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b - Generalized Pre-Closed Sets in Topological Spaces
M. Andal1, V. Thiripurasundari2

1Assistant Professor in Department of Mathematics, P.S.R.R College of Engineering, Sivakasi, Affiliated to Anna
University, Chennai

2Associate Professor in Department of Mathematics, Sri S. Ramasamy Naidu Memorial College, Sattur, Affiliated to
Maduari Kamaraj University, Madurai

DOI: https://doi.org/10.51583/IJLTEMAS.2025.1410000114

Abstract: In this paper we introduce a new class of b-generalized closed sets, b- generalized open sets in topological spaces, and
study some of its basic properties.

Keywords: g - closed, gα - closed, αg – closed, gp – closed, gb – closed, bgp – closed, bgp – open.

I. Introduction

The study of generalized closed sets in topological spaces was initiated by Levine[3] in 1970. Andrijevic[1] introduced a new class
of generalized open sets in a topological space, the so called b-open sets. Maki et.al[2] defined αg – closed set and gα – closed set
in 1994. Balachandran and Rani[5] defined the notion of generalized pre-closed sets. A.A.Omari[4] and M.S.M.Noorani
introduced and studied the concept of generalized b- closed sets in topological spaces.

Preliminaries

Definition 2.1

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

i. ∅ and X are in ��.

ii. The union of the elements of any subcollection of �� is in ��.

iii. The intersection of the elements of any finite subcollection of �� is in ��. A set X for which a topology �� has been specified is
called a topological space.

Definition 2.2

Let (X, ��) be any topological spaces. A subset S of X is said to be a closed set in (X, ��) if its complement in X, namely X\S, is open
in (X, ��).

Definition 2.3[5]

A subset A of a topological space X is called a pre - open if A ⊆ int(cl(A)) and pre - closed if cl(int(A)) ⊆ A.

Definition 2.4[5]

The pre closure of a subset A of X, denoted by pcl(A) is the intersection of all pre closed sets containing A. The pre interior of a
subset A of X, denoted by pint(A) is the union of all pre open sets contained in A.

Definition 2.5[6]

A subset A of a topological space X is called a α - open if A ⊆ int(cl(int(A))) and α - closed if cl(int(cl(A))) ⊆ A.

Definition 2.6[1]

A subset A of a topological space X is called a b - open if A ⊆ cl(int(A)) ∪ int(cl(A)) and b - closed if cl(int(A)) ∩ int(cl(A)) ⊆
A.

Definition 2.7[7]

A subset A of a topological space X is called a regular - open if A = int(cl(A)) and regular - closed if cl(int(A)) = A.

Definition 2.8[3]

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

Definition 2.9[6]

A subset A of a topological space X is called a generalized α - closed (briefly, gα - closed) if αcl(A) ⊆ U whenever A ⊆ U and U
is α - open in X.

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Definition 2.10[6]

A subset A of a topological space X is called a α - generalized closed (briefly, αg - closed) if αcl(A) ⊆ U whenever A ⊆ U and U
is open in X.

Definition 2.11[2]

A subset A of a topological space X is called a generalized pre - closed (briefly, gp - closed) if pcl(A) ⊆ U whenever A ⊆ U and U
is open in X.

Definition 2.12[4]

A subset A of a topological space X is called a generalized b - closed (briefly, gb - closed) if bcl(A) ⊆ U whenever A ⊆ U and U
is open in X.

3. b - Generalized Pre Closed sets.

Definition 3.1

A subset A of a topological space X is called a b - generalized pre closed (briefly, bgp - closed) if pcl(A) ⊆ U whenever A ⊆ U and
U is b - open in X.

The family of all bgp - closed sets in a topological space X is denoted by bgpc(X).

Example 3.2

Let X = {a, b, c} and topology �� = {X, ∅, {b}, {b, c}}, then bgpc(X) = {X, ∅, {a}, {c}, {a, c}}.

Theorem 3.3

Every closed set is a bgp - closed set.

Proof: Let A be any closed set in X such that A ⊆ U where U is b - open set in X. Since A is closed set and pcl(A) ⊆ cl(A), pcl(A)
⊆ U and U is b - open in X. Hence A is a bgp - closed set.

Remark 3.4

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

Example 3.5

Consider X = {a, b, c} with the topology �� = {X, ∅, {b}, {b, c}}. Let A = {c}, then A is bgp - closed set but not a closed set.

Theorem 3.6

Every pre - closed set is a bgp - closed set.

Proof: Let A be a pre - closed set in X such that A ⊆ U where U is b - open in X. Since A is pre - closed and pcl(A) = A, pcl(A)
⊆ U and U is b - open in X. Hence A is a bgp - closed set.

Remark 3.7

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

Example 3.8

Consider X = {a, b, c} with the topology �� = {X, ∅, {a}, {a, b}}. Let A = {a, c}, then A is bgp - closed set but not a pre closed set.

Theorem 3.9

Every regular - closed set is a bgp - closed set.

Proof: Let A be a regular closed set in X such that A ⊆ U where U is b - open in X. Since A is regular closed set and pcl(A) ⊆
rcl(A), pcl(A) ⊆ U. Hence A is a bgp - closed set.

Remark 3.10

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

Example 3.11

Consider X = {a, b, c} with the topology �� = {X, ∅, {b}, {b, c}}. Let A = {a}, then A is bgp - closed set but not a regular closed
set.

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Theorem 3.12

Every α - closed set is a bgp - closed set.

Proof: Let A be a α - closed set in X such that A ⊆ U where U is b - open in X. Since A is a α - closed set and pcl(A) ⊆ αcl(A),
pcl(A) ⊆ U. Hence A is a bgp - closed set.

Remark 3.13

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

Example 3.14

Consider X = {a, b, c} with the topology �� = {X, ∅, {a}, {a, b}}. Let A = {a, c}, then A is bgp - closed set but not a α - closed set.

Theorem 3.15

Every gα - closed set is a bgp - closed set.

Proof: Let A be a gα - closed set in X such that A ⊆ U where U is b - open in X. Since A is a gα - closed set and pcl(A) ⊆ αcl(A),
pcl(A) ⊆ U. Hence A is a bgp - closed set.

Remark 3.16

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

Example 3.17

Consider X = {a, b, c} with the topology �� = {X, ∅, {a}, {a, b}}. Let A = {a, c}, then A is bgp - closed set but not a gα - closed set.

Theorem 3.18

Every gp - closed set is a bgp - closed set.

Proof: Let A be a gp - closed set in X such that A ⊆ U where U is b - open in X. Since A is a gp - closed set and pcl(A) ⊆ U. Hence
A is a bgp - closed set.

Remark 3.19

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

Example 3.20

Consider X = {a, b, c} with the topology �� = {X, ∅, {a, b}}. Let A = {a}, then A is bgp - closed set but not a gp - closed set.

Theorem 3.21

Every g - closed set is a bgp - closed set.

Proof: Let A be a g - closed set in X such that A ⊆ U where U is b - open in X. Since A is a g - closed set and pcl(A) ⊆ cl(A),
pcl(A) ⊆ U. Hence A is a bgp - closed set.

Remark 3.22

The reverse implication of the above theorem need not be true as seen from the following example.

Example 3.23

Consider X = {a, b, c} with the topology �� = {X, ∅, {b}, {b, c}}. Let A = {c}, then A is bgp - closed set but not a g - closed set.

Theorem 3.24

Every αg - closed set is a bgp - closed set.

Proof: Let A be a g - closed set in X such that A ⊆ U where U is b - open in X. Since A is a αg - closed set and pcl(A) ⊆ αcl(A),
pcl(A) ⊆ U. Hence A is a bgp - closed set.

Remark 3.25

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

Example 3.26

Consider X = {a, b, c} with the topology �� = {X, ∅, {a, b}}. Let A = {b}, then A is bgp - closed set but not a αg - closed set.

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Remark 3.27

The Reverse implication of the above theorem need not be true summarized in the following implicit diagram.

α Closed Set α Generalized Closed Set

Generalized α Closed Set bgp - closed set

Regular Closed set Closed Set Generalized Closed Set

Pre Closed set Generalized Pre Closed Set

Diagram 1

Theorem 3.28

Let B ⊆ X be a bgp - closed set, then pcl(B) - B contains no non - empty b - closed set.

Proof: Let B be a bgp - closed set and M be a b - closed set in X such that M ⊆ pcl(B) - B, then M ⊆ pcl(B) and M ⊆ X - B implies
B ⊆ X - M. Now, B is a bgp - closed set and X – M is a b - open set containing B. It follows that pcl(B) ⊆ X - M and thus M ⊆ X
- pcl(B). This implies M ⊆ pcl(B) ∩ (X - pcl(B)) = ∅. Hence M = ∅. Therefore pcl(B) - B contains no non - empty b - closed set.

Theorem 3.29

Let A ⊆ X be a bgp - closed set, then A is pre - closed if and only if pcl(A) - A is b – closed set.

Proof: Let A ⊆ X be a bgp - closed set. Let A be a pre - closed set in X, then we have pcl(A) - A = ∅ which is b - closed set.
Therefore pcl(A) - A is b - closed set. Conversely, Assume that pcl(A) - A is b - closed set. Now, A is bgp - closed set. Since pcl(A)
- A is a b - closed subset itself. By theorem 3.27, pcl(A) - A = ∅. This implies that pcl(A) = A and so A is pre - closed. Therefore
A is pre - closed.

Theorem 3.30

If A⊆ X is both b - open and bgp - closed, then A is pre - closed in X.

Proof: Let A be b - open and bgp - closed set in X, then pcl(A) ⊆ A. But A ⊆ pcl(A) is always true. Therefore pcl(A) = A. Hence
A is a pre - closed set.

Theorem 3.31

If A ⊆ X is a bgp - closed set and A ⊆ B ⊆ pcl(A), then B is bgp – closed in X.

Proof: Let U be a b - open set in X such that B ⊆ U, then A ⊆ U. Since A is a bgp - closed, then pcl(A) ⊆ U and A ⊆ B ⊆ pcl(A).
Now, pcl(B) ⊆ pcl(pcl(A)) = pcl(A) ⊆ U. So, pcl(B) ⊆ U.

Remark 3.32

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

Example 3.33

Consider X ={a, b, c} with the topology �� = {∅, X, {a}}. Let A = {a}, B = {a, b}, then A ⊆ B ⊆ pcl(A) and B is bgp - closed set
in X but A is not a bgp - closed set in X.

Theorem 3.34

Let Y be an open subspace of a space X, and A ⊆ Y. If A is bgp - closed set in X, then A is bgp - closed set in Y.

Proof: Let U be a b - open set of Y such that A ⊆ U. Then U = Y ∩ H for some b - open set H of X. Since A is bgp - closed set in
X, we have pcl(A) ⊆ U and pclY(A) = Y ∩ pcl(A) ⊆ Y ∩ H = U. Hence A is a bgp closed set in Y.

Remark 3.35

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

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Example 3.36

Consider X ={a, b, c} with the topology �� = {∅, X, {a, b}} and let Y = {a, b} with the topology �� = {∅, X, {a, b}}. Let A = {a, b},
then A ⊆ Y ⊆ X and A is bgp - closed set relative to Y but it is not bgp - closed relative to X.

Theorem 3.37

For a space X, the following statements are equivalent:

(i) Every bgp closed set is gb - closed set.

(ii) Every pre closed set is gb - closed.

Proof: (i)→(ii):

Given, every bgp - closed set is gp - closed. By above theorem, “every pre - closed set is bgp - closed set”. By our assumption,
every pre closed set is gb - closed set.

(ii)→(i):

Given, every pre - closed set is gb - closed set. Let A be a bgp - closed set in X such that A ⊆ U where U is b - open in X, then
pcl(A) ⊆ U. Since pcl(A) is pre - closed set, then by (ii), pcl(A) is gb - closed set. Therefore cl(A) ⊆ cl(pcl(A)) ⊆ U. That is, cl(A)
⊆ U. Therefore A is a gb - closed set. Hence every bgp closed set is gb - closed set.

4. b - Generalized Pre Open Sets

Definition 4.1

A subset A of a topological space X is called a b - generalized pre open (briefly, bgp - open) set if ACis bgp - closed.

Theorem 4.2

A set A ⊆ X is bgp - open if and only if U ⊆ pint(A) whenever U is b - closed and U ⊆ A.

Proof: Let A be a bgp - open set. Suppose that U ⊆ A where U is b - closed set. Then X - A is a bgp - closed set contained in the
b-open set X - U, pcl(X - A) ⊆ X - U. Since pcl(X - A) = X - pint(A), then X - pint(A) ⊆ X – U. That is U ⊆ pint(A).Conversely,
let U ⊆ pint(A) whenever U ⊆ A and U is b - closed set, then X - pint(A) ⊆ X – U. That is pcl(X - A) ⊆ X – U. This implies X-A
is bgp - closed set and A is bgp - open in X.

Theorem 4.3

If A is bgp - open set and B is any set in X such that pint(A) ⊆ B ⊆ A, then B is bgp - open set in X.

Proof: Let A is bgp - open and B is any set in X such that pint(A) ⊆ B ⊆ A. It follows from the definition 4.1 and theorem 3.22.
Hence B is bgp - open set in X.

Theorem 4.4

If a set A ⊆ X is bgp - closed set, then pcl(A)-A is bgp-open set in X.

Proof: Suppose that A is bgp-closed set and M is a b-closed set such that M ⊆ pcl(A) - A. By theorem 3.19, M =∅. Hence M ⊆
pint(pcl(A) - A). Therefore by theorem 4.2, pcl(A) - A is bgp - open set.

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