On μ -Continuous Functions In Topological Spaces

On μ -Continuous Functions In Topological Spaces

Tuticorin 628 008

Department of Mathematics G.Venkataswamy Naidu College Kovilpatti.

In this paper, we introduce -continuous map and their relations with some generalized continuous maps. Various properties and characterizations of – continuous map are discussed by using -closure and -interior under certain conditions.

-continuous map, -closure, -interior.

2000 Mathematics Subject Code Classification : 54C05, 54C08.

1. Many others ([4] [5] [6] [13]) working in the field of general topology have shown more interest in studying the concepts of generalizations of continuous map. A weak form of continuous map called gcontinuous map was introduced by Balachandran, Sundaram and Maki [2].

   M.K.R.S.Veerakumar has introduced several generalized closed sets namely, g -closed sets, g*-closed sets, g*pclosed sets, *gclosed sets, *gclosed sets, *gs-closed sets, -closed sets, p-closed sets and s-closed sets and their continuity. The concept of,

   -closed sets was introduced by S.Pious Missier and E.Sucila [12]. In this paper we

   introduce the concept of -continuous map in topological spaces.

Throughout this paper, we consider spaces on which no separation axioms are assumed unless explicity stated . For A X, the closure and interior of A is denoted by cl(A) and int(A) respectively. The complement of A is denoted by AC.

A subset A of a topological space (X, ) is called

1. a preopen set [9] if A int(cl(A)) and preclosed if cl(int(A)) A.

2. a semiopen set [6] if A cl(int(A)) and a semiclosed set if int(cl(A)) A.

3. an -open set [11] if A int(cl(int(A)) and -closed set if cl(int(cl(A))) A.

4. a semi preopen set [1] if A cl(int(cl(A))) and a semipreclosed set if int (cl(int(A))) A.

The intersection of all semiclosed (resp. preclosed, semipreclosed, -closed) sets containing a subset A of X is called semiclosure (resp. preclosure, semipreclosure, -closure) of A is denoted by scl(A) (resp. pcl(A), spcl(A), cl(A)).

A subset A of a topological space (X, ) is called

1. a generalized closed set (briefly gclosed [7] if cl(A) U whenever A U and U is open in (X, ).

2. an -generalized closed set (briefly gclosed ) [8] if cl(A) U whenever A U and U is open in (X, ).

3. a g -closed set [15] if cl(A) U whenever A U and U is semiopen in (X, ).

4. a *gclosed set [16] if cl(A) U whenever A U and U is g -open in (X, ).

5. a g*-closed set [16] if cl(A) U whenever A U and U is g-open in (X, ).

6. a g*-preclosed set (briefly g*pclosed ) [17] if pcl(A) U whenever A U and U is gopen in (X, ).

7. a *g- semiclosed set [20] (briefly *gs-closed ) if scl(A) U whenever A U and U is g -open in (X, ).

8. a *g-closed set [22] if cl(A) U whenever A U, and U is g -open in (X, ).

9. a g*-closed set [8] if cl(A) int (U) whenever A U and U is -open in (X, ).

10. a -closed set [19] if scl(A) U whenever A U and U is sgopen in (X, ).

11. a g* -closed set [19] if cl(A) U whenever A U and U is gopen in (X, ).

12. a -closed set [21] if cl(A) U whenever A U and U is g*-open in (X, ).

13. a -preclosed set (briefly pclosed ) [22] if pcl(A) U whenever A U and U is g*- open in (X, ) .

14. a -semiclosed set (briefly sclosed) [23] if scl(A) U whenever A U and U is g*-open in (X, ).

15. a -closed set [12] if scl(A) U whenever A U and U is -open in (X, ). The complement of -closed set is called – open set. The class of all -open (resp.

-closed) subsets of X is denoted by o(X, ), (resp. c(X, )).

A map f : (X, ) (Y, ) is called

1. semicontinuous [6] if f-1(V) is semiclosed in (X, ) for every closed set V in (Y, ).

2. g-continuous [2] if f-1(V) is g-closed in (X, ) for every closed set V in (Y, ).

3. -continuous [10] if f -1(V) is -closed in (X,) for every closed set V in (Y, ).

4. gcontinuous [8] if f -1(V) is g-closed in (X, ) for every closed set V in (Y, ).

5. g -continuous [15] if f -1(V) is g -closed in (X, ) for every closed set V in (Y, ).

6. *g-continuous [15] if f -1(V) is *g-closed in (X, ) for every closed set V in (Y, ).

7. *g- continuous [15] if f -1(V) is *g-closed in (X, ) for every closed set V in (Y, ).

8. g*- continuous [16] if f -1(V) is g*-closed in (X, ) for every closed set V in (Y, ).

9. g*p- continuous [17] if f -1(V) is g*p-closed in (X, ) for every closed set V in (Y, ).

10. *gs- continuous [20] if f -1(V) is *gs-closed in (X, ) for every closed set V in (Y, ).

11. g*- continuous [19] if f -1(V) is g*-closed in (X, ) for every closed set V in

    (Y, ).

12. -continuous [21] if f -1(V) is -closed in (X, ) for every closed set V in (Y, ).

13. p- continuous [22] if f -1(V) is p-closed in (X, ) for every closed set V in (Y, ).

14. s- continuous [23] if f -1(V) is s-closed in (X, ) for every closed set V in

(Y, ).

A topological space (X, ) is called a

1. T -space [12] if every -closed set is closed.

2. T -space [12] if every -closed set is -closed.

3. sT -space [12] if every -closed set is semiclosed.

4. pT -space [12] if every -closed set is preclosed.

5. spT -space [12] if every -closed set is semipreclosed.

6. T -space [12] if every -closed set is -closed.

   1. -CONTINUITY

      We introduce the following definition.

      A function f : (X, ) (Y, ) is called -continuous if f-1(V) is -closed subset of (X, ) for every closed subset V of (Y, ).

      Every continuous (resp. semicontinuous) map is -continuous but not conversely.

      The proof follows from the fact that every closed (resp. semiclosed) set is -closed.

      Let X = Y = {a, b, c}, = {X, , {a}, {b, c}} and = {Y, , {b}}. Define a map f : X Y by f(a) = a, f(b) = b and f(c) = c is -continuous. However f is neither continuous nor semicontinuous, since for the closed set U = {a, c} in Y, f-1(U) = {a, c} which is neither closed nor semiclosed in X.

      Every – continuous map is -continuous but not conversely.

      Proof : The proof follows from the fact that every -closed set is -closed.

      Let X = Y = {a, b, c}, = {X, , {b}, {c, a}} and = {Y, , {c}, {b, c}}.Define a map f : X Y by f(a) = a, f(b) = b and f(c) = c. This map is -continuous but not

      -continuous, since for the closed set U = {a} in Y, f-1 (U) = {a} is not -closed in X.

      Thus the class of all -continuous maps properly contains the classes of continuous maps, semicontinuous maps and -continuous maps.

      The following examples shows that -continuity is independent of -continuity.

      Let X = Y= {a, b, c}, = {X, , {b}, {a, c}} and = {Y, , {b, c}}. Define f : X Y by f(a) = a, f(b) = c, f(c) = b is -continuous but not -continuous, since for the closed set U = {a} in Y, f-1(U) = {a} is not -closed in X.

      Let X = Y = {a, b, c}, = {X, , {b}, {a, b}} and = {Y, , {a}}. Let f : X Y be an identity map. Here the map f is -continuous but not -continuous, since for the closed set U ={b, c} in Y, f-1(U) = {b, c} is not -closed in X.

      The following examples shows that -continuous is independent of p continuous and s-continuous.

      Let X = Y = {a, b, c}, = {X, , {a}, {a, b}} and = {Y, , {c}}. Let f : X Y be an identity map. Then f is p continuous and s continuous but not -continuous, since for the closed set U = {a, b} in Y, f-1(U) = {a, b} is not -closed in X.

      Let X = Y = {a, b, c}, = {X, , {a}, {c}, {a, c}} and = {Y, , {b, c}}. Define a map f : X Y by f(a) = a, f(b) = c and f(c) = b. Here the map f is -continuous but not

      p-continuous, since for the closed set U = {a} in Y, f-1(U) = {a} is not p-closed in X.

      Let X = Y = {a, b, c}, = {X, , {a}, {b, c}} and = {Y, , {b}}. Let f : X Y be an identity map. Then the map f is -continuous but not s continuous, since for the closed set U = {a, c} in Y, f-1(U) = {a, c} is not s closed in X.

      The following examples shows that -continuous is independent of *g-continuous,

      *g-continuous and *gs-continuous.

      Let X = Y = {a, b, c}, = {X, , {a}, {b, c}} and = {Y, , {c}}. Define a map f : X Y by f(a) = a, f(b) = c and f(c) = b. Here the map f is -continuous but it is not

      *g-continuous, *gcontinuous and *gscontinuous. Since for the closed set U = {a, b} in Y, f-1(U) = {a,c} which is not *g-closed, *g-closed and *gs-closed in X.

      Let X = Y = {a, b, c}, = {X, , {a}, {a, b}} and = {Y, , {c}}. Let f : X Y be a map defined by f(a) = b, f(b) = c and f(c) = a. Here the map f is *g-continuous,

      *g-continuous and *gs-continuous but not -continuous, since for the closed set U = {a, b} in Y, f-1(U) = {a, c} which is not -closed in X.

      The following examples shows that -continuous is independent of g*-continuous, g-continuous, g continuous and g*p-continuous.

      Let X = Y = {a, b, c}, = {X, , {b}, {c}, {b, c}} and = {Y, , {a, b}}. Let f : X Y be an identity map. Here the map f is -continuous but not g*-continuous, g-continuous, g-continuous and g*p-continuous. Since for the closed set U = {c} in Y, f-1(U) = {c} which is not g*-closed, g-closed, g-closed and g*p-closed in X.

      Let X = Y = {a, b, c}, = {X, , {a}, {a, c}} and = {Y, , {c}}. Let f : X Y be an identity map. Then the map f is g*-continuous, g-continuous, g-continuous and g*p-continuous but not -continuous. Since for the closed set U = {a, b} in Y, f-1(U) = {a, b} is not -closed in X.

      The following examples shows that -continuous is independent of g* -continuous.

      Let X = Y = {a, b, c}, = {X, , {a}, {b, c}} and = {Y, , {b, c}}. Define a map f : X Y by f(a) = c, f(b) = a and f(c) = b. Here f is -continuous but not g* – continuous, since for the closed set U = {a} in Y, f-1(U) = {b} which is not g*-closed in X.

      Let X = Y = {a, b, c}, = {X, , {b}, {a, b}} and = {Y, , {a}}. Let f : X Y be an identity map. Then f is g* – continuous but not -continuous, since for the closed set U = {b, c} in Y, f-1(U) = {b, c} is not -closed in X.

      The following diagram shows the relationship established between -continuous function and some other continuous functions. A B (resp.A B) represents A implies B but not conversely (resp. A and B are independent of each other).

      From the above Propositions and Examples, we have the following diagram.

      s-continuous -continuous p-continuous | g*p-continuous g*-continuous

      continuous g-continuous

      -continuous -continuous g-continuous semicontinuous g*-continuous

      *g-continuous *g-continuous *gs-continuous

      The composition of two -continuous maps need not be -continuous.

      Let X = Y = Z = {a, b, c}, = {X, , {a}, {a, c}}, = {Y, , {b, c}} and

      = {Z, , {c}}. Define a map f : X Y by f(a) = b, f(b) = c and f(c) = a. Let g : Y Z be an identity map. Then both f and g are -continuous, but g f is not -continuous. Since for

      the closed set U = {a, b} in Z, (g f )-1 (U) = f-1 (g-1(U)) = f-1({a, b}) = {c, a} which is not

      -closed in X.

      If f : X Y is -continuous and g : Y Z is continuous then their composition

      g f : X Z is – continuous.

      Clearly follows from the definitions.

      A map f : (X, ) (Y, ) is -continuous if and only if f-1(U) is -open in (X, ), for every open set U in (Y, ).

      Let f : X Y be -continuous and U be an open set in Y. Then f-1(Uc) is -closed in X. But f-1(Uc) = (f-1(U))c and f-1(U) is -open in X. Converse is similar.

   2. – CLOSURE AND – INTERIOR

For every set E X we define the -closure of E to be the intersection of all

-closed sets containing E. In symbols, cl(E) = {A : E A, A C(X, )}.

For any E X, E cl(E) cl(E).

Since every closed set is -closed but not conversely.

If A B, then cl(A) cl(B)

Clearly follows from Definition 4.1.

If E is -closed, then cl(E) = E.

Clearly follows from Definition 4.1.

Let A be a subset of a topological space X. For any x X, x cl(A) if and only if U A for every -open set U containing x.

Necessity : Suppose that x cl(A). Let U be an -open set containing x such that

U A = and so A Uc. But Uc is -closed and hence cl(A) Uc. Since x Uc we obtain x cl (A) which is contrary to the hypothesis.

Sufficiency: Suppose that every -open set of X containing x meets A. If x cl(A), then there exists an -closed F of X such that A F and x F. Therefore, x Fc and Fc is an -open set containing x. But Fc A = . This is contrary to the hypothesis.

For any A X, int(A) is defined as the union of all -open sets contained in A. That is, int(A) = U{U : U A and U o(X, )}

For any set A X, int(A) int(A).

For any two subsets A1 and A2 of X,

1. If A1 A2, then int(A1) int(A2). (ii) int (A1 A2) int(A1) int(A2).

   If A is -open, then A = int(A).

   Clearly follows from the Definition 4.6.

   Let A be a subset of a space X, then the following are true.

   1. ( int(A))c = cl(Ac)

   2. int(A) = ( cl(Ac))c

   3. cl(A) = ( int(Ac))c

1. Let x ( int(A))c. Then x int(A). That is, every -open set U containing

   x such that U A. Thus every -open set U containing x is such that U Ac . By Proposition 4.5, x cl(Ac) and therefore, ( int(A))c cl(Ac).

   Conversely, let x cl(Ac). Then by Proposition 4.5, every -open set U containing x is such that U Ac . By definition 4.6, x int(A), hence x ( int(A))c and so

   cl(Ac) ( int(A))c. Thus cl(Ac) = ( int (A))c

2. Follows by taking complements in (i)

3. Follows by replacing A by Ac in (i)

For a subset A of a topological space X, the following conditions are equivalent :

1. o (X, ) is closed under any union,

2. A is -closed if and only if cl(A) = A.

3. A is -open if and only if int(A) = A.

1. (ii) : Let A be an -closed set. Then by definition of -closure, we get

   cl(A) = A. Conversely, assume that cl(A) = A. For each x Ac, x cl(A), by Proposition 4.5 there exist an -openset Gx such that Gx A = and hence x Gx Ac.

   Therefore, we obtain Ac =

   xAc

   Gx. By (i) Ac is -open and hence A is -closed.

2. (iii) = Follows by (ii) and proposition 4.9.

3. (i) = Let {U/ } be a family of -open sets of X. Put U = U U . For

each x U, there exist (x) such that x U(x) U. Since U(x) is -open, x int(U) and so U = int(U). By (iii), U is -open. Thus o (X, ) is closed under any union.

In a topologial space X, assume that o (X, ) is closed under any union. Then

cl(A) is an -closed set for every subset A of X.

set.

Since cl(A) = cl ( cl(A)) and by Proposition 4.10, we get cl(A) is an -closed

Let f : X Y be a map. Assume that o(X, ) is closed under any union. Then the following are equivalent :

1. The map f is -continuous;

2. The inverse of each open set is -open;

3. For each point x in X and each open set V in Y with f(x) V, there is an

   -open set U in X such that x U, f(U) V;

4. For each subset A of X, f( cl(A)) cl(f(A));

5. For each subset B of Y, cl(f-1(B) f-1(cl(B));

6. For each subset B of Y, f-1(int(B)) int(f-1(B)).

(i) (ii) By theorem 3.26

(i) (iii) : Suppose that (iii) holds and let V be an open set in Y and let x f-1(V).

Then f(x) V and thus there exist an -open set Ux such that x Ux and f(Ux) V. Now

x Ux f-1(V) and f-1(V) =

f is -continuous.

U Ux. By assumption f-1(V) is -open in X and therefore

x f 1 (V)

Conversely, suppose that (i) holds and let f(x) V. Then x f-1(V) which is -open in X, since f is -continuous. Let U = f-1(V). Then x U and f(U) V.

(iv) (i) Suppose that (i) holds and A be a subset of X. Since A f-1(f(A)). We have A f-1(cl(f(A))). Since cl(f(A)) is a closed set in Y, by assumption f-1(cl(f(A))) is an

-closed set containing A. Consequently, cl(A) f-1(cl(f(A))). Thus f( cl(A)) f(f-1cl(f(A))) cl(f(A)).

Conversely, suppose that (iv) holds for any subset A of X. Let F be a closed subset of Y. Then by assumption, f( cl(f-1(F))) cl(f(f-1(F))) cl(F) = F. Thus cl(f-1(F)) f-1 (F) and so f-1(F) is -closed.

1. (v) : Suppose that (iv) holds and B be any subset of Y. Then replacing A by f-1(B) in (iv) we get f( cl(f-1(B))) cl(f(f-1(B))) cl(B). Thus cl(f-1(B)) f-1(cl(B)). Conversely, suppose that (v) holds. Let B = f(A) where A is a subset of X. Then we have

   cl(A) cl(f-1(B)) f-1(cl(f(A))) and so f( cl(A)) cl(f(A)).

2. (vi) : Let B be any subset of Y. Then by (v) we have cl(f-1(Bc)) f-1(cl(Bc)) and hence ( int f-1(B))c (f-1 int (B))c. Therefore we obtain f-1(int(B)) int(f-1(B)).

3. (i) : Suppose that (vi) holds. Let F be any closed subset of Y. We have f-1(Fc) = f-1(int(Fc)) int(f-1(Fc)) = ( cl(f-1(F)))c and hence cl(f-1(F)) f-1(F). By proposition 4.10 f-1(F) is -closed in X. Hence f is -continuous.

Let f : (X, ) (Y, ) be a -continuous map. If (X, ) is a T space, then f is continuous.

The proof follows from definition.

Let f : (X, ) (Y, ) be a -continuous map. If (X, ) is a T space, then f is

-continuous.

The proof follows from definition.

Let f : (X, ) (Y, ) is -continuous map. If (X, ) is a sT space, then f is semicontinuous.

The proof follows from definition.

If f : (X, ) (Y, ) is -continuous map. If (X, ) is a pT space, then f is precontinuous.

The proof follows from definition.

If f : (X, ) (Y, ) is -continuous map. If (X, ) is a spT space, then f is semiprecontinuous.

The proof follows from definition.

If f : (X, ) (Y, ) is -continuous map. If (X, ) is a T space, then f is

-continuous.

The proof follows from definition.

1. D.Andrijevic, Semipreopen sets, Mat. Vensik, 38(1986), 24 32.

2. K. Balachandran, P.Sundaram and H. Maki, On generalized continuous maps in topological spaces, Mem. Fac.Sci. Kochi. Univ.Math., 12(1991), 5 13.

3. P.Bhattacharya and B.K.Lahiri, Semi generalized closed sets in topology, Indian J. Math.29 (1987), no.3, 375 382.

4. M.Caldas, S.Jafari and T.Noiri, Notions via g open sets, Kochi J of Maths.

5. J.Dontchev, On generalizing semi preopen sets, Mem, Fac. Sci. Kochi Univ. Ser. A.Math., 16(1995), 35 48.

6. N. Levine, Semiopen sets and semi continuity in topological spaces.,. Amer. Math.Monthly, 70(1963), 36-41.

7. N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19(1970), 89 96.

8. H. Maki, R.Devi and K. Balachandran Associated topologies of generalized

   – closed sets and – generalized closed sets, Mem. Fac., Sci., Kochi Uni. Ser.A.Math. 15(1994), 51 63.

9. A.S.Mashhour, M.E.Abd El Monsef and S.N. El Deeb, On pre continuous mapping and weak precontinuous mapping Proc.Math.Phys. Soc.Egypt,53(1982), 47 53.

10. A.S. Mashour, M.E. Abd El-Monsef and S.N. El. Deeb, – continuous and

    -open mappings Act a Math. Hung. 41(1983), No 3 4, 213 218.

11. O. Njastad, On some classes of nearly open sets, Pacific J.Math. 15(1965), 961 970.

12. S.Pious Missier and E. Sucila, On -closed sets in topological spaces. Accepted by International Journal of Mathematical Archieves.

13. P.Sundaram, Studies on generalizations of continuous maps in topological Spaces, Ph.D. Thesis, Bharathiyar University, Coimbatore (1991).

14. M.K.R.S.Veerakumar, Between semiclosed sets and semi pre closed sets. Rend. Istit. Mate. Univ. Trieste. XXXII (2000), 25 41.

15. M.K.R.S.Veerakumar, On g -closed sets in topological spaces, Bull. Allahabad Math. Soc. (18) (2003).

16. M.K.R.S.Veerakumar, Between closed sets and g*-closed sets, Mem. Fac.Sci. Kochi Univ. Ser A. Math 21 (2000), 1 19.

17. M.K.R.S. Veerakumar, g*-preclosed sets Acta ciencia Indica (Mathematics) Meerut, XXVIII (M) (1) (2002), 51 60.

18. M.K.R.S.Veerakumar., Presemiclosed, Indian J.Math, 44(2) (2002), 165 -181.

19. M.K.R.S. Veerakumar, Between -closed sets and gsp-closed sets, Antartica

    J. Math, Reprint.

20. M.K.R.S.Veerakumar, *g-semiclosed sets in topological spaces, Antartica

    J. Math.

21. M.K.R.S.Veerakumar, -closed sets in topological spaces, Antartica J.Math.

22. M.K.R.S.Veerakumar, p-closed sets in topological spaces, Antartica Journal of Math, 2(1) (2005), 31 52.

23. M.K.R.S.Veerakumar, s-closed sets in topological spaces, Antartica Journal of Math, 2(1) (2005), 91 109.