on bg – Continuous Maps and bg – Open Maps in Topological Spaces

on bg – Continuous Maps and bg – Open Maps in Topological Spaces

R. Subasree M. Maria Singam

Assistant Professor of Mathematics, Associate Professor of Mathematics Chandy College of Engineering, V.O. Chidambaram College Thoothukudi, Thoothukudi

TN, India TN, India

Abstract

Recently the author[19] defined bClosed sets and studied many basic properties. In this paper a new class of maps namely b Continuous map and b Open map were introduced in Topological Spaces and we find some of its basic properties. Further a new class of b homeomorphisms are also introduced and studied some of their relationship among other homeomorphisms.

1. Introduction

   In 1996, Andrijevic[14] introduced one such new version called b-open sets. Levine[5] introduced the concept of generalized closed sets and studied their properties. By considering the concept of gclosed sets many concepts of topology have been generalized and interesting results have been obtained by several mathematician. Veerakumar[12] introduced closed sets. Recently R.Subasree and M.MariaSingam[19] introduced bclosed sets.

   Balachandran et al[17] introduced the concept of generalized continuous maps in topological spaces. The purpose of this paper is to introduce a new version of maps called bcontinuous map and bopen map. Moreover we introduce the concept of b homeomorphism and we investigated the properties of all such transformations.

2. Preliminaries

   Throughout this paper (X, ) (or simply X) and (Y, ) (or simply Y) represents topological spaces on which no separation axioms are assumed unless otherwise mentioned. Let us recall the following definitions.

   Definition 2.1 : A subset A of a space (X,) is called a

   1. Semiopen set if A cl[Int (A)]

   2. open set if A Int[cl(Int (A))]

      1. bopen set if A cl[Int (A)] Int [cl(A)] The complement of a semiopen (resp. open,

         bopen) set is called semi-closed (resp. closed, b closed) set.

         The intersection of all semi-closed (resp. – closed, b-closed) sets of X containing A is

         called the semi-closure (resp. -closure, b- closure) and is denoted by scl(A) (resp. cl(A), bcl(A)). The family of all semi-open (resp. -open, b-open) subsets of a space X is denoted by SO(X), (resp. O(X), bO(X)).

         Definition 2.2: A subset A of a space (X,) is called a

         1. generalized closed (briefly g- closed) set[5] if cl(A)U whenever AU and U is open set in (X,).

         2. semi-generalized closed (briefly sg-closed) set[2] if scl(A)U whenever AU and U is a semi-open set in (X,).

         3. generalized semi-closed (briefly gs-closed) set[1] if scl(A) U whenever AU and U is open set in (X,).

         4. -generalized closed (briefly g-closed) set[7] if cl(A) U whenever AU and U is open set in (X,).

         5. generalized -closed (briefly g-closed) set[6] if cl(A) U whenever AU and U is – open set in (X,).

         6. -generalized closed (briefly g-closed) set[3] if cl(A)U whenever AU and U is open set in (X,).

         7. -closed set[12] if cl(A) U whenever AU and U is a semi-open set in (X,).

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

         9. gb-closed set[15] if bcl(A) U whenever AU and U is open set in (X,).

      The complement of a g-closed(resp. sg- closed, gs-closed, g-closed, g-closed, g- closed, closed and closed) set is called g-open (resp. sg-open, gs-open, g-open, g- open, g-open, open and open) set .

      Definition 2.3: A function f: (X,) (Y, ) is called

      1. Continuous[12] if is closed in (X,) for every closed set V in (Y, )

      2. gcontinuous [17] if is g closed in

         (X,) for every closed set V in (Y, )

      3. continuous[20] if is closed in (X,) for every closed set V in (Y, )

      4. g continuous[7] if is gclosed in (X,) for every closed set V in (Y, )

      5. continuous if is closed in (X,) for every closed set V in (Y, )

      6. b continuous [25] if is b closed in (X,) for every closed set V in (Y, )

      7. gbcontinuous[25] if is gbclosed in (X,) for every closed set V in (Y, )

      8. gscontinuous[22] if is gs closed in (X,) for every closed set V in (Y, )

      9. sgcontinuous[24] if is sgclosed in (X,) for every closed set V in (Y, )

      Definition 2.4: A function f: (X,) (Y, ) is called a

      1. open map[12] if is open in (Y, ) for every open set V in (X,)

      2. gopen map[23] if is gopen in (Y, ) for every open set V in (X,)

      3. open map[12] if is open in (Y, ) for every open set V in (X,)

      4. gsopen map[21] if is gsopen in (Y,) for every open set V in (X,)

      5. sg open map[21] if is sgopen in (Y,) for every open set V in (X,)

         Definition 2.5: A function f: (X,) (Y, ) is called a

         1. Homeomorphism[12] if f is both continuous map and open map.

         2. ghomeomorphism[23] if f is both g continuous map and gopen map.

         3. homeomorphism[12] if f is both continuous map and open map.

         4. sghomeomorphism[22] if f is both sg continuous map and sgopen map.

         5. gshomeomorphism[22] if f is both gs continuous map and gsopen map

3. b Continuous functions

We introduce the following definitions:

Definition 3.1: A function f: (X,) (Y, ) is said to be b continuous map if (V) is b-closed in (X,) for every closed set V of (Y, ).

Example 3.2:Let X=Y={a,b,c} = {X, , {a}} and

= {Y, , {a}, {b}, {a,b},{a,c}}

Define a function f: (X,) (Y, ) by

f(a) = a, f(b) = b, f(c) = c. Then f is b continuous, since the inverse images of a closed sets {b}, {b,c}, {a,c}, {c} in (Y, ) are

{b},{b,c}, {a,c}, {c} respectively which are b Closed in (X,).

Theorem 3.3:Every continuous map is b continuous.

Proof: Let V be a closed set in (Y, ). Since f is continuous, then (V) is closed in (X,). Since from[19] Remark 3.23 Every closed set is b Closed. Then (V) is b Closed in (X,). Hence f is b continuous.

Remark 3.4: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be a continuous map as shown in the following example.

Example 3.5:Let X=Y={a,b,c}

= {X, , {a}} and = {Y, , {b}}

Define a function f: (X,) (Y, ) by f(a) = a, f(b) = b, f(c) = c.

Then f is b continuous, but not continuous, since the inverse image of a closed set {a,c}in (Y, ) is {a,c} which is b closed but not closed in (X,).

Theorem 3.6:Every g-continuous map is b continuous.

Proof: Let V be a closed set in (Y, ). Since f is g-continuous, then (V) is g-closed in (X,). Since from[19] Proposition 3.6, Every g-closed set is b Closed. Then (V) is b Closed in (X,). Hence f is b- continuous.

Remark 3.7: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be a g- continous map as shown in the following example.

Example 3.8: Let X=Y={a,b,c}

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

Define a function f: (X,) (Y, ) by f(a) = a, f(b) = b, f(c) = c.

Then f is b continuous, but not g- continuous, since the inverse image of a closed set {b} in (Y, ) is {b} which is b closed but not g-closed in (X,).

Theorem 3.9:Every b-coninuous map is b continuous.

Proof: Let V be a closed set in (Y, ). Since f is b-continuous, then (V) is b-closed in (X,). Since from[19] Proposition 3.3, Every b-closed set is b Closed. Then (V) is b Closed in (X,).Hence f is b – continuous.

Remark 3.10: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be a b- continous map as shown in the following example.

Example 3.11:Let X=Y={a,b,c} = {X, , {a}} and

= {Y, , {a}, {b},{a,b},{a,c}}

Define a function f: (X,) (Y, ) by f(a) = a, f(b) = b, f(c) = c.

Then f is b-continuous, but not b- continuous, since the inverse image of a closed set {a,c} in (Y, ) is {a,c} which is b

Closed but not b-closed in (X,).

Theorem 3.12: Every gb-continuous map is b continuous.

Proof: Let V be a closed set in (Y, ). Since f is gb-continuous, then (V) is gb-closed in (X,). Since from[19] Proposition 3.18, Every gb-closed set is b Closed. Then

(V) is b Closed in (X,). Hence f is b

continuous.

Corollary 3.13:The converse of the above theorem is also true.

(i.e) Every b continuous is gb-continous. Proof: Let V be a closed set in (Y, ). Since f is b-continuous, then (V) is b -closed in (X,). Since from[19] Corollary 3.19, Every b -closed set is gb Closed. Then (V) is gb Closed in (X,). Hence f is gb- continuous.

Remark 3.14: The following example shows the relationship between b continuous map and gb-continous map.

b continuous gb-continous

Example 3.15:Let X=Y={a,b,c} = {X, , {a,c}}and

= {Y, , {a}, {b}, {a,b}, {a,c}}

Define a function f: (X,) (Y, ) by

f(a) = a, f(b) = b, f(c) = c.

b-closed set in X = {X, , {a}, {b}, {c},

{a,b}, {b,c}}

gb-closed set in X = {X, ,{a},{b},{c},{a,b},

{b,c}}

closed sets in Y = {Y, ,{b},{c},{a,c},{b,c}} Clearly f is both b continuous and gb-continuous.

Theorem 3.16: Every -continuous map is b Continuous.

Proof: Let V be a closed set in (Y, ). Since f is -continuous, then (V) is -closed in (X,). Since from[19] Proposition 3.9, Every -closed set is b Closed. Then

(V) is b Closed in (X,). Hence f is b

Continuous.

Remark 3.17: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be – continous map as shown in the following example.

Example 3.18:Let X=Y={a,b,c} = {X, , {a,c}}and

= {Y, , {a}, {b}, {a,b}, {a,c}}

Define a function f: (X,) (Y, ) by f(a) = b, f(b) = a, f(c) = a.

b-closed set in X = {X, ,{a},{b},{c},{a,b},

{b,c}}

-closed set in X = {X, , {b}, {a,b}, {b,c}} Then f is b continuous, but not –

continuous, since the inverse image of a closed set {b,c} in (Y, ) is {a} which is b closed but not -closed in (X,).

Theorem 3.19:Every gs-continuous map is b

continuous.

Proof: Let V be a closed set in (Y, ). Since f is gs-continuous, then (V) is gs-closed in (X,). Since from[19] Proposition 3.12 Every gs-closed set is b Closed. Then (V) is b closed in (X,). Hence f is b Continuous.

Remark 3.20: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be gs- continous map as shown in the following example.

Example 3.21:Let X=Y={a,b,c}

= {X, , {a,c}}and = {Y, , {a},{a,b}} Define a function f: (X,) (Y, ) by

f(a) = c, f(b) = a, f(c) = a.

b-closed set in X = {X, , {a}, {b}, {c},

{a,b}, {b,c}}

gs-closed set in X = {X, , {b}, {a,b}, {b,c}} Then f is b continuous, but not gs-

continuous, since the inverse image of a closed set {c} in (Y, ) is {a} which is b closed but not gs-closed in (X,).

Theorem 3.22:Every sg-continuous map is b

continuous.

Proof: Let V be a closed set in (Y, ). Since f is sg-continuous, then (V) is sg-closed in (X,). Since from [19] Remark 3.23 Every sg-closed is gs-closed and from [19] proposition (3.12) Every gs-closed set is b closed, we have Every sg-closed set is b closed . Hence (V) is b closed in (X,). Thus f is b Continuous.

Remark 3.23: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be sg- continous map as shown in the following example.

Example 3.24:Let X=Y={a,b,c}

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

Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = c.

b-closed set in X = {X, , {b}, {c}, {a,b},

{a,c}, {b,c}}

sg-closed set in X = {X, , {b}, {c}, {b,c}}

Then f is b continuous, but not sg-continuous, since the inverse image of a closed set {a,b} in (Y, ) is {a,b} which is b

• closed but not sg-closed in (X,).

  Theorem 3.25:Every g-continuous map is b

• continuous.

  Proof: Let V be a closed set in (Y, ). Since f is g-continuous, then (V) is g-closed in (X,). Since from[19] Proposition [3.15] Every g-closed is b closed, we have

  (V) is b closed in (X,).Thus f is b Continuous.

  Remark 3.26: The converse of the above theorem need not be true.

  (i.e) Every b continuous need not be g- continous map as shown in the following example.

  Example 3.27:Let X=Y={a,b,c} = {X, , {a}, {b}, {a,b}} and

  = {Y, , {b}}

  Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = b.

  b-closed set in X = {X, , {a}, {b}, {c},

  {a,c}, {b,c}}

  g-closed set in X = {X, , {c}, {a,c}, {b,c}}

  Then f is b continuous, but not g-continuous, since the inverse image of a closed set {a,c} in (Y, ) is {a} which is b closed but not g-closed in (X,).

  Theorem 3.28:Every -continuous map is b

• continuous.

Proof: Let V be a closed set in (Y, ). Since f is -continuous, then (V) is -closed in (X,). Since from[19] Proposition [3.20] Every -closed is b closed, we have

(V) is b closed in (X,). Thus f is b Continuous.

Remark 3.29: The converse of the above theorem need not be true.

(i.e) Every b continuous need not be – continous map as shown in the following example.

Example 3.30:Let X=Y={a,b,c} = {X, , {a},{b,c}}and

= {Y, , {a},{b},{a,b}}

Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = c.

b-closed set in X = {X, , {a}, {b}, {c},

{a,b},{a,c}, {b,c}}

-closed set in X = {X, , {a}, {b,c}}

Then f is b continuous, but not – continuous, since the inverse image of a closed set {c} in (Y, ) is {c} which is b closed but not -closed in (X,).

Remark 3.31:The following diagram shows the relationships of b continuous map with other known existing maps. A B represents A implies B but not conversely.

4 Applications

Remark 4.1: The composition of two b continuous functions need not be b continuous. For we consider the following example.

Example 4.2: Let X={a,b,c}

= {X, , {a},{b},{a,b}}, = {X, , {b}} and = {X, , {a}}

Define a function f: (X,) (X, ) by f(a) =a, f(b) = b, f(c) = c and

Define a function g: (X,) (X,) by g(a) =b, g(b) = c, g(c) = a.

Clearly f and g are b continuous. But for a closed set {b,c} in (X,)

{b,c} = =

= {a,b} which is not b closed in (X,). Hence g is not b-continuous.

Definition 4.3: A function f: (X,) (Y, ) is said to be b irresolute if (V) is b-closed in (X,) for every b-closed set V of (Y, ).

Remark 4.4: The composition of two b irresolute functions is again b irresolute.

5 b open maps and b closed maps

We introduce the following definitions:

Definition 5.1: Let X and Y be two topological spaces. A map f: (X,) (Y, ) is called b open map if the image of every open set in X is b-open in (Y, ).

Definition 5.2: Let X and Y be two topolgical spaces. A map f: (X,) (Y, ) is called b closed map if the image of every closed set in X is b-closed in (Y, ).

Theorem 5.3: Every open map is b-open map. Proof: Let f: (X,) (Y, ) is a open map and V be a open set in X, then f(V) is a open set in Y. Since[19] Proposition(3.3), Every open set is b- open set, we have f(V) is a b-open set in Y. Thus f is b-open map.

Remark 5.4: The converse of the above theorem need not be true.

(i.e) Every b-open map need not be a open map as shown in the following example.

Example 5.5: Let X=Y={a,b,c}

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

Define a function f: (X,) (Y, ) by f(a) =b, f(b) = b, f(c) = c.

Open sets in X = {X, , {a,c}}

b open set in Y = {Y, , {a}, {b}, {c},

{a,b},{b,c}}

Here f is b open map, but not a open map, since the image of a open set {a,c} in (X, ) is {b,c} which is b open but not open in (Y, ) .

Theorem 5.6: Every gopen map is bopen map.

Proof: Let f: (X,) (Y, ) is a gopen map and V be a open set in X, then f(V) is a gopen set in Y. Since from[19] Proposition(3.6), Every gopen set is b open set, we have f(V) is a bopen set in Y. Thus f is a bopen map.

Remark 5.7: The converse of the above theorem need not be true.

(i.e) Every bopen map need not be a gopen map as shown in the following example.

Example 5.8: Let X=Y={a,b,c}

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

Define a function f: (X,) (Y, ) by f(a) =b, f(b) = c, f(c) = b.

Open sets in X = {X, , {a}, {b},{a,b},{a,c}}

b open set in Y = {Y, , {a}, {c},

{a,b},{a,c},{b,c}}

gopen set in Y = {Y, , {a}, {c}, {a,c}} Here f is b open map, but not a g

open map, since the image of a open set {a,c} in (X, ) is {a,b} which is b open but not g open in (Y, ) .

Theorem 5.9: Every open map is bopen map.

Proof: Let f: (X,) (Y, ) is a open map and V be a open set in X, then f(V) is a open set in Y. Since from[19] Proposition(3.9), Every open set is b open set, we have f(V) is a bopen set in Y. Thus f is a bopen map.

Remark 5.10: The converse of the above theorem need not be true.

(i.e) Every bopen map need not be a open map as shown in the following example.

Example 5.11: Let X=Y={a,b,c}

= {X, , {a}, {b},{a,b}}and = {Y, ,{a}}

Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = c.

Open sets in X = {X, , {a}, {b},{a,b}}

b open sets in Y = {Y, , {a},{b}, {c},

{a,b},{a,c}}

open sets in Y = {Y, , {a}}

Here f is b open map, but not a open map, since the image of a open set {a,b} in (X, ) is {a,b} which is b open set but not open set in (Y, ) .

Theorem 5.12: Every sgopen map is a b open map.

Proof: Let f: (X,) (Y, ) is a sgopen map and V be a open set in X, then f(V) is a sg open set in Y. Since from[19] Remark(3.23), Every sgopen set is bopen set, we have f(V) is a bopen set in Y. Thus f is a bopen map.

Remark 5.13: The converse of the above theorem need not be true.

(i.e) Every bopen map need not be sgopen map as shown in the following example.

Example 5.14: Let X=Y={a,b,c}

= {X, ,{a,c}}and = {Y, ,{a}}

Define a function f: (X,) (Y, ) by f(a) =b, f(b) = a, f(c) = b.

Open sets in X = {X, ,{a,c}}

b open sets in Y = {Y, , {a},{b}, {c},

{a,b},{a,c}}

sgopen sets in Y = {Y, , {a},{a,b},{a,c}}

Here f is b open map, but not a sgopen map, since the image of a open set

{a,c} in (X, ) is {b} which is b open set but not sgopen set in (Y, ) .

Theorem 5.15: Every gsopen map is a b open map.

Proof: Let f: (X,) (Y, ) is a gsopen map and V be a open set in X, then f(V) is a gs open set in Y. Since from[19] Proposition(3.12),Every gsopen set is b open set, we have f(V) is a bopen set in Y. Thus f is a bopen map.

Remark 5.16: The converse of the above theorem need not be true.

(i.e) Every bopen map need not be a gs open map as shown in the following example.

Example 5.17: Let X=Y={a,b,c}

= {X, ,{a},{b,c}}and = {Y, ,{a,c}} Define a function f: (X,) (Y, ) by f(a) =a, f(b) =b, f(c) = c.

Open sets in X = {X, ,{a},{b,c}}

b open sets in Y = {Y, , {a},{c},

{a,b},{a,c},{b,c}}

gsopen sets in Y = {Y, , {a},{c},{a,c}}

Here f is b open map, but not a gsopen map, since the image of a open set

{b,c} in (X, ) is {b,c} which is b open set but not gsopen set in (Y, ) .

Remark 5.18: The following diagram shows the relationships of b open map with other known existing open maps. A B represents A implies B but not conversely.

1. b open map 2. Open map

3. gopen map 4. open map

5. sgopen map 6. gs open map

1. b Homeomorphisms

   Definition 6.1:A bijection f: (X,) (Y,) is called a b homeomorphism if f is both

   b continuous map and b open map.

   Example 6.2: Let X=Y={a,b,c}

   = {X, ,{a}}and = {Y, ,{b}}

   Define a function f: (X,) (Y, ) by f(a) =c, f(b) = a, f(c) = b.

   b closed sets in X = {X,,{b},{c},{a,b},

   {a,c},{b,c}}

   b open sets in Y =

   {Y,,{a},{b},{c},{a,b},{b,c}}

   sg closed sets in X = {X,,{b},{c},{b,c}}

   sg open sets in Y = {Y,,{b},{a,b},{b,c}} Here the inverse image of a closed set

   {a,c} in Y is {a,b} which is b closed in X and the image of a open set {a} in X is {c} which is b open in Y . Hence f is b homeomorphism.

   Theorem 6.3:Every homeomorphism is a b homeomorphism

   Proof: Follows from theorem 3.3 Every Continuous map is b continuous and from theorem 5.3 Every open map is b open map.

   Remark 6.4:The converse of the above theorem need not be true.

   (i.e) Every bhomeomorphism need not be a homeomorphism as shown in the following example.

   Example 6.5: Let X=Y={a,b,c}

   = {X, ,{a},{b,c}}and = {Y, ,{a,c}} Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = c.

   b closed sets in X

   ={X,,{a},{b},{c},{a,b}, {a,c},{b,c}}

   open sets in Y = {Y,,{a,c}} closed sets in X = {X,,{a},{b,c}} b open sets in Y =

   {Y,,{a},{c},{a,b},{a,c},{b,c}}

   Here the inverse image of a closed set

   {b} in Y is {b} which is b closed in X but not closed in X and the image of a open set

   {a} in X is {a} which is b open in Y but not open in Y.

   Hence f is b homeomorphism, but not a homeomorphism, since f is not a open- map and not a continuous map.

   Theorem 6.6:Every sghomeomorphism is a b homeomorphism

   Proof: Follows from theorem 3.22 Every sg continuous map is b continuous and by theorem 5.12 Every sgopen map is b open map.

   Remark 6.7:The converse of the above theorem need not be true.

   (i.e) Every bhomeomorphism need not be a sghomeomorphism as shown in the following example.

   Example 6.8: Let X=Y={a,b,c}

   = {X, ,{a}}and = {Y, ,{b}}

   Define a function f: (X,) (Y, ) by f(a) =c, f(b) = a, f(c) = b.

   b closed sets in X = {X,,{b},{c},{a,b},

   {a,c},{b,c}}

   b open sets in Y =

   {Y,,{a},{b},{c},{a,b},{b,c}}

   sg closed sets in X = {X,,{b},{c},{b,c}}

   sg open sets in Y = {Y,,{b},{a,b},{b,c}} Here the inverse image of a closed set

   {a,c} in Y is {a,b} which is b closed in X but not sgclosed in X and the image of a open set {a} in X is {c} which is b open in Y but not sgopen in Y.

   Hence f is b homeomorphism, but not sghomeomorphism, since f is not sg continuous and sgopen map.

   Theorem 6.9:Every gshomeomorphism is a b homeomorphism

   Proof: Follows from theorem 3.19 Every gs continuous map is b continuous and by theorem 5.15 Every gsopen map is b open map.

   Remark 6.10:The converse ofthe above theorem need not be true.

   (i.e) Every bhomeomorphism need not be a gshomeomorphism as shown in the following example.

   Example 6.11: Let X=Y={a,b,c}

   = {X, ,{a,c}}and = {Y, ,{a},{b,c}} Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = c.

   b closed sets in X =

   {X,,{a},{b},{c},{a,b},{b,c}}

   b open sets in Y =

   {Y,,{a},{b},{c},{a,b},{a,c},{b,c}}

   gs closed sets in X = {X,,{b},{a,b},{b,c}}

   gs open sets in Y =

   {Y,,{a},{b},{c},{a,b},{a,c},{b,c}}

   Here the inverse image of a closed set {a} in Y is {a} which is b closed in X but not gsclosed in X hence f is not gs continuous, however f is a gsopen map. Hence f is a b homeomorphism but not gs homeomorphism.

   Theorem 6.12:Every ghomeomorphism is a b homeomorphism

   Proof: Follows from theorem 3.6 Every g continuous map is b continuous and by theorem 5.6 Every gopen map is b open map.

   Remark 6.13:The converse of the above theorem need not be true.

   (i.e) Every bhomeomorphism need not be a ghomeomorphism as shown in the following example.

   Example 6.14: Let X=Y={a,b,c}

   = {X, ,{a},{a,b}}and = {Y, ,{a,c}} Define a function f: (X,) (Y, ) by f(a) =a, f(b) = b, f(c) = c.

   b closed sets in X=

   {X,,{b},{c},{a,c},{b,c}}

   b open sets in Y =

   {Y,,{a},{c},{a,b},{a,c},{b,c}}

   g closed sets in X = {X,,{c},{a,c},{b,c}}

   g open sets in Y = {Y,,{a},{c},{a,c}}

   Here the inverse image of a closed set {b} in Y is {b} which is b closed in X but not gclosed in X and for the image of a

   open set {a,b} in X is {a,b} which is b open in Y but not gopen in Y hence f is not g continuous and gopen map. Thus f is a b homeomorphism but not ghomeomorphism .

   Theorem 6.15: Every homeomorphism is a b homeomorphism

   Proof: Follows from theorem 3.16 Every continuous map is b continuous and by theorem 5.9 Every open map is b open map.

   Remark 6.16:The converse of the above theorem need not be true.

   (i.e) Every bhomeomorphism need not be a homeomorphism as shown in the following example.

   Example 6.17: Let X=Y={a,b,c} = {X, ,{a},{a,b}}and

   = {Y, ,{a},{b},{a,b}}

   Define a function f: (X,) (Y, ) by f(a) =b, f(b) = a, f(c) = c.

   b closed sets in X =

   {X,,{b},{c},{a,c},{b,c}}

   b open sets in Y =

   {Y,,{a},{b},{a,b},{a,c},{b,c}}

   closed sets in X = {X,,{c},{b,c}}

   open sets in Y = {Y,,{a},{b},{a,b}}

   Here the inverse image of a closed set {b,c} in Y is {a,c} which is b closed in X but not closed in X, hence f is not continuous, however f is a open in Y.

   Thus f is a b homeomorphism but not homeomorphism.

   Remark 6.18: The following diagram shows the relationships of b homeomorphism with other known existing homeomorphisms.

   A B represents A implies B but not conversely.

   1. b homeomorphism

   2. homeomorphism

   3. g homeomorphism

   4. homeomorphism

   5. sg homeomorphism

   6. gs homeomorphism

2. REFERENCES

   1. S.PArya and T Nour, Characterizations of S-normal spaces, Indian J.Pure.Appl.MAth.,21(8)(1990), 717-719.

   2. P Bhattacharya and B.K Lahiri, Semi-generalized closed sets in topology, Indian J.Math., 29(1987), 375-382.

   3. JDontchev and M Ganster, On – generalized closed sets and T3/4-spaces, Mem.Fac.Sci.KochiUniv.Ser.A, Math., 17(1996),15-31.

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

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

   6. H Maki, R Devi and K Balachandran, Generalized -closed sets in topology, Bull.FukuokaUni.Ed part III, 42(1993), 13-21.

   7. H Maki, R Devi and K Balachandran, Associated topologies of Generalized – closed sets and -generalized closed sets, Mem.Fac. Sci.Kochi Univ. Ser. A. Math., 15(1994), 57-63.

   8. A.S.Mashhour, M. E Abd El-Monsef and S.N. El-Debb, On precontinuous and weak precontinuous mappings, Proc.Math. andPhys.Soc. Egypt 55(1982), 47-53.

   9. M. E Abd El-Monsef, S.Rose Mary and M. LellisThivagar, On -closedsets in topological spaces, Assiut University Journal of Mathematics and Computer Science,Vol 36(1),P-P.43-51(2007).

   10. ONjastad, On some classes of nearly open sets, Pacific J Math., 15(1965), 961-970.

   11. M Stone, Application of the theory of Boolian rings to general topology, Trans. Amer. Math. Soc., 41(1937), 374-481.

   12. M.K.R.S. Veera Kumar, -closed sets in topological spaces, Bull. Allah Math.Soc, 18(2003), 99-112.

   13. N.V. Velicko, H-closed topological spaces, Amer. Math.Soc. Transl., 78(1968), 103-118.

   14. D. Andrijevic, On b-open sets, Mat. Vesnik 48(1996), no. 1-2, 59-64.

   15. Ahmad Al. Omari and Mohd.SalmiMD.Noorani, On Generalized b-closed sets, Bull. Malaysian Mathematical Sciences Society(2) 32(1) (2009), 19-30

   16. M. LellisThivagar, B. Meera Devi and E. Hatir, -closed sets in Topological

       Spaces, Gen. Math. Notes, Vol 1, No.2, December 2010, PP 17-25.

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

   18. M. Caldas and S. Jafari, On some applications of b-open sets in topological spaces, Kochi J.Math 2(2007),11-19

   19. R.Subasree, M. Maria singam, On b-closed sets in topological spaces, International journal of Mathematical Archive, 4(7), 2013, 168-173.

   20. Veera Kumar, -closed sets and GLC functions,

       Indian Journal Math.43(2)(2001), 231-247

   21. R.Devi, H.Maki and K. Balachandran, Semi generalized Closed maps and generalized semi-closed maps, MEM. Fac.sci.Kochi Univ. Sec A Math 14(1993) 41-54.

   22. R.Devi, H.Maki and K. Balachandran, Semi generalized homeomorphism and generalized semi- homeomorphism , Indian Journal of Pure and Applied Math. 26(1995) 271-284.

   23. H. Maki , P. Sundaram and K. Balachandran, On generalized homeomorphisms in Topological Spaces , bull Fukuoka Univ Ed. Part III 40(19991) 13-21.

   24. P. Sundaram, H.Maki and K. Balachnadran, Semi generalized continuous maos and Semi T1/2 spaces Bull. Fukuoka Univ Ed Part III 40(1991) 33-40.

   25. M. Caldas and E. Ekici, continuous functions Bol. Soc, Parana Mat (3)22(2004) No.2,63-74.