But the converse part is not true Example 3(b)7: Let X ={a, b, c }, = {, x, {a},{b},{a, b}, {a, c}, {b, c}} Let A ={a, b} A is a s osed, sp osed and p osed set but not in -osed set of (X, ) Theorem 3(b)8: Union of two osed sets of (X, ) is 701 opyriht 2016 Vandana Publications All Rihts Reserved

6 also a osed set of (X, ) Example 3(b)8: Let X ={a, b, c, d }, = {, x, {a},{a, b, c}, {a, b, d}} Here osed sets are {{, x,{b}, {c}, {d}, {b, c}, {b, d}, {c, d}} Hence all A and B are osed sets Then A B is also a osed sets of (X, ) Theorem 3(b)9: Intersection of two osed sets of (X, ) is also a osed set of (X,) Example 3(b)9: Let X ={a, b, c, d }, = {, x,{a}, {a, b, c}, {a, b, d}} Here osed sets of (X, ) are {{, x, {b}, {c}, {d}, {b, c}, {b, d}, {c, d}} Hence all A and B are osed sets of (X,) Then A Bis also a osed sets of (X, ) The above results can be represented in the followin fiure Where A B (rep: B) represents A implies B (resp: A&B are independent) IV -ONTINUOUS AND - IRRESOLUTE MAPS We roduce the followin definitions Definition 41: A function f: (X, ) is called continuous if is a osed set of (X, ) for every osed set of Theorem 42: Every continuous map is continuous Proof: Let f: (X, ) be continuous and let F be any osed set of Then is osed in (X, ) Since every osed set is osed, is osed Therefore f is continuous But the converse part of the above theorem need not be true Example 43: Let X = Y = {p, q, r}, = {, x,{p}} = {{, Y, {q}} f: (X, ) is defined as the identity map The inverse imae of all the osed sets of are osed in (X, ) Therefore f is continuous but not continuous Theorem 44: Every continuous map is continuous, and hence -continuous, r - continuous, continuous, maps pr -continuous, w - continuous, s - sp - continuous and p - continuous Proof: Let f: (X, ) be a V be a osed set of then is a Since f is osed set of (X, ) continuous map Let By proposition (35), (36), (37), (38) and (39), is osed and hence w osed, and p osed, osed set of (X, ) continuous, and r osed, osed, s -osed, sp osed Example 45: Let X=Y = {p, q, r}, = {, x, {p}, p,r}}, = {{, Y, {p, q}} Let f: (X, ) be the identity map Then ={r} is not osed set in X But {r} is osed and hence r osed, w osed, osed, osed, s -osed, sp osed and p osed set of (X, ) Therefore f is continuous, and hence - continuous, pr - continuous w -continuous, r - continuous, s -continuous, sp - continuous and p - continuous maps but f is not a continuous Thus the ass of all continuous maps is properly contained in the asses of continuous, and hence - continuous, r - continuous, pr continuous w - continuous, s - continuous, sp - continuous and p - continuous maps 702 opyriht 2016 Vandana Publications All Rihts Reserved

7 The followin example shows that the compositions of two continuous maps need not be a continuous map Example 46: Let X=Y = Z={p, q, r} and let f: (X, ), : ( ) be the identity maps = {,x,{p},{p, r}}, = {{,Y,{p}}, = {{,Z,{q}} is not osed set in (X, )But f and are continuous maps Theorem 47: Every continuous map is continuous map Proof: Let f: (X, ) be continuous and let V be a osed set of Y Then is a osed and we know that every osed set is and it is a osed Hence f is part of the above theorem need not be true continuous But the converse Example 48: Let X = Y = {l, m, n}, = {, x, {l}}, = {{, Y, {m}} Let f: (X, ) be the identity map A= {l, n} is osed in ( and is osed in (X, ) but not not osed in (X, ) Therefore f is continuous continuous but Definition 49: A function f: (X, ) is called irresolute if is a osed set of (X, ) for every osed set V of Theorem 410: (i) Every (ii) Every irresolute function is irresolute function is continuous continuous We et the proof of this theorem from the above definition But the converse part of the above theorems need not be true Example 411: Let X=Y= {i, j, k}, = {,x,{i},{j},{i, j},{i, k}}, = {{,Y,{i, j}} Define f: (X,) by f(i)= j, f(j)=k and f(k)=i{k} is the only osed set of Y ={j} is continuous{j, k} is ={i, j} is not irresolute Therefore f is irresolute {j, k} is a osed set of (X, ) Therefore f is osed set of Y but osed in (X, ) Therefore f is not continuous but not osed set in Y but = {i, j} is not irresolute and f is osed in (X, ) Therefore f is not continuous but not V ONLUSION irresolute Throuhout the current paper, we roduced the notations of - osed sets, - osed sets, - continuous - irresolute, - continuous maps, - irresolute maps in Stron eneralized Topoloical space The new findins in this paper will enhance and promote the further study on eneralized osed sets in terms of an ideal I in stron eneralized Topoloical space to carry out a eneral framework for their applications in practical life Generalized topoloy is important in many fields of applied sciences as well as branches of Mathematics REFERENES [1] A saszar Generalized Topoloy, Generalized continuity, Acta Math, Hunar, Vol 96, pp , 2002 [2] A saszar, Generalized open sets in Generalized Topoloies, Acta Math, Hunar, Vol 106, pp53-66, 2005 [3] N Levine, Semi open sets and semi continuity in topoloical spaces, Amer Math, Monthly, Vol 70, pp 36-41, 1963 [4] W K Min, Generalized continuous functions defined by Generalized open sets on Generalized Topoloical space, Acta Math Hunar, 2009, DOI: S [5] R Ramesh, R Mariyappan, Generalized open sets in Heriditary Generalized Topoloical space, J Math omput Sci Vol5, pp , 2015 [6] M S Sarasak, Some properties of Generalized open sets in Generalized Topoloical space, Demons ratio, Math 2013 [7] Asaszar, Modifications of Generalized Topoloies, Acta Math, Hunery, Vol 115, 2007, pp [8] Vishwamvhar roy, a type of eneralized open sets in Applied General Topoloy, universided politinicavalecia, Vol 12, no: 2, pp , 2011 [9] R Khayyeri, R Mohamadian, On base for Generalized Topoloical space IND J ontemp, Math Sciences, Vol 6, no2 48, pp , 2011 [10] Govindappa Navalai, properties of s osed set and s osed set in Topoloy International journal of communication in Topoloy, Vol1, No1, pp 31-4, Jan- Jun 2013 [11] R Parimelazhaan, V Subramonia Pillai, Stronly - osed sets in Topoloical spaces, Vol2, pp 37-42, Jun opyriht 2016 Vandana Publications All Rihts Reserved

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