EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1809-1816
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

b∗J Sets and b∗J-Compact Ideal Spaces

Michael P. Baldado Jr.

Mathematics Department, Negros Oriental State University, Dumaguete City, Philippines

Abstract. We came up with the concept b∗-open set which has stricter condition with respect to
the notion b-open sets, introduced by Andrijevic [2] as a generalization of Levine’s [7] generalized
closed sets. The condition imposes equality instead of inclusion.
In this study, we gave some important properties of b∗-open sets with respect to an ideal, and
b∗-compact spaces.

2020 Mathematics Subject Classifications: 54D30

Key Words and Phrases: b∗-open sets, b∗J -open sets, ideals, b∗-compact space, b∗J -compact space

1. Introduction

Topology is a relatively new branch of mathematics, being introduced in the 19th
century. But topology is already seen in many areas of science [10]. It is applied in
biochemistry [3] and information systems [15]. Topology as a mathematical system is
fundamentally comprised of sets together with the operations union and intersection. Over
time, open sets (elements of topology) were generalized in different directions. To name a
few, Stone [16] presented regular open set. Levine [6] presented semi-open sets. Njasted
[12] presented α-open sets. Mashhour et al. [8] presented pre-open sets. Abd El-Monsef
et al. [1] presented β-open set.

It was in the year 1970, when Levine [7] presented the concept of generalized closed
sets, and achoring on this notion, Andrijevic [2] presented yet another generalization of
open sets called b-open sets. This study uses the notion of b-open sets to come up with a
new concept called b∗-open sets.

The concept ideal topological spaces (or simply, ideal space) was first seen in [5].
Vaidyanathaswamy [19] investigated this concept in point set topology. Tripathy and
Shravan [13, 14], Tripathy and Acharjee [17], Tripathy and Ray [18], Catalan et al. [4]
among others, also made investigations in ideal topological spaces.

Several concepts in topology were generalized using this structure. One of which is
the concept b∗-open sets. Consequently, using the notion of b∗-open sets, we introduced

DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4855

Email address: michael.baldadojr@norsu.edu.ph (M. Baldado Jr.)

https://www.ejpam.com 1809 © 2023 EJPAM All rights reserved.



M. Baldado Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1809-1816 1810

the concepts b∗-compact sets, compatible b∗J -compact sets, countably b∗J -compact sets,
b∗J -connected sets, in ideal generalized topological spaces.

Let W be a non-empty set. An ideal J on a set W is a non-empty collection of subsets
of W which satisfies:

1. B ∈ J and D ⊆ B implies D ∈ J .

2. B ∈ J and D ∈ J implies B ∪D ∈ J .

Let W be a topological space and B be a subset of W . We say that B is b∗-open set
if B = cl(int(B)) ∪ int(cl(B)). For example, consider W = {a, b, c} and the topology
ς = {∅, {a}, {b}, {a, b},W} on W . Then the b∗-open subsets are ∅, {a, b}, {c} and W .

Let W be a topological space and B be a subset of W . The set B is called b∗-open
relative to an ideal J (or b∗J -open), if there is an open set P with P ⊆ Int(B), and a closed
set S with Cl(B) ⊆ S such that

1. (Int(S) ∪ Cl(Int(B)))\B ∈ J , and

2. B\(Int(Cl(B)) ∪ Cl(P )) ∈ J .

In addition, we say that a set B is a b∗J -close set if BC is b∗J -open.
Consider the ideal space ({q, r, s}, {∅, {q}, {r}, {q, r}, {q, r, s}}, {∅, {r}}). Then B =

{r, s} is a b∗-open with respect to the ideal J = {∅, {r}}. To see this, we let P be the open
set {r} and S be the closed set {r, s}. Then Int(S) ∪ cl(int({r, s}))\{r, s} = int({r, s}) ∪
cl({r})\{r, s} = {r} ∪ {r, s}\{r, s} = {r, s}\{r, s} = ∅ ∈ J . Also, Int(cl({r, s}) ∪
cl(P )\{r, s} = int({r, s}) ∪ cl({r})\{r, s} = {r} ∪ {r, s}\{r, s} = {r, s}\{r, s} = ∅ ∈ J .
This shows that B = {r, s} is a b∗J -open.

Let W be a topological space and B be a subset of W . The set B is called nearly b∗-
open relative to an ideal J (or nearly b∗J -open) if there is an open set P with P ⊆ Int(B),
and a closed set S with Cl(B) ⊆ S such that

1. (Int(S) ∪ Cl(Int(B)))\Cl(B) ∈ J , and

2. B\(Int(Cl(B)) ∪ Cl(P )) ∈ J .

Consider the ideal topological space ({1, 2, 3}, {∅, {1}, {2}, {1, 2}, {1, 2, 3}},
{∅, {2}}). Then B = {2, 3} is a nearly b∗-open with respect to the ideal J (or nearly
b∗J -open). To see this, we let P be the open set {2} and S be the closed set {2, 3}. Then
Int(S)∪cl(int({2, 3}))\cl({2, 3}) = int({2, 3})∪cl({2})\cl({2, 3}) = {2}∪{2, 3}\cl({2, 3}) =
{2, 3}\{2, 3} = ∅ ∈ J . Also, Int(cl({2, 3}) ∪ cl(P )\{2, 3} = int({2, 3}) ∪ cl({2})\{2, 3} =
{2} ∪ {2, 3}\{2, 3} = {2, 3}\{2, 3} = ∅ ∈ J . This shows that B = {2, 3} is a nearly
b∗J -open.

The set B is said to be b∗-compact if every cover of B by b∗-open sets, containingW , has
a smaller finite sub-cover. The spaceW is said to be a b∗-compact space ifW is b∗-compact
set. Consider the topological space (W = {a, b, c}, {∅, {a}, {b, c},W}, J = {∅, {a}}).
Then B = {a} is a b∗-compact set, while D = {a, b} is not. To see this, we note that the



M. Baldado Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1809-1816 1811

b∗-open sets of W are ∅, {a}, {b, c} and W . Observe that the covering of B containing
W is {{a},W}. Thus, {{a}} is a smaller cover. Hence, B = {a} is a b∗-compact set.

On the other hand, observe that the covering of D containing W are {{a}, {b, c},W}
and {{b, c},W}. Since {{b, c},W} has no smaller subcover, D = {a, b} is not a b∗-compact
set.

The set B is called b∗J -compact if every cover of B by b∗J -open sets which contains W ,
has a smaller finite sub-cover. The space W is called b∗J -compact space if it is b∗J -compact
set. Consider the ideal topological space (W = {x, y, z}, {∅, {x}, {y}, {x, y},W}, {∅, {y}}).
Then B = {y, z} is a b∗J -compact set where J = {∅, {y}}. To see this, we note that the
b∗J -open sets of W are ∅, {y, z} and W . Hence, every cover {Pψ : ψ ∈ Ψ} of B by b∗J -open
set must contain {y, z} or W . Thus, each of the following is a covering of B: {{y, z}};
{{y, z},W}; and {W}. Note that {{y, z},W} is a covering of B which has a smaller
subcover {{y, z}}. This shows that B = {y, z} is a b∗J -compact set.

Now, consider the ideal topological space (W = {l,m, n}, {∅, {l}, {m}, {l,m},W},
{∅, {m}}). Then B = {l,m} is a not b∗J -compact set where J = {∅, {m}}. To see this,
we note again that the b∗J -open sets of W are ∅, {m,n} and W . Hence, every cover
{Pψ : ψ ∈ Ψ} of B by b∗J -open set must contain W . Thus, each of the following is a
covering of B: {{m,n},W}; and {W}. Note that {{m,n},W} has no smaller. This
shows that B = {l, y} is not a b∗J -compact set.

The set B is said to be compatible b∗J -compact (or simply cb∗J -compact) if any cover
{Pψ : ψ ∈ Ψ} of B by b∗J -open sets containing W , Ψ has a smaller finite subset Ψ0

such that B\
⋃
{Uψ : ψ ∈ Ψ0} ∈ J . The topological space W is said to be a cb∗J -

compact space if it is cb∗J -compact as a set. Consider the ideal topological space (Z, ς, J) =
({h, i, j}, {∅, {h}, {i, j}, Z}, {∅, {i}}). Then {h, i} is a compatible b∗J -compact where J =
{∅, {i}}. To see this, we observe that the b∗J -open sets of Z are ∅, {h}, {i, j} and Z.
Hence, every cover {Pψ : ψ ∈ Ψ} of Z by b∗J -open set must contain {h}, {i, j} or Z. Thus,
{Pψ : ψ ∈ Ψ} is {{h}, {i, j}} or {{h}, Z} or {Z, {i, j}, {h}} or {Z, {i, j}}. In the first 3
cases, there is a smaller subset {{h}} such that {h, i}\{h} = {i} ∈ J , and for the last
case, there exist a smaller subset {{h, i}} such that {h, i}\{h, i} = ∅ ∈ J . This shows
that {h, i} is a compatible b∗J -compact set. Next, consider the ideal topological space
(V = {q, r, s}, {∅, {q}, {r, s}, V }, {∅, {s}}). Then {q, r} is not compatible b∗J -compact.
To see this, we note that the b∗J -open sets of V are ∅, {q}, {r, s} and V . Hence, every
cover {Pψ : ψ ∈ Ψ} of {q, r} by b∗J -open set must contain {q}, {r, s} or V . Thus, {Pψ :
ψ ∈ Ψ} is {{q}, {r, s}} or {{q}, V } or {V, {r, s}, {q}} or {V, {r, s}}. Consider the open
cover {{q}, {r, s}}. Note that its smaller covers are {{q}} and {{r, s}}. Observe that
{q, r}\{q} = {r} /∈ J and {q, r}\{r, s} = {q} /∈ J . This shows that {q, r} is not a
compatible b∗J -compact set.

2. Results

We present some of the important properties of b∗-open sets and b∗J -open sets.
Lemma 1 is a characterization of b∗-open sets.



M. Baldado Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1809-1816 1812

Lemma 1. Let (Y, ς, J) be an ideal space and B be a subset of Y . Then B is an b∗-open
set precisely when there is an open set P with P ⊆ Int(B) and there is a close set S with
Cl(B) ⊆ S such that Int(S) ∪ Cl(Int(B)) ⊆ B ⊆ Int(Cl(B)) ∪ Cl(P ).

Proof. Necessity. Let B is a b∗-open set. Then B = Int(Cl(B)) ∪ Cl(Int(B)). Take
the open set P = Int(B) and the close set S = Cl(B). Note that Int(S) ∪ Cl(Int(B)) ⊆
Int(Cl(B)) ∪ Cl(Int(B)) = B, and Int(Cl(B)) ∪ Cl(P ) ⊇ Int(Cl(B)) ∪ Cl(Int(B)) = B.
Hence, Int(S) ∪ Cl(Int(B)) ⊆ B ⊆ Int(Cl(B)) ∪ Cl(P ).

Sufficiency. Next, let P be an open set with P ⊆ Int(B) and let S be a closed
set with Cl(B) ⊆ S such that Int(S) ∪ Cl(Int(B)) ⊆ B ⊆ Int(Cl(B)) ∪ Cl(P ). Then
B ⊇ Int(S) ∪ Cl(Int(B)) ⊇ Int(Cl(B)) ∪ Cl(Int(B)), and B ⊇ Int(Cl(B)) ∪ Cl(P ) ⊆
Int(Cl(B)) ∪ Cl(Int(B)).

Therefore, B = Int(Cl(B)) ∪ Cl(Int(B)), that is B is a b∗-open set.

An open set is nearly b∗J -open. The next lemma, Lemma 2, shows this idea.

Lemma 2. Let (Y, ς, J) be an ideal space. Then every open set is a b∗J -open set.

Proof. Let B be an open set, and consider S = ∅ = P . Then S and P are both open
and closed. Observed that int(cl(B))∪ cl(P ) ⊇ int(B)∪ cl(∅) = int(B)∪∅ = int(B) = B,
and int(S) ∪ cl(int(B)) = int(∅) ∪ cl(int(B)) ⊆ ∅ ∪ cl(B) = cl(B).

Hence, we have B\int(cl(B))∪ cl(P ) = ∅ ∈ J , and int(S)∪ cl(int(B))\cl(B) = ∅ ∈ J ,
that is, B is nearly b∗J -open.

An element of ideal J is nearly b∗J -open set. The next lemma, Lemma 3, shows this
idea. Please see [9] and [4] to have more insights.

Lemma 3. Let (Y, ς, J) be an ideal space. Then each element of J is b∗J -open.

Proof. Let B ∈ J . Since B − int(cl(B)) ∪ cl(B) ⊆ B, we have int(cl(B)) ∪ cl(B) ∈ J .
Next, consider S = ∅. Then int(S) ∪ cl(int(B))\cl(B) = int(∅) ∪ cl(int(B))\cl(B) =
∅ ∪ cl(int(B))\cl(B) = cl(int(B))\cl(B) = ∅ ∈ J . Therefore, B is nearly b∗J -open.

Lemma 4 says that each b∗-open set is b∗J -open.

Lemma 4. Let (Y, ς, J) be an ideal space. Then a b∗-open set is b∗J -open.

Proof. Let B be a b∗-open set. Then int(cl(B))∪cl(int(B)) = B. Consider P = int(B)
and S = cl(B). Then P is open with P ⊆ int(B), and S is closed with S ⊆ cl(B).
Observed that int(cl(B)) ∪ cl(P ) = int(B) ∪ cl(int(B)) = B, and int(S) ∪ cl(int(B)) =
int(cl(B)) ∪ cl(int(B)) = B.

Hence, we have B\int(cl(B)) ∪ cl(P ) = ∅ ∈ J , and int(S) ∪ cl(int(B))\B = ∅ ∈ J ,
that is, B is b∗J -open.



M. Baldado Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1809-1816 1813

Lemma 5. Let (Y, ς, J) be an ideal space with J = {∅}. Then B is b∗-open precisely if
B is b∗J -open.

Proof. Necessity. Let B be b∗J -open. Then there is an open set P such that P ⊆ int(B),
and there is a close set S such that S ⊆ cl(B). Hence, B ⊆ int(cl(B)) ∪ cl(P ), and
int(S) ∪ cl(int(B)) ⊆ B. Thus, int(cl(B)) ∪ cl(int(B)) = int(S) ∪ cl(int(B)) ⊆ B, and
int(cl(B)) ∪ cl(int(B)) = int(cl(B)) ∪ cl(P ) ⊇ B. Therefore, int(cl(B)) ∪ cl(int(B)) = B,
that is, B is b∗-open.

Sufficiency. The converse follows from Lemma 4.

If J is the minimal ideal, then the notions b∗-compact, b∗J -compact and cb∗J-compact
are the same. Theorem 1 shows this idea.

Theorem 1. Let (Y, ς, J) be an ideal space with J = {∅}. Then the following are equiv-
alent.

(i). (Y, ς, J) is a b∗-compact ideal space.

(ii). (Y, ς, J) is a b∗J -compact ideal space.

(iii). (Y, ς, J) is a cb∗J -compact ideal space.

Proof. (i) implies (ii): Let {Uψ : ψ ∈ Ψ} be a b∗J -open covering Y . By Lemma 5,
{Uψ : ψ ∈ Ψ} is also a b∗-open covering Y . Since Y is a b∗-compact ideal space, Ψ has
a smaller finite subset, say Ψ0, with {Uψ : ψ ∈ Ψ0} still covering Y . Thus, by Lemma 5,
{Uψ : ψ ∈ Ψ0} is a smaller finite b∗J -covering of Y . This shows that Y is a b∗J compact set.

(ii) implies (iii): Let {Uψ : ψ ∈ Ψ} be a b∗J -open covering Y . Since Y is a b∗J -compact
ideal space, Ψ has a smaller finite subset, say Ψ0, with {Uψ : ψ ∈ Ψ0} still covering Y .
Thus, Y −

⋃
ψ∈Ψ0

Uψ = ∅ ∈ J . Therefore, Y is cb∗J compact set.

(iii) implies (i): Let {Uψ : ψ ∈ Ψ} be a b∗-open covering Y . y Lemma 5, {Uψ : ψ ∈ Ψ}
is also a b∗J -open covering Y . Since Y is a cb∗J -compact ideal space, Ψ has a smaller finite
subset, say Ψ0, with Y −

⋃
ψ∈Ψ0

Uψ = ∅ ∈ J , that is, {Uψ : ψ ∈ Ψ0} is a smaller finite

b∗-covering of Y . Therefore, Y is b∗ compact set.

Another characterization of b∗J -compact topological spaces is presented in Theorem 2.

Theorem 2. Let (Y, ς, J) be an ideal space. Then statement (i) is a necessary and suffi-
cient condition for statement (ii).

i. (Y, ς, J) is a b∗J -compact space.

ii. If {Sψ : ψ ∈ Ψ} is a class of b∗J -closed sets with
⋂
{Sψ : ψ ∈ Ψ} = ∅, then Ψ has a

smaller finite subset, say Ψ0, with
⋂
{Sψ : ψ ∈ Ψ0} = ∅.



M. Baldado Jr. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1809-1816 1814

Proof. (i) implies (ii): Let {Sψ : ψ ∈ Ψ} be a class of b∗J -closed sets with
⋂
{Sψ : ψ ∈

Ψ} = ∅. Then Y = ∅C = (
⋂
{Sψ : ψ ∈ Ψ})C =

⋃
{SCψ : ψ ∈ Ψ}. Hence, {SCψ : ψ ∈ Ψ} is

a class of b∗J -open sets which covers of Y . By assumption, Ψ has a smaller finite subset,
say Ψ0, with the property

⋃
{SCψ : ψ ∈ Ψ0} = X. Hence, (

⋂
{Sψ : ψ ∈ Ψ0} =

⋃
{SCψ : ψ ∈

Ψ0})C = Y C = ∅.
(ii) implies (i): Let {Pψ : ψ ∈ Ψ} be a b∗J -open covering of Y , i.e.

⋃
{Pψ : ψ ∈ Ψ} = Y .

Then
⋂
{PCψ : ψ ∈ Ψ} = (

⋃
{Pψ : ψ ∈ Ψ})C = ∅. Note that PC is b∗J -close since P is

b∗J -open. By assumption, Ψ has a smaller finite subset, say Ψ0, with the property that⋂
{PCψ : ψ ∈ Ψ0} = ∅. Note that

⋃
{Pψ : ψ ∈ Ψ0} = (

⋂
{PCψ : ψ ∈ Ψ0})C = Y . Hence,

{Pψ : ψ ∈ Ψ0} is a class of b∗J -open sets that covers Y .

Another characterization of cb∗J -compact topological spaces is presented in Theorem 3.

Theorem 3. Let (Y, ς, J) be an ideal topological space. Then (i) is a necessary and
sufficient condition for statement (ii).

i. (Y, ς, J) is cb∗J -compact.

ii. If {Sψ : ψ ∈ Ψ} is a class of b∗J -closed sets with
⋂
{Sψ : ψ ∈ Ψ} = ∅, then Ψ has a

smaller finite subset, say Λ0, with the property that
⋂
{Fλ : λ ∈ Λ0} ∈ I.

Proof. (i) implies (ii): Let {Sψ : ψ ∈ Ψ} be a class of b∗J -closed sets such that⋂
{Sψ : ψ ∈ Ψ} = ∅. Note that

⋃
{SCψ : ψ ∈ Ψ} = (

⋂
{Sψ : ψ ∈ Ψ})C = Y . Hence,

{SCψ : ψ ∈ Ψ} is a class of b∗J -open sets covering Y . By assumption, Ψ has a finite subset,

say Ψ0, with Y −
⋃
{SCλ : ψ ∈ Ψ0} ∈ J , i.e.

⋂
{Sψ : ψ ∈ Ψ0} ∈ J .

(ii) implies (i): Let {Pψ : ψ ∈ Ψ} be a b∗J -open covering of Y , i.e.
⋃
{Pψ : ψ ∈ Ψ} = Y .

Note that
⋂
{PCψ : ψ ∈ Ψ} = (

⋃
{Pψ : ψ ∈ Ψ})C = ∅. By assumption, Ψ has a smaller

finite subset, say Ψ0, with
⋂
{PCψ : ψ ∈ Ψ0} ∈ J , i.e. Y −

⋃
{Pψ : ψ ∈ Ψ0} ∈ J .

Remark 1. [11] Let (Y, ς, J) and (W, ξ,K) be ideal topological spaces, and ζ : Y →W be
a mapping. Then:

i. ζ(J) = {ζ(B) : B ∈ J} is an ideal in W ; And,

i. if ζ is a one to one correspondence, then ζ−1(K) = {ζ−1(D) : D ∈ K} is an ideal
in Y .

Definition 1. Let (Y, ς, J) and (W, ξ,K) be ideal spaces. A mapping ζ : Y →W is

i. b∗J -open if ζ(B) is b∗K-open for every b∗J -open set B in Y , and

ii. b∗J -irresolute if ζ−1(D) is b∗J -open for each b∗K-open set D in W .

If the domain of a b∗-irresolute map is cb∗J -compact with respect to an ideal, then so
is the image. We show this idea in Theorem 4.



REFERENCES 1815

Theorem 4. Let (Y, ς, J) and (W, ξ,K) be ideal spaces, and ζ : Y →W be a b∗J -irresolute
function with ζ(J) = K. If Y is a cb∗J -compact, then ζ(Y ) is cb∗K-compact.

Proof. Let {Pψ : ψ ∈ Ψ} be a b∗K-open covering of ζ(Y ). Since ζ is b∗J -irresolute,
{ζ−1(Pψ) : ψ ∈ Ψ} is a b∗J -open covering Y . By assumption, Ψ has a smaller fi-
nite subset, say Ψ0, with Y −

⋃
{ζ−1(Pψ) : ψ ∈ Ψ0} ∈ J . And so by Remark 1

ζ(Y )\
⋃
{Pψ : ψ ∈ Ψ0} = ζ(Y −

⋃
{ζ−1(Pψ) : ψ ∈ Ψ0}) ∈ K.

If the co-domain of a b∗-open and onto map is cb∗J -compact with respect to an ideal,
then so is the domain. We show this idea in Theorem 5.

Theorem 5. Let (Y, ς, J) and (W, ξ,K) be ideal spaces, and ζ : Y →W be a b∗J -open and
onto map with ζ(J) = K. If W is cb∗K-compact, then Y is cb∗K-compact.

Proof. Let {Pψ : ψ ∈ Ψ} be a b∗J -open covering of Y . Since ζ is a b∗J -open and
onto, {ζ(Pψ) : ψ ∈ Ψ} is a b∗K-open covering of W . By assumption, Ψ has a smaller
finite subset, say Ψ0, with W\

⋃
{ζ(Pψ) : ψ ∈ Ψ0} ∈ K. Thus, Y \

⋃
{Pψ : ψ ∈ Ψ0} =

ζ−1(W\
⋃
{ζ(Pψ) : ψ ∈ Ψ0}) ∈ J .

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