Contents

ABSTRACT 2

1 INTRODUCTION 3
1.1 Introduction to Topology . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Introduction to Semi and Semi-pre closed sets . . . . . . . . 4
1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 β # GENERALIZED CLOSED AND OPEN SETS 10

2.1 β # generalized closed sets . . . . . . . . . . . . . . . . . . . . . 10
2.2 β # generalized open sets . . . . . . . . . . . . . . . . . . . . . . 19

3 β # g - CONTINUOUS FUNCTIONS AND CONTRA - β # g - CON-

TINUOUS FUNCTIONS 20
3.1 β # g - Continuous Functions . . . . . . . . . . . . . . . . . . . . 20
3.2 Contra - β # g - continuous functions . . . . . . . . . . . . . . . 29

4 β # g - CONNECTED AND β # g - COMPACT 33

4.1 β # g - Connected Spaces . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 β # g - Compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

APPLICATION OF TOPOLOGY 41

CONCLUSION 46

BIBLIOGRAPHY 47

PUBLICATION 50

1
ABSTRACT

In this dissertation a new class of generalized sets, namely β # g - closed sets and
β # g - open sets are introduced in Topological spaces and we also discuss some of
the properties of these sets. We also define a continuous mapping namely, β # g -
continuous, using the new generalized sets. From β # g - continuous function we
further extend it to contra - β # g - continuous function and analyze its properties.
Further we develop the concepts to define two new spaces, namely, β # g - connected
and β # g - compact.

2
Chapter 1
INTRODUCTION

1.1 Introduction to Topology

Topology is one of the major areas of abstract mathematics. The word topology
is obtained from the Greek words “t̀opos” and “l̀ogos” meaning study of spaces.

Topology, as a well-defined mathematical discipline, originated in the early

part of the twentieth century, but some isolated results can be traced back several
centuries. Among these are certain questions in geometry investigated by Leon-
hard Euler. His paper on the Seven Bridges of Königsberg is regarded as one of
the first practical applications of topology. He discovered the polyhedron formula
V E + F = 2 where V, E, and F respectively indicate the number of vertices, edges,
and faces of the polyhedron. Some authorities regard this analysis as the first the-
orem, signaling the birth of topology.

The English form “topology” was used in 1883 in Listing’s obituary in the
journal Nature to distinguish “qualitative geometry from the ordinary geometry in
which quantitative relations chiefly are treated”.

In 1895, Henri Poincaré published his ground-breaking paper on Analysis Si-

tus, which introduced the concepts now known as homotopy and homology, which
are now considered part of algebraic topology.

3
 Maurice Fréchet introduced the metric space in 1906. A metric space is now
considered a special case of a general topological space, with any given topological
space potentially giving rise to many distinct metric spaces. In 1914, Felix Haus-
dorff coined the term “topological space” and gave the definition for what is now
called a Hausdorff space. Currently, a topological space is a slight generalization of
Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.

Modern topology depends strongly on the ideas of set theory, developed by

Georg Cantor in the later part of the 19th century.

1.2 Introduction to Semi and Semi-pre closed sets

In mathematics, a topological space is a geometrical space in which closeness

is defined but, generally, cannot be measured by a numeric distance. More specifi-
cally, a topological space is a set of points, along with a set of neighbourhoods for
each point, satisfying a set of axioms relating points and neighbourhoods.

A topological space is the most general type of a mathematical space that

allows for the definition of limits, continuity, and connectedness. Other spaces, such
as Euclidean spaces, metric spaces and manifolds, are topological spaces with extra
structures, properties or constraints.

Although very general, topological spaces are a fundamental concept that is

used virtually in every branch of modern mathematics. The branch of mathemat-
ics that studies topological spaces in their own right is called point-set topology or
general topology.

In 1963, N.Levine [10],[11] introduced semi- open sets and semi-continuity in

a topological space. He also introduced the concepts of generalized closed sets in
topology, in 1970.

4
 In 1987, Bhattacharyya and Lahiri defined and studied about semi- general-
ized closed sets and U.D. Tapi, et.al [20] have made a study on more concepts of
semi- generalized closed sets.

In 2012, A. Robert and S. Pious Missier [14],[15] have introduced semi*-closed

sets and made a detailed study on it which led to an extension of semi*-connected
and semi*-compact spaces.

S. Saranya and K. Bageerathi [17], in 2016, defined a new generalized closed

sets namely Semi# generalized closed sets in Topological spaces and have also ex-
tended it to open sets.

P. Sundaram,et.al [19], in 1991 discussed about Semi-Generalized Continuous

Maps in Topological Spaces which led to further extensions like Semi Generalized
b- Continuous maps by D.Iyappan and N. Nagaveni [6], in 2012.

5
1.3 Preliminaries

Definition 1.3.1. [7]

A topology on a set X is a collection τ of subsets of X having the following properties:

1. φ and X belong to τ

2. The union of the elements of any subcollection of τ is in τ

3. The intersection of the elements of any finite subcollection of τ is in τ

A set X with a topology τ is called a topological space and it is denoted by (X,τ ).

Definition 1.3.2. [7]

Let (X,τ ) represents a topological space with topology τ . The members of τ are
called open sets of X.

Definition 1.3.3. [7]

A subset A of a topological space X is said to be closed if X - A is open.

Definition 1.3.4. [7]

Let A be a subset of a topological space (X,τ ). Then

• Interior of A is defined to be the union of all open sets contained in A, and

• Closure of A is defined to be the intersection of all closed sets containing A.

Here the interior of A is denoted by int(A) and the closure of A is denoted by cl(A).

Definition 1.3.5. [14]

Let (X,τ ) be a topological space. A subset A of X is said to be generalized closed
(briefly g-closed) if cl(A) ⊆ U whenever, A ⊆ U and U is open in (X,τ ).

Definition 1.3.6. [17]

Let (X,τ ) be a topological space and A ⊆ X. The generalized closure of A denoted
by cl*(A) and is defined by the intersection of all g-closed sets containing A and
generalized interior of A, denoted by int*(A) and is defined by the union of all g-open
sets contained in A.

6
Definition 1.3.7.
Let (X,τ ) be a topological space. A subset A of the space X is said to be

1. semi-open [11] if A ⊆ cl(int(A)) and semi-closed, if int(cl(A)) ⊆ A.

2. α -open [12] if A ⊆ int(cl(int(A))) and α -closed, if cl(int(cl(A))) ⊆ A.

3. pre-open [13] if A ⊆ int(cl(A)) and pre-closed, if cl(int(A)) ⊆ A.

4. semi*-open [14] if A ⊆ cl*(int(A)) and semi*-closed, if int*(cl(A)) ⊆ A.

5. semi-pre open [5] if A ⊆ cl(int(cl(A))) and semi-pre closed if int(cl(int(A))) ⊆

A.

Definition 1.3.8.
Let (X,τ ) be a topological space and A ⊆ X. The semi-closure of A, denoted by
scl(A) is defined to be the intersection of all semi-closed sets containing A

Definition 1.3.9.
Let (X,τ ) be a topological space. A subset A of the space X is said to be

1. semi-generalized closed [5] (briefly sg-closed) if scl(A) ⊆ U, whenever A ⊆ U

and U is semi-open in (X,τ ).

2. α-generalized closed [12] (briefly αg-closed) if αcl(A) ⊆ U, whenever A ⊆ U

and U is open in (X,τ ).

3. semi*generalized closed [14] (briefly semi*g-closed) if s*cl(A) ⊆ U, whenever

A ⊆ U and U is semi*-open in (X,τ ).

4. semi# generalized closed [17] (briefly s# g-closed) if scl(A) ⊆ U, whenever A ⊆

U and U is semi*-open in (X,τ ).

7
Definition 1.3.10.
A map f : X → Y is said to be

1. continuous [6] if for every closed set F of Y, the inverse image f −1 (F ) is closed
in X.

2. semi-continuous [19] if for every closed set F of Y, the inverse image f −1 (F )

is semi-closed in X.

3. α-continuous [6] if for every closed set F of Y, the inverse image f −1 (F ) is

α-closed in X.

4. pre continuous [21] if for every closed set F of Y, the inverse image f −1 (F ) is
pre closed in X.

5. semi-pre continuous [9] if for every closed set F of Y, the inverse image f −1 (F )
is semi-pre closed in X.

Definition 1.3.11.
A function f : X → Y is said to be

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

2. contra - semi - continuous [8] if f −1 (V ) is semi - closed in X for every open set
V in Y.

3. contra - α - continuous [4] if f −1 (V ) is α - closed in X for every open set V in

Y.

4. contra - pre - continuous [16] if f −1 (V ) is pre - closed in X for every open set
V in Y.

5. contra - semi-pre - continuous [1] if f −1 (V ) is semi-pre - closed in X for every

open set V in Y.

8
Definition 1.3.12. [18]
A topological space X is said to be connected, if X cannot be expressed as union of
any two disjoint non-empty open sets in X.

Definition 1.3.13. [18]

A topological space X is said to be separable, if it is not connected.

Definition 1.3.14. [15]

A collection B of open (resp. semi- open) sets in X is called an open (resp. semi-
open) cover of A ⊆ X if A ⊆ ∪{Uα : Uα ∈ B} holds.

Definition 1.3.15. [15]

A space X is said to be compact (resp. semi- compact) if every open (resp. semi-
open) cover of X has a finite subcover.

Definition 1.3.16. [15]

A space X is said to be Lindelof (resp. semi- Lindelof) if every cover of X by open
(resp. semi- open) sets contains a countable sub cover.

9
Chapter 2
β # GENERALIZED CLOSED AND OPEN SETS

In this chapter we introduce a set namely, β # g - closed set and we also introduce
and discuss the properties of β # g - open sets and β # g - continuous functions in a
Topological space.

2.1 β # generalized closed sets

In this section we introduce the set namely, β # g - closed in Topological spaces

and discuss some of its properties.

Definition 2.1.1.
A subset A of a topological space (X,τ ) is called semi-pre# generalized closed (briefly
β # g - closed) if βcl(A) ⊆ U whenever A ⊆ U and U is semi*- open in (X,τ ).

Theorem 2.1.2.
Every closed set is β # g - closed.

Proof.
Let A be a closed set in (X,τ ).
Let A ⊆ U, and U is semi*- open in X.
Since A is closed, cl(A) = A.
But A ⊆ U.
This implies, cl(A) = A ⊆ U.
But βcl(A) ⊆ cl(A).
Thus, we have, βcl(A) ⊆ U whenever, A ⊆ U and U is semi*- open in X.
Therefore, A is β # g - closed.

10
Remark 2.1.3.
The converse of the above theorem is not true, as seen from the following example.

Example 2.1.4.
Let X={a,b,c} with the topology τ ={φ, X, {a}}.
Let A={a}.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed, but A is not closed.

Theorem 2.1.5.
Every semi-closed set is β # g - closed.

Proof.
Let A be semi-closed in (X,τ ). Let A ⊆ U and U be semi*- open in X.
Since A is semi- closed, scl(A) = A.
But A ⊆ U.
This implies, scl(A) ⊆ U.
But, βcl(A) ⊆ scl(A) holds true always.
Thus we have, βcl(A) ⊆ U whenever, A ⊆ U and U is semi*- open in X.
Therefore, A is β # g - closed.

Remark 2.1.6.
The converse of the above theorem is not true, as seen from the following example.

11
Example 2.1.7.
Let X={a,b,c} with the topology τ ={φ, X, {a}}.
Let A={a}.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed, but A is not semi- closed, since int(cl(A)) ⊆ A does not
hold true.

Theorem 2.1.8.
Every semi*-closed set is β # g - closed.

Proof.
Let A be semi*-closed in (X,τ ).
Let A ⊆ U and U be semi*- open in X.
Here A is semi*-closed, and
since every semi*-closed sets are semi-closed, we have A is semi-closed.
Thus, scl(A) = A.
But, βcl(A) ⊆ scl(A) holds true always.
Thus we have, βcl(A) ⊆ U whenever A ⊆ U and U is semi*-open in X.
Therefore, A is β # g - closed.

Remark 2.1.9.
The converse of the above theorem is not true, as seen from the following example.

12
Example 2.1.10.
Let X={a,b,c} with the topology τ ={φ, X, {a}}.
Let A={a}.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed, but A is not semi*- closed, since int*(cl(A)) ⊆ A does not
hold true.

Theorem 2.1.11.
Every pre-closed set is β # g - closed.

Proof.
Let A be pre-closed in (X,τ ).
Let A ⊆ U and U be semi*-open in X.
Since A is pre-closed, pcl(A) = A.
But, βcl(A) ⊆ pcl(A) always holds true.
Thus, βcl(A) ⊆ U, whenever A ⊆ U and U is semi*-open in X.
Therefore, A is β # g - closed.

Remark 2.1.12.
The converse of the above theorem is not true, as seen from the following example.

Example 2.1.13.
Let X={a,b,c} with the topology τ ={φ, X, {a}}.
Let A={a}.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed, but A is not pre-closed, since cl(int(A)) ⊆ A does not hold
true.

13
Theorem 2.1.14.
Every α - closed set is β # g - closed.

Proof.
Let A be α - closed in (X,τ ).
Let A ⊆ U and U be semi*-open in X.
Since every α - closed set is pre-closed,
A is pre-closed.
Thus by above theorem,
A is β # g - closed.

Remark 2.1.15.
The converse of the above theorem is not true, as seen from the following example.

Example 2.1.16.
Let X={a,b,c} with the topology τ ={φ, X, {a}}.
Let A={a}.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed, but A is not α - closed, since cl(int(cl(A))) ⊆ A does not
hold true.

Theorem 2.1.17.
Every semi-pre closed set is β # g - closed.

Proof.
Let A be semi-pre closed in (X,τ ).
Let A ⊆ U and U be semi*-open in X.
Since A is semi-pre closed, βcl(A) = A.
But, A ⊆ U.
Thus, βcl(A) ⊆ U whenever A ⊆ U and U is semi*-open in X.
Therefore, A is β # g - closed.

14
Remark 2.1.18.
The converse of the above theorem is not true, as seen from the following example.

Example 2.1.19.
Let X={a,b,c} with the topology τ ={φ, X, {a}}.
Let A={a}.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed, but A is not semi- pre closed, since int(cl(int(A))) ⊆ A does
not hold true.

The above theorems are represented in the diagrammatical process.

Remark 2.1.20.
Union of any two β # g - closed sets need not be β # g - closed.

15
Example 2.1.21.
Let X={a,b,c} with the topology τ ={φ, X,{a},{b,c}}.
Let A={a} and B={b}.
First let us prove that A is β # g - closed.
The sets which are β-closed are {{a},{b},{c},{a,b},{b,c},{a,c}}.
Then, βcl(A) = ∩{{a},{a,b},{a,c}}.
=⇒ βcl(A) = {a}.
Let U = {a} ∈ τ .
The sets which are g - closed are {{a},{b},{c},{b,c}}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed.
Now let us prove, B is β # g - closed.
βcl(B) = ∩{{b},{a,b},{b,c}}.
=⇒ βcl(B) = {b}.
Let U = {b,c} ∈ τ .
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus B is β # g - closed.
Now, A∪B = {a}∪{b} = {a,b}.
Let C = {a,b}.
Then, βcl(C)= {a,b}.
Here there is no open set U in τ , other than X, which satisfy {a,b} ⊆ U.
Therefore, {a,b} is not β # g - closed.

Theorem 2.1.22.
If a subset of a topological space X is β # g - closed such that A ⊆ B ⊆ βcl(A), then
B is also β # g - closed.

Proof.
Let A be β # g - closed in X and let A ⊆ B ⊆ βcl(A).
Let U be semi*-open in X, and B ⊆ U.
Then A ⊆ U.
Since A is β # g - closed,
βcl(A) ⊆ U.

16
By hypothesis,
βcl(B) ⊆ βcl(βcl(A)) = βcl(A) ⊆ U.
Thus βcl(B) ⊆ U where U is semi*-open and B ⊆ U.
Therefore, B is β # g - closed.

Remark 2.1.23.
The converse of the above theorem need not be true as seen from the following
example.

Example 2.1.24.
Let X={a,b,c} with the topology τ ={φ, X,{a},{b,c}}.
Let A = {b} and B = {b,c}.
First let us prove that A is β # g - closed.
The sets which are β-closed are {{a},{b},{c},{a,b},{b,c},{a,c}}.
βcl(A) = ∩{{b},{a,b},{b,c}}.
=⇒ βcl(A) = {b}.
Let U = {b,c} ∈ τ .
The sets which are g - closed are {{a},{b},{c},{b,c}}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed.
Now let us prove, B is β # g - closed.
βcl(B) = {b,c}.
Let U = {b,c} ∈ τ .
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus B is β # g - closed.
Therefore, A and B are β # g - closed.
But A ⊆ B 6⊆ βcl(A), since {b} ⊆ {b,c} 6⊆ {b}.

17
Theorem 2.1.25.
If A is both open and g - closed in X, then it is β # g - closed in X.

Proof.
Let A be an open and g - closed set in X.
Let A ⊆ U and U be semi*-open in X.
Now, A ⊆ A.
By hypothesis,
Since A is g - closed and A is open, cl(A) ⊆ A.
But, βcl(A) ⊆ cl(A) holds true always.
Thus, βcl(A) ⊆ A.
But A ⊆ U.
Thus βcl(A) ⊆ U where A ⊆ U and U is semi*-open in X.
Therefore, A is β # g - closed.

Remark 2.1.26.
If A is both open and β # g - closed in X, then A need not be g - closed, which is
seen from the following example.

Example 2.1.27.
Let X = {a,b,c} and τ = {φ, X, {a}}.
Let A = {a}.
Then A is open.
The sets which are β-closed are {{b},{c},{b,c}}.
βcl(A) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open, since U ⊆ cl*(int(U)).
Thus A is β # g - closed.
But A is not g - closed since cl(A) ⊆ U does not hold true.

18
2.2 β # generalized open sets

In this section we introduce the set namely, β # g - open in Topological spaces and
discuss some of its properties.

Definition 2.2.1.
A subset A of a topological space (X,τ ) is called semi-pre# generalized open (briefly
β # g - open) if its complement X\A is β # g - closed.

Theorem 2.2.2.
Every open set is β # g - open.

Proof.
The proof is obvious from the theorem 2.1.2.

Theorem 2.2.3.
Every semi-open set is β # g - open.

Proof.
The proof of this theorem is similar to the proof of the theorem 2.1.5.

Theorem 2.2.4.
Every semi*-open set is β # g - open.

Proof.
The proof of this theorem is obvious from theorem 2.1.8. since, the complement of
β # g - closed is β # g - open.

Theorem 2.2.5.
Every pre-open set is β # g - open.

Proof.
The proof of this theorem is similar to the proof of the theorem 2.1.11.

Theorem 2.2.6.
Every semi-pre open set is β # g - open.

Proof.
The proof of this theorem is obvious from theorem 2.1.17. since β # g - closed sets
are the complement of β # g - open

19
Chapter 3
β # g - CONTINUOUS FUNCTIONS AND CONTRA - β # g -
CONTINUOUS FUNCTIONS

In this chapter we analyze the concept of β # g - continuous functions and contra

- β # g - continuous functions and discuss some of their properties.

3.1 β # g - Continuous Functions

In this section we discuss about β # g - continuous functions and analyze some of

its properties

Definition 3.1.1.
A map f : X → Y is said to be β # g - continuous if for every closed(resp. open) set
F of Y, the inverse image f −1 (F ) is β # g - closed(resp. open) in X.

Definition 3.1.2.
A map f : X → Y is said to be β # g - irresolute if and only if the inverse image of
every β # g - closed(resp. open) set in Y, is β # g - closed(resp. open) in X.

20
Theorem 3.1.3.
If a map f : X → Y is continuous, then it is β # g - continuous.

Proof.
Let f : X → Y be continuous and let V be closed in Y.
Since f is continuous,
f −1 (V ) is closed in X.
Since, every closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Therefore, f is β # g - continuous.

Remark 3.1.4.
The converse of the above theorem is not true, as seen from the following example.

Example 3.1.5.
Let X = {a, b, c} = Y.
Let f : X → Y be defined as,
f(a) = a, f(b) = b, f(c) = c.
Let τ = {φ, X, {a}} and σ = {φ, Y, {b, c}} be the topologies of X and Y respectively.
The sets which are closed in X are {φ, X, {b, c}}.
The sets which are closed in Y are {φ, Y, {a}}.
Let V = {a} which is closed in Y.
Then, f −1 (V ) = {a} is in X.
The sets which are β - closed in X are {{b}, {c}, {b, c}}.
βcl(f −1 (V )) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open in X, since U ⊆ cl*(int(U)).
Thus, f −1 (V ) = {a} is β # g - closed in X but it is not closed in X.
Then, f is β # g - continuous but not continuous.

21
Theorem 3.1.6.
If a map f : X → Y is semi- continuous, then it is β # g - continuous.

Proof.
Let f : X → Y be semi- continuous and let V be closed in Y.
Since f is semi- continuous,
f −1 (V ) is semi- closed in X.
Also, since every semi- closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Therefore, f is β # g - continuous.

Remark 3.1.7.
The converse of the above theorem is not true, as seen from the following example.

Example 3.1.8.
Let X = {a, b, c} = Y.
Let f : X → Y be defined as,
f(a) = a, f(b) = b, f(c) = c.
Let τ = {φ, X, {a}} and σ = {φ, Y, {b, c}} be the topologies of X and Y respectively.
The sets which are closed in X are {φ, X, {b, c}}.
The sets which are closed in Y are {φ, Y, {a}}.
Let V = {a} which is closed in Y.
Then, f −1 (V ) = {a} is in X.
The sets which are β - closed in X are {{b}, {c}, {b, c}}.
βcl(f −1 (V )) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open in X, since U ⊆ cl*(int(U)).
Thus, f −1 (V ) = {a} is β # g - closed in X.
Then, f is β # g - continuous.
But int(cl(f −1 (V ))) ⊆ f −1 (V ) does not hold in X.
Thus f −1 (V ) = {a} is not semi- closed in X.
Then, f is not semi- continuous.

22
Theorem 3.1.9.
If a map f : X → Y is α - continuous, then it is β # g - continuous.

Proof.
Let f : X → Y be α - continuous and let V be closed in Y.
Since f is α - continuous,
f −1 (V ) is α - closed in X.
Also, since every α - closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Therefore, f is β # g - continuous.

Remark 3.1.10.
The converse of the above theorem is not true, as seen from the following example.

Example 3.1.11.
Let X = {a, b, c} = Y.
Let f : X → Y be defined as,
f(a) = a, f(b) = b, f(c) = c.
Let τ = {φ, X, {a}} and σ = {φ, Y, {b, c}} be the topologies of X and Y respectively.
The sets which are closed in X are {φ, X, {b, c}}.
The sets which are closed in Y are {φ, Y, {a}}.
Let V = {a} which is closed in Y.
Then, f −1 (V ) = {a} is in X.
The sets which are β - closed in X are {{b}, {c}, {b, c}}.
βcl(f −1 (V )) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open in X, since U ⊆ cl*(int(U)).
Thus, f −1 (V ) = {a} is β # g - closed in X.
Then, f is β # g - continuous.
But cl(int(cl(f −1 (V )))) ⊆ f −1 (V ) does not hold in X.
Thus f −1 (V ) = {a} is not α - closed in X.
Then, f is α - continuous.

23
Theorem 3.1.12.
If a map f : X → Y is pre- continuous, then it is β # g - continuous.

Proof.
Let f : X → Y be pre- continuous and let V be closed in Y.
Since f is pre- continuous,
f −1 (V ) is pre- closed in X.
Also, since every pre- closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Therefore, f is β # g - continuous.

Remark 3.1.13.
The converse of the above theorem is not true, as seen from the following example.

Example 3.1.14.
Let X = {a, b, c} = Y.
Let f : X → Y be defined as,
f(a) = a, f(b) = b, f(c) = c.
Let τ = {φ, X, {a}} and σ = {φ, Y, {b, c}} be the topologies of X and Y respectively.
The sets which are closed in X are {φ, X, {b, c}}.
The sets which are closed in Y are {φ, Y, {a}}.
Let V = {a} which is closed in Y.
Then, f −1 (V ) = {a} is in X.
The sets which are β - closed in X are {{b}, {c}, {b, c}}.
βcl(f −1 (V )) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open in X, since U ⊆ cl*(int(U)).
Thus, f −1 (V ) = {a} is β # g - closed in X.
Then, f is β # g - continuous.
But cl(int(f −1 (V ))) ⊆ f −1 (V ) does not hold in X.
Thus f −1 (V ) = {a} is not pre- closed in X.
Then, f is not pre- continuous.

24
Theorem 3.1.15.
If a map f : X → Y is semi-pre continuous, then it is β # g - continuous.

Proof.
Let f : X → Y be semi-pre continuous and let V be closed in Y.
Since f is semi-pre continuous,
f −1 (V ) is semi-pre closed in X.
Also, since every semi-pre closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Therefore, f is β # g - continuous.

Remark 3.1.16.
The converse of the above theorem is not true, as seen from the following example.

Example 3.1.17.
Let X = {a, b, c} = Y.
Let f : X → Y be defined as,
f(a) = a, f(b) = b, f(c) = c.
Let τ = {φ, X, {a}} and σ = {φ, Y, {b, c}} be the topologies of X and Y respectively.
The sets which are closed in X are {φ, X, {b, c}}.
The sets which are closed in Y are {φ, Y, {a}}.
Let V = {a} which is closed in Y.
Then, f −1 (V ) = {a} is in X.
The sets which are β - closed in X are {{b}, {c}, {b, c}}.
βcl(f −1 (V )) = φ.
Let U = {a}, where φ ⊆ {a}.
Here U is semi*-open in X, since U ⊆ cl*(int(U)).
Thus, f −1 (V ) = {a} is β # g - closed in X.
Then, f is β # g - continuous.
But int(cl(int(f −1 (V )))) ⊆ f −1 (V ) does not hold in X.
Thus f −1 (V ) = {a} is not semi-pre closed in X.
Then, f is not semi-pre continuous.

25
Theorem 3.1.18.
If a map f : X → Y is β # g - irresolute, then it is β # g - continuous.

Proof.
Let f : X → Y be β # g - irresolute.
Let V be closed in Y.
Since every closed set is β # g - closed,
V is β # g - closed in Y.
Since f is β # g - irresolute,
f −1 (V ) is β # g - closed in X.
Thus, for a set V which is closed in Y, f −1 (V ) is β # g - closed in X.
Therefore, f is β # g - continuous.

Theorem 3.1.19.
If f : X → Y and g : Y → Z are both β # g - irresolute, then g◦ f : X → Z is β # g -
irresolute.

Proof.
Let f : X → Y and g : Y → Z be β # g - irresolute.
Let A ⊂ Z and let A be β # g - closed in Z.
Since g is β # g - irresolute,
g −1 (A) is β # g - closed in Y.
Since f : X → Y is β # g - irresolute,
f −1 (g −1 (A)) is β # g - closed in X.
Thus (g◦ f )−1 (A) = f −1 (g −1 (A)) is β # g - closed in X.
Therefore, g◦ f : X → Z is β # g - irresolute.

26
Theorem 3.1.20.
Let X,Y and Z be any topological spaces. For any β # g - irresolute map f : X → Y
and any β # g - continuous map g : Y → Z, the composition g◦ f : X → Z is β # g -
continuous.

Proof.
Let f : X → Y be β # g - irresolute and let g : Y → Z be β # g - continuous.
Let A ⊂ Z and let A be closed in Z.
Since g is β # g - continuous,
g −1 (A) is β # g - closed in Y.
Since f : X → Y is β # g - irresolute,
f −1 (g −1 (A)) is β # g - closed in X.
Thus (g◦ f )−1 (A) = f −1 (g −1 (A)) is β # g - closed in X.
Therefore, g◦ f : X → Z is β # g - continuous.

Theorem 3.1.21.
Let X,Y and Z be any topological spaces. For any β # g - continuous map f : X → Y
and any continuous map g : Y → Z, the composition g◦ f : X → Z is β # g -
continuous.

Proof.
Let f : X → Y be β # g - continuous and let g : Y → Z be continuous.
Let A ⊂ Z and let A be closed in Z.
Since g is continuous,
g −1 (A) is closed in Y.
Since f : X → Y is β # g - continuous,
f −1 (g −1 (A)) is β # g - closed in X.
Thus (g◦ f )−1 (A) = f −1 (g −1 (A)) is β # g - closed in X.
Therefore, g◦ f : X → Z is β # g - continuous.

27
Thus the above theorems can be represented in a tabular form as

f g g◦ f
β # g - continuous continuous β # g - continuous
β # g - irresolute β # g - continuous β # g - continuous
β # g - irresolute β # g - irresolute β # g - irresolute

28
3.2 Contra - β # g - continuous functions

In this section we define contra - β # g - continuous functions and analyze some of

its properties.

Definition 3.2.1.
A function f : X → Y is said to be contra - β # g - continuous if f −1 (V ) is β # g -
closed in X for each open set V of Y.

Definition 3.2.2.
A space (X,τ ) is said to be β # g - locally indiscrete if every β # g - open set in it is
closed.

Theorem 3.2.3.
Every contra - continuous function is contra - β # g - continuous.

Proof.
Let f : X → Y be a contra - continuous function.
Let V be open in Y.
Since f is contra - continuous,
f −1 (V ) is closed in X.
Since every closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Thus for a set V open in Y, f −1 (V ) is β # g - closed in X.
Therefore, f is contra - β # g - continuous.

Remark 3.2.4.
If a function is contra - β # g - continuous then it need not be a contra - continuous
function.

29
Theorem 3.2.5.
Every contra - semi - continuous function is contra - β # g - continuous.

Proof.
Let f : X → Y be a contra - semi - continuous function.
Let V be open in Y.
Since f is contra - semi - continuous,
f −1 (V ) is semi - closed in X.
Since every semi - closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Thus for a set V open in Y, f −1 (V ) is β # g - closed in X.
Therefore, f is contra - β # g - continuous.

Remark 3.2.6.
If a function is contra - β # g - continuous then it need not be a contra - semi -
continuous function.

Theorem 3.2.7.
Every contra - pre - continuous function is contra - β # g - continuous.

Proof.
Let f : X → Y be a contra - pre - continuous function.
Let V be open in Y.
Since f is contra - pre - continuous,
f −1 (V ) is pre-closed in X.
Since every pre-closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Thus for a set V open in Y, f −1 (V ) is β # g - closed in X.
Therefore, f is contra - β # g - continuous.

Remark 3.2.8.
If a function is contra - β # g - continuous then it need not be a contra - pre -
continuous function.

30
Theorem 3.2.9.
Every contra - α - continuous function is contra - β # g - continuous.

Proof.
Let f : X → Y be a contra - α - continuous function.
Let V be open in Y.
Since f is contra - α - continuous,
f −1 (V ) is α - closed in X.
Since every α - closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Thus for a set V open in Y, f −1 (V ) is β # g - closed in X.
Therefore, f is contra - β # g - continuous.

Remark 3.2.10.
If a function is contra - β # g - continuous then it need not be a contra - α - continuous
function.

Theorem 3.2.11.
Every contra - semi-pre - continuous function is contra - β # g - continuous.

Proof.
Let f : X → Y be a contra - semi-pre - continuous function.
Let V be open in Y.
Since f is contra - semi-pre - continuous,
f −1 (V ) is semi-pre - closed in X.
Since every semi-pre - closed set is β # g - closed,
f −1 (V ) is β # g - closed in X.
Thus for a set V open in Y, f −1 (V ) is β # g - closed in X.
Therefore, f is contra - β # g - continuous.

Remark 3.2.12.
If a function is contra - β # g - continuous then it need not be a contra - semi-pre -
continuous function.

31
Theorem 3.2.13.
If a function f : X → Y is β # g - continuous and X is β # g - locally indiscrete, then
f is contra - continuous.

Proof.
Let f : X → Y be β # g - continuous and let X be β # g - locally indiscrete.
Let V be open in Y.
Since f is β # g - continuous,
f −1 (V ) is β # g - open in X.
Since X is β # g - locally indiscrete,
f −1 (V ) is closed in X.
Thus if V is open in Y, then f −1 (V ) is closed in X.
Therefore, f is contra - continuous.

32
Chapter 4
β # g - CONNECTED AND β # g - COMPACT

In this chapter we introduce and discuss the concepts of β # g - connected and β # g

- compact spaces.

4.1 β # g - Connected Spaces

In this section we dicuss about the space which is β # g - connected.

Definition 4.1.1.
A topological space X is said to be β # g - connected if X cannot be expressed as
union of any two disjoint non-empty β # g - open sets in X.

Theorem 4.1.2.
If a space X is β # g - connected, then it is connected.

Proof.
Let X be β # g - connected.
Suppose X is not connected.
Then there exists two disjoint non-empty open sets A and B, such that A ∩ B = φ
and X = A ∪ B.
Since every open set is β # g - open,
A and B are β # g - open.
Then A and B forms a separation of X.
This is a contradiction to the fact that X is β # g - connected.
Therefore, X is connected.

33
Theorem 4.1.3.
For a topological space X, the following are equivalent

1. X is β # g - connected.

2. X and φ are the only subsets of X which are both β # g - open and β # g - closed.

Proof.
First, we prove,
(1) =⇒ (2)
Let X be β # g - connected.
Suppose S is a proper subset of X which is both β # g -open and β # g - closed.
Then its complement X - S is also both β # g - open and β # g - closed.
Thus X = S ∪ (X - S), is the union of two disjoint non-empty β # g - open sets.
This is a contradiction, since X is β # g - connected.
Therefore, S = φ or X are the only subsets of X which are both β # g - open and β # g
- closed.
Now, we prove,
(2) =⇒ (1)
Suppose X = A ∪ B, where A and B are disjoint non-empty β # g - open subsets of
X.
Let A be both β # g - open and β # g - closed.
By (2), A = φ or X.
If A = φ then B = X, or if A = X then B = φ.
Therefore, X is β # g - connected.

34
Theorem 4.1.4.
If f : X → Y is β # g - continuous onto and X is β # g - connected, then Y is connected.

Proof.
Let X be β # g - connected and let f : X → Y be β # g - continuous.
Suppose Y is not connected.
Then Y = A ∪ B,
where A and B are disjoint non-empty open sets in Y.
Since f is β # g - continuous and onto,
f −1 (Y ) = f −1 (A) ∪ f −1 (B)
That is, X = f −1 (A) ∪ f −1 (B),
where f −1 (A) and f −1 (B) are β # g - open in X.
This is a contradiction, since X is β # g - connected.
Therefore, Y is connected.

Theorem 4.1.5.
If f : X → Y is β # g - irresolute onto and X is β # g - connected, then Y is β # g -
connected.

Proof.
Let f : X → Y be β # g - irresolute and let X be β # g - connected.
Suppose Y is not β # g - connected.
Then Y = A ∪ B, where A and B are disjoint non-empty β # g - open sets in Y.
Since f is β # g - irresolute and onto,
f −1 (Y ) = f −1 (A) ∪ f −1 (B)
That is, X = f −1 (A) ∪ f −1 (B),
where f −1 (A) and f −1 (B) are disjoint non-empty β # g - open sets in Y.
This is a contradiction, since X is β # g - connected.
Therefore, Y is β # g - connected.

35
Theorem 4.1.6.
If the sets C and D are β # g - open and forms a separation of X and if Y is a β # g -
connected subspace of X, then Y lies entirely in C or in D.

Proof.
Let Y be a β # g - connected subspace of X.
Let C and D form a separation of X, where both C and D are β # g - open.
Then C ∩ Y and D ∩ Y are β # g - open in Y.
These two sets are disjoint.
Also, Y = (C ∩ Y)∪(D ∩ Y).
If both C ∩ Y and D ∩ Y are non-empty, then it forms a separation of Y.
This is a contradiction, since Y is β # g - connected.
Thus either the set C ∩ Y or D ∩ Y must be empty.
Therefore, Y lies entirely in C or in D.

36
4.2 β # g - Compact

In this section we dicuss about spaces which are β # g - compact.

Definition 4.2.1.
A collection A of β # g - open sets in X is called a β # g - open cover of B, which is a
subset of X, if B ⊆ ∪ {Uα : Uα ∈A} holds.

Definition 4.2.2.
A space X is said to be β # g - compact if every β # g - open cover of X has a finite
subcover.

Definition 4.2.3.
A subset B of X is said to be β # g - compact relative to X if for every β # g - open
cover A of B, there is a finite subcollection of A that covers B.

Definition 4.2.4.
A space X is said to be β # g - Lindelof if every cover of X by β # g - open sets contains
a countable subcover.

Remark 4.2.5.

• Every countable space is β # g - Lindelof.

• Every finite space is β # g - compact.

Theorem 4.2.6.
Let f : X → Y be a β # g - open function and let Y be β # g - compact. Then X is
compact.

Proof.
Let Y be β # g - compact.
Let {Vα } be an open cover of X.
Then {f (Vα )} is a cover of Y consisting of β # g - open sets.
Since Y is β # g - compact,
{f (Vα )} contains a finite subcover.
(i.e.,) {f (Vα1 ), f (Vα2 ), . . . , f (Vαn )} forms a finite subcover of f (Vα ).

37
=⇒ {Vα1 , Vα2 , . . . , Vαn } is a finite subcover of {Vα }.
Therefore, X is compact.

Theorem 4.2.7.
Let f : X → Y be a β # g - continuous surjection and let X be β # g - compact. Then
Y is compact.

Proof.
Let f : X → Y be a β # g - continuous surjection and let X be β # g - compact.
Let {Vα } be an open cover of Y.
Then {f −1 (Vα )} is a cover of X, consisting of β # g - open sets.
Since X is β # g - compact,
{f −1 (Vα )} contains a finite subcover.
(i.e.,) {f −1 (Vα1 ), f −1 (Vα2 ), . . . , f −1 (Vαn )} is a finite subcover of {f −1 (Vα )}.
=⇒ {Vα1 , Vα2 , . . . , Vαn } is a finite subcover of {Vα }.
Therefore, Y is compact.

Theorem 4.2.8.
If f : X → Y is β # g - irresolute and onto and if X is β # g - compact, then Y is β # g
- compact.

Proof.
Let f : X → Y be β # g - irresolute and onto and let X be β # g - compact.
Let {Vα } be a β # g - open cover of Y.
Then, {f −1 (Vα )} is a cover of X, consisting β # g - open sets.
Since X is β # g - compact,
{f −1 (Vα )} contains a finite subcover.
(i.e.,) {f −1 (Vα1 ), f −1 (Vα2 ), . . . , f −1 (Vαn )} is a finite subcover of {f −1 (Vα )}.
=⇒ {Vα1 , Vα2 , . . . , Vαn } is a finite subcover of {Vα }.
Therefore, Y is β # g - compact.

38
Theorem 4.2.9.
Let f : X → Y be β # g - continuous onto and let X be β # g - Lindelof. Then, Y is
Lindelof.

Proof.
Let f : X → Y be β # g - continuous onto and let X be β # g - Lindelof.
Let {Vα } be an open cover of Y.
Then {f −1 (Vα )} is a cover of X, consisting of β # g - open sets.
Since X is β # g - Lindelof,
{f −1 (Vα )} contains a countable subcover.
(i.e.,) {f −1 (Vαn )} is a countable subcover of {f −1 (Vα )}.
=⇒ {Vαn } is a countable subcover of {Vα }.
Therefore, Y is Lindelof.

Theorem 4.2.10.
Let f : X → Y be β # g - irresolute onto and let X be β # g - Lindelof. Then, Y is β #
- Lindelof.

Proof.
Let f : X → Y be β # g - irresolute onto map and let X be β # g - Lindelof.
Let {Vα } be an β # g - open cover of Y.
Then {f −1 (Vα )} is a cover of X, consisting of β # g - open sets.
Since X is β # g - Lindelof,
{f −1 (Vα )} contains a countable subcover.
(i.e.,) {f −1 (Vαn )} is a countable subcover of {f −1 (Vα )}.
=⇒ {Vαn } is a countable subcover of {Vα }.
Therefore, Y is β # g - Lindelof.

39
Theorem 4.2.11.
Let f : X → Y be a β # g - open mapping and let Y be a β # g - Lindelof space, then
X is Lindelof.

Proof.
Let Y be β # g - Lindelof.
Let {Vα } be an open cover of X.
Then {f (Vα )} is a cover of Y which consists of β # g - open sets.
Since Y is β # g - Lindelof,
{f (Vα )} contains a countable subcover.
(i.e.,) {f (Vαn )} is a countable subcover of f (Vα ).
=⇒ {Vαn } is a countable subcover of {Vα }.
Therefore, X is Lindelof.

40
Application of Topology

Introduction
Topology in general is deliberated to be one of the headstones of modern abstract
mathematics.
Initially, the results in topology were inspired by real-life problems, and after
its official emergence, the prominence curved to its abstract growth. Nonetheless,
from past few eras there has been a momentous improvement of the applied topology
to other fields also. Nowadays, mathematicians and scientists use topology to model
and comprehend the real world occurrences.
Topology basically has emerged out of geometry, where the concept of distances
and angles are excluded. Here objects are taken and treated as if they are made of
rubber which one can stretch, crumble, twist and even deform but without cutting
and pasting.
In topology, the objects we take are called topological spaces.
Topologists cannot distinguish between a circle and a square, sphere and a
cube, because they share topological properties throughout the deformation math-
ematically called Homeomorphism (topological isomorphism). The properties that
remain unchanged during homeomorphism are what we call topological properties
or topological invariants.
Here we describe some applications of topology in other fields of Science and
Technology. Here we discuss topological applications to Biology, and Geographic
Information System (GIS)

41
Applications
1. Biology
Topology has influenced the world of Science and Technology with much great effects.
Topology besides being a branch of mathematics, it plays a very good role in Biology
also. Here we will discuss of how topology has influenced the study of DNA.
Deoxyribonucleic acid (DNA) is a molecule composed of two polynucleotide
chains that coil around each other to form a double helix carrying genetic instructions
for the development, functioning, growth and reproduction of all known organisms
and many viruses.
The two DNA strands are known as polynucleotides as they are composed of
simpler monomeric units called nucleotides. Each nucleotide is composed of one of
four nitrogen-containing nucleobases
i. Cytosine [C]
ii. Guanine [G]
iii. Adenine [A]
iv. Thymine [T],
a sugar called deoxyribose, and a phosphate group.
The nitrogenous bases of the two separate polynucleotide strands are bound
together, according to base pairing rules (A with T and C with G), with hydrogen
bonds to make double-stranded DNA. The complementary nitrogenous bases are
divided into two groups, pyrimidines and purines. In DNA, the pyrimidines are
thymine and cytosine; the purines are adenine and guanine.
Since genotypes-phenotypes are of primary importance in biology, we see how
topology is even useful in sequencing the right nucleotides in DNA strand.
The four nucleotides are arranged in a manner that they resemble a sequence.
The sequence of nucleotides on every single chain of DNA decides the sequence of
the other chain. The problem found in DNA research is in the comparison of distinct
DNA sequences.
In topology, we define something called Metric, which is basically a distance
function used to measure the distance between the elements of a set. Here the sets

42
 Figure 4.1:

on which a metric can be defined are called metric spaces.

For example, real line R has a usual metric defined as d : R × R → R such
that (x,y) = |x − y|.
So using the concept of metric spaces (a special type of topological spaces), we
measure the distance between two sequences, where the distance function(metric)
gives the intuition towards the nature of evolutionary history of species.
Let us consider x and y to be the two sequences of letters A, C, G and T.
To calculate the distance between x and y, we fix the number of operations
on x to turn it into y. We can apply three operations that is insertion, deletion and
replacement for the sequence, say P, to turn x into y.
Let iP , dP and rP be the number of insertions, deletions and replacements
respectively.
So the entire number of operations to turn x into y is iP + dP + rP .
Since there are many choices of operations to turn x into y, we calculate the
distance between x and y by:

DL (x,y) = min{ iP + dP + rP },

where the minimum taken is from all sequences P that turn x into y.
Thus by measuring the distance between DNA strands using Topology, com-
parison of distinct DNA sequences are being done and hence Topology plays an

43
important role in DNA modelling by metric spaces.

2. Geographic Information System (GIS):

A Geographic Information System (GIS) is a framework for gathering, managing,
and analyzing data. Rooted in the science of geography, GIS integrates many types
of data. It analyzes spatial location and organizes layers of information into visual-
izations using maps and 3D scenes. With this unique capability, GIS reveals deeper
insights of data, such as patterns, relationships, and situationshelping users to make
smarter decisions.
The field of geographic information systems (GIS) started in the 1960s as com-
puters and early concepts of quantitative and computational geography emerged.
Roger Tomlinson’s pioneering work to initiate, plan, and develop the Canada Geo-
graphic Information System resulted in the first computerized GIS in the world in
1963. While at Northwestern University in 1964, Howard Fisher created one of the
first computer mapping software programs known as SYMAP. In 1965, he estab-
lished the Harvard Laboratory for Computer Graphics. In 1969, Jack Dangermonda
member of the Harvard Lab founded Environmental Systems Research Institute,
Inc. (Esri). The companys early work demonstrated the value of GIS for problem
solving. Esri went on to develop many of the GIS mapping and spatial analysis
methods now in use.
One of the example of a GIS software is Geo-tagging.
Geo-tagging is the process of adding geographic information about digital con-
tent, within “metadata” tags - including latitude and longitude coordinates, place
names and/or other positional data. Once geo-tagged, media such as photos, im-
ages, videos, websites, blog posts or RSS feeds can be easily displayed on an online
map or cross-referenced with other information about that area or location.
There are two ways to geo-tag a medium such as a digital photograph:

• First, by looking at a map and working out exactly where the medium is
located and then entering that information manually to the image.

44
 Figure 4.2:

• The second option is to use a satellite-based navigation system (i.e., a Global

Positioning System or GPS) to log the location of the photograph and then
adding that information either automatically or manually to it.

Topology in GIS is used in analysing the spatial relationships among different

regions in an area. It is used in analysing how points, lines and polygons share a
boundary.
Topology provides suitable models to distinguish between different regions, as
sometimes the image may show the intersection of two land areas but on zooming
it, they may have no intersection, mostly when the cameras are with less resolution.
In order to get rid of those vague results, topology provides us valuable mod-
elling tools. One such model was published by Egenhofer, that practices topological
concepts to differentiate the relationships among the pairs of geographic areas. In
that model, the author presents an idea of considering two closed sets say and in
a topological space , and uses the topological concepts to understand different ways
of checking the relationship among and in .
Thus by the result of the relationship we can conclude whether the regions A
and B are intersected or not.
Thus the Topological concepts are used medically and geographically.

45
Conclusion
Thus we have introduced a new closed set β # g - closed in a Topological space
and we have extended this concept to β # g - open sets. Using these sets we have
dicussed about β # g - continuous and contra - β # g - continuous functions. Also we
have defined two new spaces β # g -connected and β # g - compact.

46
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49
PUBLICATION

[1] J.Merlin Swetha and Dr.A.Arokia Lancy, “Introduction of β # g closed sets,

β # g open sets and β # g continuous functions in Topological spaces”,
The International journal of analytical and experimental modal analysis, Vol.XIII,
Issue III, March \2021.

50