Slightly ω b Continuous Functions in Topological Spaces

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1 Slightly Continuous Functions in Topological Spaces Raja Mohammad Latif, Muhammad Rafiq Raja 2, Muhammad Razaq 3 Department of Mathematics and Natural Sciences Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia 2 Department of Mathematics, University of Sargodha, Mandi Bahauddin Campus, Pakistan 3 Department of Mathematics, Postgraduate Horizon Cllege, Chakwal, Pakistan Abstract In this paper, slightly continuity is introduced and studied Furthermore, basic properties and presentation theorems of slightly continuous functions are investigated and relationships between slightly continuous functions and graphs are studied and investigated Keywords, Topological space, open set, continuity, slightly continuity, slightly continuity 200 Mathematics Subject Classification: 54B05, 54C08 I INTRODUCTION Generalized open sets play a very important role in General Topology and they are now the research topics of many topologists worldwide Semi open sets, preopen sets, α sets, and β open sets play an important role in the researchers of generalizations of continuity in topological spaces Indeed a significant theme in General Topology and Real Analysis concerns the variously modified forms of continuity, separation aioms etc by utilizing generalized closed sets Functions and of course continuous functions are stated among the most important and most researched points in the whole of the Mathematical Sciences Many different forms of continuous functions have been introduced over the years Various interesting problems arise when one considers continuity Its importance is significant in various areas of mathematics and related sciences Some of them are strongly α irresoluteness, α irresoluteness, α continuity, precontinuity, semi continuity, γ continuity and slightly continuity In 980 Jain [9] introduced the notion of slightly continuous functions In 995 Nour [28] defined slightly semi continuous functions as week form of slight continuity and investigated the functions In 2000 Noiri and Chae [24] have further investigated slightly semi continuous functions On the other hand, Pal and Bhattacharyya [29] defined a function to be faintly precontinuous if the preimages of each clopen set of the codomain is preopen and obtained many properties of such functions Slight continuity implies both slight semi-continuity and faint precontinuity but not conversely In 2009 Noiri, Al-Omari and Noorani [27] introduced a new week form of both slightly and continuous, called slightly continuous, and studied basic properties and preservation theorems of slightly continuous functions In 202 Chakraborty [4] introduced continuous and studied the relations of slightly continuous functions wih other forms of continuous functions In 2008 Noiri, Al-Omari and Noorani [26] introduced and investigated properties of a new generalization of class of open set and open set called open sets The aim of this paper is to introduce and study a new weaker form of continuity called slightly continuity Moreover, basic properties and preservation theorems of slightly continuous functions are investigated and relationships between slightly continuous DOI : /IJRTER OHM56 72

2 functions and graphs are investigated In section 3, the notion of slightly continuous functions is also introduced and characterizations and some relationships of continuous functions and basic properties of slightly continuous functions are investigated and obtained The relationships between slightly continuity and connectedness are investigated In Section 4 and in Section 5, the relationships between slightly continuity and compactness and the relationships between slightly continuity and separation aioms and graphs are obtained II PRELIMINARIES Throughout this paper, ( X, τ and (or simply, X and Y denote topological spaces on which no separation aioms are assumed unless eplicitly stated For a subset A of a space ( X, τ, Cl ( A, ( Int A and X A denote the closure of A, the interior of A and the complement of A in X, respectively Recently, as generalization of closed sets, the notion of closed sets were introduced and studied by Hdeib [7] Let ( X, τ be a topological space and let A be a subset of X A point X is called a condensation point of A if for each U τ with U, the set U I A is uncountable A subset A is said to be closed [7] if it contains all its condensation points The complement of an closed sets is said to be an open set It is well known that a subset W of a space ( X, τ is open if and only if for each W, there eists U τ such that U and U W is countable The family of all open subsets of a topological space ( X, τ, denoted by ( τ O X, τ, forms a topology on X which is finer than containing a point X is denoted by O( X, The complement of an open set is said to be closed The intersection of all closed sets of X containing A is called the closure of A and is denoted by Cl ( A The union of all open sets of X contained in A is called τ The set of all open sets of ( X, τ int erior of A and is denoted by Int ( A The family of all open open, closed, clopen, clopen sets of X is denoted by O( X, τ, Cl ( X, τ, CO( X, τ, CO( X, τ τ A subset A of a topological space X is said to be open [] if A Int Cl ( A U Cl Int ( A The complement of a open set is called closed The intersection of all closed sets of X containing A is called the closure of A and is denoted by bcl ( A The union of all open sets of X contained in A is called the int erior of A and is denoted by bint ( A A = subset A of X is said to be regular open if A Int Cl ( A closed, clopen, b clopen, regular open sets in X is denoted by ( BCO X RO( X CO( X, (, The family of all open (resp BO X (resp BC ( X, DEFINITION 2 A subset A of a space X is said to be open if for every A, there eists a open subset U X containing such that U A is countable The complement of an open subset is said to be closed sets in a topological space ( X, τ is denoted by b O( X, τ subsets in a topological space (, The family of all b open O ( X The family of all b closed ( τ C X, or C ( X For any X, we present O( X = { U X U and U is open in X}, : or X τ is denoted All Rights Reserved 73

3 LEMMA 22 For a subset of a topological space, both openness Proof ( that ( ( openness and openness imply Assume A is open, then for each A, there is an open set U containing such U A is countable set Since every open set is open, A is open 2 Let A be open For each A, there eists a open set U = A such that U and U A = φ Therefore, A is open The following diagram shows the implications for properties of subsets open set open set open set open set The converses need not be true as shown by the eamples 23 and 24 in [26] LEMMA 23 [26] A subset A of a space X is open if and only if for every A, there eists a open subset U containing and a countable subset C such that U C A The intersection of two open sets is not always open EXAMPLE 24 Let X = R with the usual topology τ Let A = Q be the set of all rational numbers and = [ 0 B, Then A and B are open, but AI B is not open, since each open set containing 0 is uncountable set PROPOSITION 25 The union of any family of open sets is open PROOF Let { A α : α } U A α, A β α such that ( be a collection of open subsets of X, then for every for some β Hence there eists a open subset U of X containing U A is countable Now as U ( U A ( U A β α α β and thus U ( A α α ` U is countable Therefore, U A α α is open The intersection of all closed sets of X containing A is called the closure of A and is denoted by Cl ( A The union of all open sets of X contained in A is called the int erior of A and is denoted by Int ( A PROPOSITION 26 [26] The intersection of an open set with an open set is open III SLIGHTLY CONTINUOUS FUNCTIONS In this section, the notion of slightly continuous functions is introduced and characterizations and some relationships of continuous functions and basic properties of slightly continuous functions are investigated and obtained DEFINITION 3 A function f : ( X, τ X if for each clopen subset V in Y containing ( X containing such that f ( U V DEFINITION 32 A function f : ( X, τ slightly continuous at each point of X is called slightly continuous at a point f, there eists an open subset U in is called slightly continuous if it is THEOREM 33 Let ( X, τ and be topological spaces and let f : ( X, τ function Then the following statements are equivalent f is slightly continuous; ( be All Rights Reserved 74

4 ( ( ( 2 for every clopen set V Y, 3 for every clopen set V, 4 for every clopen set V, PROOF ( ( : f ( V Y f ( V Y f ( V is open; is closed; is clopen 2 Let V be a clopen subset of Y and let f ( V Since f ( V, by (, there eists an open set U in X such that U and U f ( V We obtain that { } f ( V = U U : f ( V Thus, f ( V is open ( 2 ( 3 : Let V be a clopen subset of Y Then, ( f ( Y V = X f ( V is an open set in X Thus, f ( V b ( 3 ( 4 : Let V be a clopen subset of Y Then, by (3, f ( V Note that Y V is also clopen in closed set in Y V is clopen By (2, is closed is an closed set in X Y Hence by (3, it follows that f ( Y V = X f ( V X Thus, f ( V is clopen in X f By (4, f ( V ( 4 ( : Let V be a clopen subset of Y containing ( X Take U f = ( V Then, f ( U V Hence, f is slightly continuous THEOREM 34 Let f : ( X, τ be a function and Σ = { Ui : i I} X such that U O( X, τ for each i I is a is clopen in Σ = be a cover of i If f U i is slightly continuous for each i I, then f is a slightly continuous function PROOF Suppose that V is any clopen set of Y Since f U i is slightly continuous for each it follows that ( f U ( V O( U τ U i I, { i }, i i i ( = U ( I : ( ( f V f V U i I ( f ( V O ( X, τ THEOREM 35 Let f : ( X, τ function of f, defined by g(, f ( { f Ui V i I} We have = U : We obtain which means that f is slightly continuous be a function and let g : X X Y be the graph ( if and only if f is slightly continuous PROOF Let (, then ( = ( ( τ = for every X Then g is slightly continuous V CO Y Then X V CO( X Y Since g is slightly continuous, f V g X V b O X, Thus, f is slightly continuous Conversely, let X and let W be a clopen subset of X Y containing g( Then W I ({ } Y is clopen in { } Y containing g( Also { } { y Y : (, y W } continuous, { f ( y : (, y W } U f ( y : (, y W g W Hence g ( W Y is homeomorphic to Y Hence is a clopen subset of Y Since f is slightly U is an open subset of X Further { } ( continuous DEFINITION 36 A function f : ( X, τ open subset G of Y, f ( G is open in X is open Then g is slightly is called irresolute if for All Rights Reserved 75

5 DEFINITION 37 A function f : ( X, τ subset A of X, f ( A is open in Y DEFINITION 38 A function f : ( X, τ open set in X for each open set V of Y is called open if for every open is said to be b continuous if f ( V DEFINITION 39 A function f : ( X, τ is slightly continuous if f ( V is is open set in X for each clopen set V of Y THEOREM 30 Let f : ( X, τ and g : ( Z, be functions Then, the following properties hold: If f is irresolute and g is slightly b continuous, g ο f : X, τ Z, is ( slightly continuous ( 2 If f is b irresolute continuous and g is b continuous, ( 3 If f is b irresolute continuous then ( ( then g ο f : ( X, τ ( Z, is slightly and g is slightly continuous, then g ο f : ( X, τ ( Z, PROOF ( Let V be any clopen set in Z Since g is slightly b continuous, open, Since f is b irresolute, gο f is slightly continuous ( 2 and ( 3 can be obtained similarly f g ( V = ( gο f ( V THEOREM 3 Let f : ( X, τ and g : ( Z, is slightly g ( V is is open Therefore be functions If f is open and surjective and gο f : X Z is slightly continuous, then g is slightly continuous PROOF Let V be any clopen set in Z Since gο f is slightly continuous, ( gο f ( V f g ( V g ( V is b open = is open Since f is b open, Hence, g is slightly continuous ( then f f g ( V = Combining the previous two theorems, we obtain the following result f : X, τ Y, be surjective, irresolute and open and THEOREM 32 Let ( ( g : ( Z, be a function Then g ο f : ( X, τ ( Z, if and only if g is slightly continuous is slightly continuous DEFINITION 33 Let ( X, τ be a topological space Then a filter base Λ is said to be convergent to a point X if for any U O( X, τ containing, there eists a B Λ such that B U X τ be a topological space A filter base Λ is said to be DEFINITION 34 Let (, co convergent to a point in X if for any U CO( X, τ B Λ such that B U THEOREM 35 If a function f : ( X, τ containing, there eists a is slightly continuous, then for each point X and each filter base Λ in X converging to, co convergent to f ( f Λ is the filter base All Rights Reserved 76

6 PROOF Let X and Λ be any filter base in X converging to Since f is slightly continuous, then for any V CO containing f (, there eists a U O( X, τ containing such that f ( U V Since Λ is b converging that B U This means that f ( B V and therefore the filter base ( to f ( to, there eists a B Λ such f Λ is co convergent DEFINITION 36 A topological space ( X, τ is called connected provided that X is not the union of two disjoint nonempty open sets THEOREM 37 If a function f : ( X, τ is slightly continuous surjective function and X is connected space, then Y is connected space PROOF Suppose that Y is not connected space Then there eists nonempty disjoint open sets U and V such that Y = U U V Therefore, U and V are clopen sets in Y Since f is slightly continuous, then f ( U f and f ( V ( U and f ( V are nonempty disjoint and X f = ( U f ( V connected This is a contradiction Hence, Y is connected DEFINITION 38 A topological space (, base consisting of clopen sets THEOREM 39 Suppose that f : ( X, τ 0 dim ensional space, then f is continuous are closed and open in X Moreover, U This shows that X is not X τ is called 0 dim ensional if its topology has a is slightly continuous and Y is PROOF Let X and V be any open subset of Y containing f ( Since Y is a 0 dim ensional space, there eists a clopen set U containing f ( such that U V Since f is slightly continuous, then there eists an open subset G in X containing such that ( f G U V Thus, f is continuous DEFINITION 320 A subset M of a topological space ( X, τ is said to be dense in X if there is no proper closed set C in X such that M C X PROPOSITION 32 A subset M of a topological space ( X, τ is said to be dense in X if for any nonempty open set U in X, U I M φ THEOREM 322 Let f : ( X, τ be a surjective function and 0 dim ensional space Then the following statements are equivalent: ( f is slightly continuous ( 2 If C is a clopen subset of Y such that ( f C D f C X D of X such that ( ( 3 If M is an dense subset of X, then ( be, then there is a proper closed subset f M is a dense subset of Y PROOF ( ( 2 : Let C be a clopen subset of Y such that f ( C X clopen set in Y such that f ( Y \ C = X \ f ( C φ Y such that V φ and V f ( Y \ C = X \ f ( C X \ V = D is a proper closed set in X ( 2 ( 3 : Let M be an dense set in Then ( Y C is a By (, there eists an open set V in X Suppose that ( This shows that f ( C ( X \ V and f M is not dense in Y Then there eists a nonempty proper closed set C in Y such that f ( M C Y Clearly f ( C X All Rights Reserved 77

7 U = Y \ C is a nonempty proper open subset of Y Since Y is 0 dim ensional So there eists a nonempty proper clopen set E in Y such that E U = Y \ C Then C Y \ E = F and F = Y E is a nonempty proper clopen set in f M F Y and f ( F φ By (2, there eists a Y Also ( nonempty proper closed set D such that M f ( F D X that M is dense in X This is a contradiction to the fact ( 3 ( : Suppose that f is not a slightly continuous function, then there eits a nonempty proper clopen set U in Y such that int erior of f ( U is empty, that is X f ( U is ( dense in X, while f X f ( U = Y U PROPOSITION 323 Let f : ( X, τ is not dense in Y This is a contradiction be a function and X = AU B, where A, B τ If the restriction functions f : ( A, τ and f : ( B, τ A A continuous, then f is slightly continuous B B are slightly PROOF Let X and let U be any clopen subset of Y such that f ( U Now f ( U Then ( f ( U or ( f ( U or both ( f ( U and ( f ( U Suppose ( f ( U A A B Since f A is slightly continuous, there eists an open set V in A such that V and V ( f ( U f ( U A A Since V is open in A and A is open in X Hence V is open in X Thus we find that f is slightly continuous IV COVERING PROPERTIES In this section, the relationship between slightly continuous functions and compactness are investigated X, τ is said to be mildly compact if for every clopen DEFINITION 4 A topological space ( cover of X has a finite subcover DEFINITION 42 A topological space (, X τ is said to be compact if for every open cover of X has a finite subcover X, τ be a topological space Then a subset A of X is said to be mildly DEFINITION 43 Let ( compact (respectively compact relative to X if every cover of A by clopen (resp open sets of X has a finite subcover DEFINITION 44 A subset A of a space X is said to be mildly compact (respectively compact if the subspace A is mildly compact (resp compact THEOREM 45 If a function f : ( X, τ compact relative to X, then ( PROOF Let { Hα : α I} be any cover of ( is slightly continuous and K is f K is mildly compact in Y f K by clopen sets of the subspace f ( K For each α I, there eists a clopen set K α of Y such that H α = K α I f ( K For each K, there eists α I, such that f ( K α and there eists U (, O X τ containing such that ( f U K Since the family α { U : K} eists a finite subset K 0 of K such that K { U K } is a cover of K by open sets of K, there U : 0 Therefore, we obtain All Rights Reserved 78

8 f ( K U { f ( U : K 0 } which is a subset of U K : K α Thus 0 f ( K = U H : K α and hence f 0 ( K is mildly compact f : X, τ Y, is slightly continuous surjection and X is COROLLARY 46 If ( ( compact, then Y is mildly compact DEFINITION 47 A topological space ( X, τ said to be mildly countably compact if every clopen countable cover of X has a finite subcover X, τ is said to be mildly Lindelof if every cover of X by DEFINITION 48 A topological space ( clopen sets has a countable subcover DDEFINITION 49 A topological space (, countable open cover of X has a finite subcover X τ said to be countably compact if every DEFINTION 40 A topological space ( X, τ said to be Lindeloff if every open cover of X has a countable subcover DEFINTION 4 A topological space (, closed cover of X has a finite subcover DEFINTION 42 A topological space (, every countable cover of X by closed sets has a finite subcover DEFINTION 43 A topological space (, of X by closed sets has a countable subcover X τ said to be closed compact if every X τ said to be countably closed compact if X τ said to be closed Lindeloff if every cover THEOREM 44 Let ( X, τ be Lindeloff and let f : ( X, τ continuous surjection Then Y is mildly Lindelof PROOF Let γ = { Vα : α I} then ( ( : be a slightly = be any clopen cover of Y Since f is slightly continuous, { α } λ = f γ = f V α I is an open cover of X Since X is Lindeloff, { } there eists a countable subset I 0 of I such that X = f ( Vα : α I0 Y = U { Vα : α I 0 } and Y is mildly Lindelof THEOREM 45 Let ( X, τ be countably compact and let f : ( X, τ slightly continuous surjection Then Y is mildly countably compact PROOF Let γ = { Vα : α I} continuous, then ( ( U Thus, we have be a = be a countable open cover of X Since f is slightly { α : } λ = f γ = f V α I is a countable open cover of X Since X is countably compact, there eists a finite subset I 0 of I such that U { ( α : α 0} Thus, we have Y = { Vα : α I 0 } X = f V I U and Y is mildly countably compact The proofs of the net three theorems can be obtained similarly as the previous two theorems f : X, τ Y, be a slightly b continuous THEOREM 46 Let ( ( be closed compact, and Then Y is mildly compact THEOREM 47 Let (, X τ be closed Lindeloff slightly continuous surjection Then Y is mildly Lindelof surjection Let ( X, τ and let f : ( X, τ be All Rights Reserved 79

9 THEOREM 48 Let (, X τ be countably closed compact and let f : ( X, τ be a slightly continuous surjection Then Y is mildly countably compact V SEPARATION AXIOMS In this section, the relationships between slightly continuous functions and separation aioms are investigated X, τ said to be T if for each pair of distinct points DEFINITION 5 A topological space ( and y of X, there eist open sets U and V containing and y respectively such that y U and V DEFINITION 52 A topological space ( X, τ said to be b T 2 ( b Hausdorff if for each pair of distinct points and y in X, there eist disjoint open sets U and V in X such that U and y V DEFINITION 53 A topological space ( X, τ said to be clopen T if for each pair of distinct points and y of X, there eist clopen sets U and V containing and y respectively such that y U and V DEFINITION 54 A topological space ( X, τ said to be clopen T 2 (clopen Hausdorff or ultra Hausdorff if for each pair of distinct points and y in X, there eist disjoint clopen sets U and V in X such that U and y V PROPOSITION 55 A topological space (, closed sets X τ is T if and only if the singletons are PROPOSITION 56 A topological space ( X, τ is T 2 ( b Hausdorff if and only if the intersection of all closed neighbourhoods of each point of X is reduced to that point THEOREM 57 If a function f : ( X, τ is slightly continuous injection and Y is clopen T, then X is T PROOF Suppose that Y is T For any distinct points and y in X, there eist V, W CO such that f ( V, f ( y V, f ( W and f ( y W slightly continuous, f ( V and f ( W f ( V, y f ( V, f ( W and y f ( W b THEOREM 58 If f : ( X, τ is a slightly b continuous Since f is are open subsets of X such that This shows that X is T injection and Y is clopen T 2, then X is T 2 PROOF For any pair of distinct points and y in X, there eist disjoint clopen sets U and V in and f ( y V Since f is slightly continuous, f ( U in X containing and y respectively Therefore ( I ( Y such that f ( U f ( V are b open and f U f V = φ because U I V = φ This shows that X is T 2 f : X, τ Y, is a slightly continuous function and THEOREM 59 If ( ( g : ( X, τ E = X : f ( = g ( is clsed in X is slightly continuous function and Y is clopen Hausdorff, then { All Rights Reserved 80

10 PROOF If (, Since Y is clopen Hausdorff, there X E then it follows that f ( g( eist f ( V CO and g( W CO such that V W = φ continuous and g is slightly continuous, then f ( V is open and g ( W X with f ( V and g ( W Set O f = ( V g ( W open We notice that O and f ( O g( O φ that Cl ( E This proves that E is closed in X I Since f is slightly is open in I By Corollary 29 of [26], O is I = It follows that O I E = φ Thus it shows DEFINITION 50 Let f : ( X, τ {(, f ( : X} X Y is called the graph of f and is denoted by G ( f DEFINITION 5 A graph G ( f = {(, f ( : X} of a function f : ( X, τ said to be strongly co closed if for each (, y ( X Y \ G ( f, U CO( X, τ containing and V CO containing y such that ( U V I G ( f = φ LEMMA 52 A graph G ( f =, f ( : X of a function f : ( X, τ be a function Then the subset {( } is there eist is strongly co closed in X Y if and only if for each (, y ( X Y G ( f, there eist U CO( X, τ containing and V CO containing y such that f ( U I V = φ THEOREM 53 If f : ( X, τ is slightly continuous function and Y is clopen T, then G ( f is strongly co closed in X Y PROOF Let (, y ( X Y \ G ( f, then f ( y and there eists a clopen set V of Y such that f ( V and y V Since f is slightly b continuous, then f ( V CO( X, τ containing Take U f = ( V and ( (, X Y Y V CO Y containing We have f ( U V Therefore, we obtain f ( U I ( Y V = φ y This shows that G ( f is strongly co closed in COROLLARY 54 If f : ( X, τ clopen Hausdorff, then G ( f is strongly co closed in X Y THEOREM 55 Suppose that the function f : ( X, τ is slightly continuous function and Y is is an injection and has a strongly co closed graph G ( f Then X is T PROOF Let and y be any two distinct points of X Then, we have (, ( ( ( V CO such that, f ( y U Y and f ( U I V = φ Hence ( f y X Y G f By Lemma 52, there eists an clopen set U of X and ( y U This implies that X is b T THEOREM 56 Suppose that the function f : ( X, τ strongly co closed graph G ( f Then Y is T 2 U I f V = φ and is a surjection and has a PROOF Let y and y 2 be any distinct points of f = y for some X and (, y2 ( X Y \ G ( f By Definition 5, there eists an clopen set U of X Y Since f is surjective So ( and V CO such that (, y2 U V and ( U V G ( f = φ I Then, we All Rights Reserved 8

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