EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 3, No. 2, 2010, 295-302
ISSN 1307-5543 – www.ejpam.com

On supra b−open sets and supra b-continuity on topological
spaces

O. R. Sayed1∗ and Takashi Noiri2

1 Department of Mathematics, Faculty of Science, Assiut University,Assiut 71516, Egypt
& Department of Mathematics, University’s College, Umm-Al-Qurah University, Makkah, Saudi
Arabia
2 2949-1 Shiokita-cho, Hinagu, Yatsushiri-shi, Kumamoto-ken, 869-5142 Japan

Abstract. In this paper, we introduce and investigate a new class of sets and maps between topological
spaces called supra b−open sets and supra b−continuous maps, respectively. Furthermore, we intro-
duce the concepts of supra b-open maps and supra b-closed maps and investigate several properties of
them.
2000 Mathematics Subject Classifications: 54A10, 54A20

Key Words and Phrases: Supra b−open set, supra b−continuity, supra b-open map, supra b-closed
map and supra topological space

1. Introduction

In 1983, A. S. Mashhour et al. [6] introduced the supra topological spaces and studied
s-continuous maps and s?−continuous maps. In 1996, D. Andrijevic’ [2] introduced and
studied a class of generalized open sets in a topological space called b-open sets. This class
of sets contained in the class of β -open sets [1] and contains all semi-open sets [4] and all
pre-open sets [5]. In 2008, R. Devi et al. [3] introduced and studied a class of sets and
maps between topological spaces called supra α−open sets and supra α−continuous maps,
respectively. Now, we introduce the concept of supra b−open sets and study some basic
properties of it. Also, we introduce the concepts of supra b−continuous maps, supra b-open
maps and supra b-closed maps and investigate several properties for these classes of maps.
In particular, we study the relation between supra b-continuous maps and supra b-open maps
(supra b-closed maps).

Throughout this paper, (X ,τ) , (Y,σ) and (Z ,υ) (or simply, X , Y and Z) denote topolog-
ical spaces on which no separation axioms are assumed unless explicitly stated. For a subset

∗Corresponding author.

Email addresses: o_r_sayed@yahoo.com (O. R. Sayed), t.noiri@nifty.com (T. Noiri)

http://www.ejpam.com 295 c© 2010 EJPAM All rights reserved.



O. R. Sayed, T. Noiri / Eur. J. Pure Appl. Math, 3 (2010), 295-302 296

A of (X ,τ), the closure and the interior of A in X are denoted by Cl(A) and Int(A), respec-
tively. The complement of A is denoted by X − A. In the space (X ,τ), a subset A is said
to be β -open (resp. b-open, semi-open, pre-open, α-open [7]) if A ⊆ Cl(Int(Cl(A))) (resp.
A ⊆ Cl(Int(A)) ∪ Int(Cl(A)), A ⊆ Cl(Int(A)), A ⊆ Int(Cl(A)), A ⊆ Int(Cl(Int(A))). The
family of all β-open (resp. b-open, semi-open, preopen, α-open) sets of (X ,τ) is denoted by
β (X ) (resp. B (X ) , SO(X ), PO(X ),α (X )). A subcollection µ ⊂ 2X is called a supra topology
[6] on X if X ∈ µ and µ is closed under arbitrary union. (X ,µ) is called a supra topological
space. The elements of µ are said to be supra open in (X ,µ) and the complement of a supra
open set is called a supra closed set. The supra closure of a set A, denoted by Clµ(A), is the
intersection of supra closed sets including A. The supra interior of a set A,denoted by Intµ(A),
is the union of supra open sets included in A. The supra topology µ on X is associated with
the topology τ if τ ⊂ µ. A set A is called a supra α-open set [3] (resp. supra semi-open set
[6]) if A⊆ Intµ(Clµ(Intµ(A))) (resp. A⊆ Clµ(Intµ(A))).

Before we study the basic properties of supra b-open sets we have the following correction
in [3].

(1) Definition 6 should be written as we stated before.

(2) Example 3.2 is not correct and we will state instead of it.

(3) The proof of Theorem 3.3 (ii) is not correct as τ∗ is not supa-topology as well as Theo-
rem 3.4 (ii).

2. Supra b-open sets

In this section, we introduce a new class of generalized open sets called supra b-open sets
and study some of their properties.

Definition 1. Let (X ,µ) be a supra topological space. A set A is called a supra b -open set
if A ⊆ Clµ(Intµ(A))

⋃
Intµ(Clµ(A)). The complement of a supra b-open set is called a supra

b-closed set.

Theorem 1. Every supra semi-open set is supra b-open.

Proof. Let A be a supra semi-open set in (X ,µ). Then A ⊆ Clµ(Intµ(A)). Hence, A ⊆
Clµ(Intµ(A))

⋃
Intµ(Clµ(A)) and A is supra b-open in (X ,µ).

The converse of the above theorem need not be true as shown by the following example.

Example 1. Let (X ,µ) be a supra topological space, where X = {a, b, c} and µ= {X ,φ, {a}, {a, b}, {b, c}}.
Here {a, c} is a supra b-open set, but it is not supra semi-open.

In [3], the author proved that every supra α-open set is supra semi-open. The following
example (Instead of Example 3.2 [3]) shows the converse need not be true.

Example 2. Let (X ,µ) be a supra topological space, where X = {a, b, c, d} and µ= {X ,φ, {a}, {b}, {a, b}}.
Here {b, c} is a supra semi-open set, but it is not supra α-open.



O. R. Sayed, T. Noiri / Eur. J. Pure Appl. Math, 3 (2010), 295-302 297

From Theorems 3.1 and 3.2 in [3], the above theorem, Example 3.1 [3], and the above
two examples, we have the following diagram in which the converses of the implications need
not be true:

(DIAGRAM 1)
supra− open→ supra α− open→ supra semi− open → supra b− open

Theorem 2.

(i) Arbitrary union of supra b-open sets is always supra b-open.

(ii) Finite intersection of supra b-open sets may fail to be supra b-open.

(iii) X is a supra b-open set.

Proof.

(i) Let A and B be two supra b-open sets. Then, A ⊆ Clµ(Intµ(A))
⋃

Intµ(Clµ(A)) and
B ⊆ Clµ(Intµ(B))

⋃
Intµ(Clµ(B)). Then, A∪B ⊆ Clµ(Intµ(A∪B))

⋃
Intµ(Clµ(A∪B)).

Therefore, A∪ B is supra b-open set.

(ii) In Example 1, both {a, c} and {b, c} are supra b-open sets, but their intersection {c} is
not supra b-open.

Theorem 3.

(i) Arbitrary intersection of supra b-closed sets is always supra b-closed.

(ii) Finite union of supra b-closed sets may fail to be supra b-closed.

Proof.

(i) This follows immediately from Theorem 2.

(ii) In Example 1, both {a} and {b} are supra b-closed sets, but their union {a, b} is not
supra b-closed.

Definition 2. The supra b−closure of a set A, denoted by Clµb (A), is the intersection of supra
b−closed sets including A. The supra b−interior of a set A, denoted by Intµb (A), is the union of
supra b−open sets included in A.

Remark 1. It is clear that Intµb (A) is a supra b-open set and Clµb (A), is a supra b−closed set.

Theorem 4.

(i) A⊆ Clµb (A); and A= Clµb (A) iff A is a supra b-closed set;

(ii) Intµb (A)⊆ A; and Intµb (A) = A iff A is a supra b-open set;



O. R. Sayed, T. Noiri / Eur. J. Pure Appl. Math, 3 (2010), 295-302 298

(iii) X − Intµb (A) = Clµb (X − A);

(iv) X − Clµb (A) = Intµb (X − A).

Proof. Obvious.

Theorem 5.

(a) Intµb (A)∪ Intµb (B)⊆ Intµb (A∪ B);

(b) Clµb (A∩ B)⊆ Clµb (A)∩ Clµb (B).

Proof. obvious.

Proposition 1. The intersection of a supra α-open set and a supra b-open set is a supra b-open
set.

3. Supra b-continuous maps

In this section, we introduce a new type of continuous maps called a supra b-continuous
map and obtain some of their properties and characterizations.

Definition 3. Let (X ,τ) and (Y,σ) be two topological spaces and µ be an associated supra
topology with τ. A map f : (X ,τ) → (Y,σ) is called a supra b-continuous map if the inverse
image of each open set in Y is a supra b-open set in X .

Theorem 6. Every continuous map is supra b-continuous.

Proof. Let f : (X ,τ)→ (Y,σ) be a continuous map and A is open in Y . Then f −1(A) is an
open set in X . Since µ is associated with τ, then τ ⊆ µ. Therefore, f −1(A) is supra open in X
and it is supra b-open in X . Hence f is supra b-continuous.

The converse of the above theorem is not true as shown in the following example.

Example 3. Let X = {a, b, c} and τ =
�

X ,φ, {a, b}	 be a topology on X . The supra topology
µ is defined as follows: µ =

�
X ,φ, {a}, {a, b}	. Let f : (X ,τ) → (X ,τ) be a map defined as

follows: f (a) = a, f (b) = c, f (c) = b. The inverse image of the open set {a, b} is {a, c} which is
not an open set but it is a supra b-open.Then f is supra b-continuous but it is not continuous.

The following example shows that supra b-continuous maps need not be supra semi-
continuous.

Example 4. Consider the set X = {a, b, c, d} with the topology τ = {X ,φ, {a, c}, {b, d}} and
the supra topology µ =

�
X ,φ, {a, c}, {b, d}, {a, c, d}	. Also, suppose Y = {x , y, z} with the

topology σ = {Y,φ, {z}}. Define the map f : (X ,τ)→ (Y,σ) by: f (a) = y, f (b) = f (c) = z,
f (d) = x. The inverse image of the open set {z} is {b, c} which is a supra b-open set but it is not
a supra semi-open set.Then f is supra b-continuous but it is not supra semi-continuous map.



O. R. Sayed, T. Noiri / Eur. J. Pure Appl. Math, 3 (2010), 295-302 299

Therefore, from diagram 1 we have the following diagram in which the converses of the
implications need not be true by the above discussion.

(DIAGRAM 2) supra-continui t y → supra α-continui t y → supra semi-
continui t y → supra b-continui t y

Theorem 7. Let (X ,τ) and (Y,σ) be two topological spaces and µ be an associated supra topol-
ogy with τ. Let f be a map from X into Y . Then the following are equivalent:

(1) f is a supra b-continuous map;

(2) The inverse image of a closed set in Y is a supra b-closed set in X ;

(3) Clµb ( f
−1(A))⊆ f −1(Cl(A)) for every set A in Y ;

(4) f (Clµb (A))⊆ Cl( f (A)) for every set A in X ;

(5) f −1(Int(B))⊆ Intµb ( f
−1(B)) for every B in Y .

Proof.

• (1)⇒(2): Let A be a closed set in Y , then Y −A is an open set in Y . Then f −1(Y −A) =
X − f −1(A) is a supra b-open set in X . It follows that f −1(A) is a supra b-closed subset
of X .

• (2)⇒(3): Let A be any subset of Y . Since Cl(A) is closed in Y, then f −1(Cl(A)) is supra
b-closed in X . Therefore, Clµb ( f

−1(A))⊆ Clµb ( f
−1(Cl(A))) = f −1(Cl(A)).

• (3)⇒(4): Let A be any subset of X . By (3) we have f −1(Cl( f (A)))⊇ Clµb ( f
−1( f (A)))⊇

Clµb (A).Therefore, f (Clµb (A))⊆ Cl( f (A)).

• (4)⇒ (5):Let B be any subset of Y . By (4), f (Clµb (X − f −1(B))) ⊂ Cl( f (X − f −1(B)))
and f (X−Intµb ( f

−1(B)))⊂ Cl(Y−B) = Y−Int(B). Therefore,we have X−Intµb ( f
−1(B))⊂

f −1(Y − Int(B)) and f −1(Int(B))⊂ Intµb ( f
−1(B)).

• (5)⇒(1): Let B be an open set in Y and f −1(Int(B)) ⊆ Intµb ( f
−1(B)). Then, f −1(B) ⊆

Intµb ( f
−1(B)). But, Intµb ( f

−1(B))⊆ f −1(B). Hence, f −1(B) = Intµb ( f
−1(B)). Therefore,

f −1(B) is supra b-open in X .

Theorem 8. Let (X ,τ), (Y,σ) and (Z ,υ) be three topological spaces. If a map f : (X ,τ) →
(Y,σ) is supra b-continuous and g : (Y,σ)→ (Z ,υ) is a continuous map, then g ◦ f : (X ,τ)→
(Z ,υ) is supra b-continuous.

Proof. Obvious.

Theorem 9. Let (X ,τ) and (Y,σ) be two topological spaces and µ and ν be the associated supra
topologies with τ and σ, respectively. Then f : (X ,τ)→ (Y,σ) is a supra b-continuous map, if
one of the following holds:



O. R. Sayed, T. Noiri / Eur. J. Pure Appl. Math, 3 (2010), 295-302 300

(1) f −1(Intνb(B))⊆ Int( f −1(B)) for every set B in Y.

(2) Cl( f −1(B))⊆ f −1(Clνb (B)) for every set B in Y .

(3) f (Cl(A))⊆ Clµb ( f (A)) for every set A in X .

Proof. Let B be any open set of Y . If condition (1) is satisfied, then f −1(Intνb(B)) ⊆
Int( f −1(B)). We get f −1(B) ⊆ Int( f −1(B)). Therefore, f −1(B) is an open set. Every open
set is supra b-open. Hence, f is a supra b-continuous map.
If condition (2) is satisfied, then we can easily prove that f is a supra b-continuous map.
Let condition (3) be satisfied and B be any open set of Y. Then f −1(B) is a set in X and
f (Cl( f −1(B))) ⊆ Clµb ( f ( f

−1(B))). This implies f (Cl( f −1(B))) ⊆ Clµb (B). This is nothing but
condition (2). Hence f is a supra b-continuous map.

4. Supra b-open maps and supra b-closed maps

Definition 4. A map f : (X ,τ) → (Y,σ) is called a supra b-open (resp. supra b-closed) if the
image of each open (resp. closed) set in X is supra b-open (resp. supra b-closed) in (Y,ν).

Theorem 10. A map f : (X ,τ)→ (Y,σ) is supra b-open if and only if f (Int(A))⊆ Intνb( f (A))
for each set A in X.

Proof. Suppose that f is a supra b-open map. Since Int(A) ⊆ A, then f (Int(A)) ⊆ f (A).
By hypothesis, f (Int(A)) is a supra b-open set and Intνb( f (A)) is the largest supra b-open set
contained in f (A). Hence f (Int(A))⊆ Intνb( f (A)).

Conversely, suppose A is an open set in X . Then, f (Int(A)) ⊆ Intνb( f (A)). Since Int(A) =
A, then f (A) ⊆ Intνb( f (A)). Therefore f (A) is a supra b-open set in (Y,ν) and f is a supra
b-open map.

Theorem 11. A map f : (X ,τ)→ (Y,σ) is supra b-closed if and only if C lνb ( f (A)) ⊆ f (Cl(A))
for each set A in X.

Proof. Suppose f is a supra b-closed map. Since for each set A in X , Cl(A) is closed
set in X , then f (Cl(A)) is a supra b-closed set in Y . Also, since f (A) ⊆ f (Cl(A)), then
Clνb ( f (A))⊆ f (Cl(A)).

Conversely, Let A be a closed set in X . Since Clνb ( f (A)) is the smallest supra b-closed set
containing f (A), then f (A) ⊆ Clνb ( f (A)) ⊆ f (Cl(A)) = f (A). Thus, f (A) = Clνb ( f (A)). Hence,
f (A) is a supra b-closed set in Y . Therefore, f is a supra b-closed map.

Theorem 12. Let (X ,τ), (Y,σ) and (Z ,υ) be three topological spaces and f : (X ,τ)→ (Y,σ)
and g : (Y,σ)→ (Z ,υ) be two maps. Then,

(1) if g ◦ f is supra b-open and f is continuous surjective, then g is a supra b-open map.



REFERENCES 301

(2) if g ◦ f is open and g is supra b-continuous injective, then f is a supra b-open map.

Proof.

(1) Let A be an open set in Y . Then, f −1(A) is an open set in X . Since g ◦ f is a supra
b-open map, then (g ◦ f )( f −1(A)) = g( f ( f −1(A))) = g(A) (because f is surjective) is a
supra b-open set in Z . Therefore, g is a supra b-open map.

(2) Let A be an open set in X . Then, g( f (A)) is an open set in Z . Therefore, g−1(g( f (A))) =
f (A) (because g is injective) is a supra b-open set in Y . Hence, f is a supra b-open map.

Theorem 13. Let (X ,τ) and (Y,σ) be two topological spaces and f : (X ,τ) → (Y,σ) be a
bijective map. Then the following are equivalent:

(1) f is a supra b-open map;

(2) f is a supra b-closed map;

(3) f −1 is a supra b-continuous map.

Proof.

• (1) =⇒ (2): Suppose B is a closed set in X . Then X − B is an open set in X and by (1),
f (X−B) is a supra b-open set in Y . Since f is bijective, then f (X−B) = Y− f (B).Hence,
f (B) is a supra b-closed set in Y . Therefore, f is a supra b-closed map.

• (2) =⇒ (3): Let f is a supra b-closed map and B be closed set in X . Since f is bijective,
then ( f −1)−1(B) = f (B) which is a supra b-closed set in Y . Therefore, by Theorem 7, f
is a supra b-continuous map.

• (3) =⇒ (1): Let A be an open set in X . Since f −1 is a supra b-continuous map, then
( f −1)−1(A) = f (A) is a supra b-open set in Y . Hence, f is a supra b-open map.

References

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[2] D. Andrijevic’, On b-open sets, Mat. Vesnik, 48 (1996), 59- - 64.

[3] R. Devi, S. Sampathkumar and M. Caldas, On supra α-open sets and sα-continuous maps,
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[4] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,
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[6] A. S. Mashhour, A. A. Allam, F. S. Mahmoud and F. H. Khedr, On supra topological spaces,
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[7] O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961- -970.