EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 358-369
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On B-Open Sets

Layth M. Alabdulsada

Institute of Mathematics, University of Debrecen, H-4002 Debrecen, P.O. Box 400, Hungary

Abstract. The aim of this paper is to define and study B-open sets and related properties. A
B-open set is, roughly speaking, a generalization of a b-open set, which is in turn a generalization of
a pre-open set and a semi-open set. Using B-open sets, we introduce a number of concepts such as
B-dense, B-Frechet, contra-B-closed graph and contra-B-continuity. Also, we define a bi-operator
topological space (X, τ, T1, T2) which involves two operators T1 and T2, which are used to define
B-open sets.

2010 Mathematics Subject Classifications: 54C05, 54C08, 54C10

Key Words and Phrases: Operator topological space, bi-operator topological space, B-open
sets, T ∗-open sets, contra-B-continuous, Urysohn space, weakly Hausdorff space

1. Introduction

Over the past years, an amount of generalizations of open sets has been considered.
The first notion due to Levine [12] in 1963 was semi-open sets, while in 1965 Nj̊astad [15]
introduced some classes of nearly open sets, more precisely, they investigated the structure
of α-open set and gave some applications. Mashhour et al. in 1982 [5] introduced and
studied pre-open sets and pre-continuous functions. In 1983 Abd El-Monsef et al. [13]
introduced the new topological notions, β-open sets, β-continuous mappings and β-open
mappings. In 1996 [3], Andrijević introduced and studied a new class of generalized open
sets in a topological space, called b-open sets. All of these above concepts were defined
similarly using the closure operator Cl and the interior operator Int.

This research area (which is fertile in information) still takes a significant part of the
investigations because it has a clear effect on the development of the topological space
through the experience of many theories and characteristics of different types of open sets,
for instance see ([1], [17], [7], [8], [10], [9] and [21]).

We work on circulating the b-openness from a different point of view than previously
stated since our generalization depends entirely on operators attached with topology τ on
X to define the B-open sets. More accurately, let P (X) be the power set of X and functions
T1, T2 : P (X)→ P (X) are operators associated with topology τ on X. Then the quadruple

DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3401

Email address: layth.muhsin@science.unideb.hu (L. M. Alabdulsada)

http://www.ejpam.com 358 c© 2019 EJPAM All rights reserved.



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 359

(X, τ, T1, T2) is called a bi-operator topological space. However, if T1(S) = Cl(Int(S)) and
T2(S) = Int(Cl(S)), then the notion of B-open sets became exactly the same as the
definition of the b-open sets. The use of the operator topological spaces for the first time
goes back to H. J. Mustafa et al. [11], [14], and recently Alabdulsada [2].

In this paper, first we introduce and study the new notion of bi-operator topological
spaces and its related properties. Our generalization of open sets in topological space is
called B-open sets, which linked to bi-operator topological spaces. First we recall several
concepts and definitions that contributed to constructing our definition, namely B-open
which generalizes b-open sets in a topological space. Afterwards, we apply B-open sets to
define some further new concepts, and show some remarks and examples for B-open sets.

Our main results are given in Section 3, where we present and study several different
spaces as well as functions which are based on B-open sets. Also, we investigate the
relationships between these types of functions, besides we check the relationships with
some special spaces such as Urysohn space or weakly Hausdorff space. To be precise, we
prove that, among others, if the function f : (X, τ, T1, T2)→ (Y, σ) has a contra-B-closed
graph, then the inverse image of a contra-compact set S of Y is B-closed in X. In addition,
if f : (X, τ, T1, T2)→ (Y, σ) is contra-B-continuous from a B-connected space onto Y , then
Y is not a discrete space. Another new result says if f : (X, τ, T1, T2) → (Y, σ) is a
contra-B-continuous surjective function and X is B-compact, then Y is contra-compact.
Furthermore, a number of important related properties are stated and proved.

2. Background

In this section, we recall and introduce some of the definitions and the fundamental
notions that play a key role in this paper. Throughout, (X, τ), (Y, σ) are arbitrary topo-
logical spaces and S ⊆ X. The closure of S will be denoted by Cl(S). The interior of S
will be denoted by Int(S).

Definition 1. A subset S of a topological space (X, τ) is said to be:

(i) regular open, if S = Int(Cl(S)), regular closed if S = Cl(Int(S)) [20].

(ii) pre-open, if S ⊆ Int(Cl(S)), the complement of a pre-open is pre-closed [5].

(iii) semi-open, if S ⊆ Cl(Int(S)), the complement of a semi-open is semi-closed [12].

(iv) α-open, if S ⊆ Int(Cl(Int(S))), the complement of an α-open is α-closed [15].

(v) β-open, if S ⊆ Cl(Int(Cl(S))), the complement of a β-open is β-closed [13].

(vi) b-open, if S ⊆ Cl(Int(S)) ∪ Int(Cl(S)), the complement of a b-open is b-closed [3].

In particular, the β-closure of a set S denoted by βCl(S), is the intersection of all β-
closed sets containing S. The β-interior of a set S denoted by βInt(S), is the union of all
β-open sets contained in S. The preclosure, preinterior, semiclosure, semiinterior, b-closure
and b-interior of a set S denoted by pCl(S),pInt(S), sCl(S), sInt(S), bCl(S) and bInt(S),
respectively, are defined analogously.



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 360

Proposition 1. [3] Let S be a subset of a space X. Then:

(i) pInt(S) = S ∩ Int(Cl(S)),

(ii) pCl(S) = S ∪ Cl(Int(S)),

(iii) sInt(S) = S ∩ Cl(Int(S)),

(iv) sCl(S) = S ∪ Int(Cl(S)).

Definition 2. [11] Let (X, τ) be a topological space and P (X) be the power set of X. A
function

T : P (X)→ P (X)

is said to be an operator associated with topology τ on X if U ⊆ T (U) for all U ∈ τ and
the triple (X, τ, T ) is called an operator topological space.

Definition 3. Let (X, τ, T ) be an operator topological space and S ⊆ X, then

(i) S is said to be T -open [11], if for each x ∈ S there exists U ∈ τ such that x ∈ U ⊆
T (U) ⊆ S. The complement of T -open is called T -closed.

(ii) S is said to be T ∗-open [14], if S ⊆ T (S) (observe that S not necessarily open). The
complement of T ∗-open is called T ∗-closed.

Remark 1. T1Cl(S), T2Cl(S) are the intersection of all T1-closed, T2-closed sets, resp.,
in X containing S. Now, if T1(S) = Int(Cl(S)) and T2(S) = Cl(Int(S)) where S ⊆ X,
then T1-open set is exactly the pre-open set and T2-open set is exactly the semi-open set.
In addition, we have that T1Cl(S) ≡ pCl(S) and T2Cl(S) ≡ sInt(S).

Definition 4. Let (X, τ) be a topological space and T1, T2 be two operators associated with
the topology τ on X that is U ⊆ T1(U) and U ⊆ T2(U) for each U ∈ τ . The quadruple
(X, τ, T1, T2) is called a bi-operator topological space.

Example 1.

(i) If T1, T2 are the identity operators, i.e. T1(S) = S and T2(S) = S, then the quadru-
ple (X, τ, T1, T2) will reduces to (X, τ), thus the bi-operator topological space is the
ordinary topological space.

(ii) Let (X, τ) be any topological space and T1, T2 : P (X)→ P (X) be functions such that
T1(S) := Int(Cl(S)) and T2(S) := Cl(Int(S)) for any S ⊆ X. Notice that if U is
open in X, then U ⊆ Int(Cl(U)) = T1(U) and U ⊆ Cl(Int(U)) = T2(U). Thus, T1, T2

are operators associated with the topology τ on X and the quadruple (X, τ, T1, T2) is
a bi-operator topological space.

Definition 5. Let (X, τ, T1, T2) be a bi-operator topological space and S ⊆ X. The set S
is said to be a B-open set if

S ⊆ T1(S) ∪ T2(S).



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 361

The complement of a B-open set is B-closed. Moreover, if T1(S) = Cl(Int(S)) and
T2(S) = Int(Cl(S)), then S is B-open if and only if S is b-open, so the concepts of B-
openness reduces to the concepts of b-openness in this case. Cf. Definition 1.

Remark 2.

(i) As an example of B-open set, one can consider a bi-operator topological space (R, τu,
T1, T2) such that R stands for the set of real numbers and τu for the usual topology.
Let S ⊆ R and T1(S) = Int(Cl(S)) and T2(S) = Cl(Int(S)). If S = [0, 1]∪

(
(1, 2)∩Q

)
,

Q denotes the set of the rational numbers then S is B-open but neither T ∗1 -open nor
T ∗2 -open set. On other hand, if E = [0, 1) ∪ Q, then E is T ∗1 -open but not T ∗2 -open
while E is B-open.

(ii) The intersection of two B-open sets is not necessarily B-open. So, the collection of
all B-open sets is not necessarily a topology on X.

(iii) The intersection of any collection of B-closed sets is B-closed. BCl(S) is the intersec-
tion of all B-closed sets containing S, i.e. BCl(S) := ∩ {U | U is B-closed, U ⊇ S}.

(iv) BInt(S) is the union of all B-open sets contained in S, i.e. BInt(S) := ∪ {U |
U is B-open, U ⊆ S}.

(v) Every T ∗1 -open (T ∗2 -open) set is B-open because if we assume that S is T ∗1 -open then
S ⊆ T1(S) ⊆ T1(S) ∪ T2(S), therefore, S is B-open and the same for the T ∗2 -open.
More precisely, if we put T ∗12-open instead of B-open, then we have
T1-open → open → T ∗1 -open → T ∗12-open → T ∗123-open → ... T ∗123...n-open.

Similarly,

T2-open → open → T ∗2 -open → T ∗12-open → T ∗123-open → ... T ∗123...n-open,

means that S is T ∗123-open if

S ⊆ T1(S) ∪ T2(S) ∪ T3(S),

and T ∗123...n-open means analogously

S ⊆ T1(S) ∪ T2(S) ∪ T3(S) ∪ ... ∪ Tn(S).

Definition 6. The graph G(f) of a function from a bi-operator topological space (X, τ, T1,
T2) into a topological space (Y, σ) is said to be

(i) B-regular graph, if for every (x, y) ∈ X × Y \ G(f), there exists U which is B-
closed in X containing x and a regular open set V in Y containing y such that
(U × V ) ∩G(f) = ∅.

(ii) contra-B-closed graph, if for each (x, y) ∈ X × Y \ G(f), there exists a B-closed
set U in X containing x and a regular closed set V in Y containing y such that
f(U) ∩ V = ∅.



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 362

Definition 7. [6] A function f : (X, τ)→ (Y, σ) is said to be contra-continuous, if f−1(V )
is closed in X for each open subset V of Y .

Definition 8. A function f : (X, τ, T1, T2) → (Y, σ) is said to be contra-B-continuous, if
f−1(V ) is B-closed in X for each open subset V of Y .

Definition 9. Let (X, τ, T1, T2) be a bi-operator topological space, then X is called a B-
Frechet, if for each pair of distinct points x1, x2 of X, there exists B-open sets U and V
containing x1 and x2, respectively where x2 /∈ U and x1 /∈ V. This is equivalent to saying
that each single {x} is B-closed.

Definition 10. A topological space (X, τ) is said to be (see [1], [7], [21] [22] and [18]):

(i) compact, if for every open cover of X has finite subcover.

(ii) contra-compact, if for every closed cover of X has finite subcover.

(iii) R-compact, if for every regular open cover of X has finite subcover.

(iv) contra-R-compact, if for every regular closed cover of X has finite subcover.

(v) R-Lindelöf, if for every regular open cover of X has countable subcover.

(vi) contra-R-Lindelöf, if for every regular closed cover of X has countable subcover.

(vii) countable-R-compact, if for every countable regular open cover of X has finite sub-
cover.

(viii) contra countable-R-compact, if for every countable regular closed cover of X has
finite subcover.

Definition 11. We call the bi-operator topological space (X, τ, T1, T2):

(i) B-compact, if for every B-open cover of X has finite subcover.

(ii) B-Lindelöf, if for every B-open cover of X has countable subcover.

(iii) countable-B-compact, if for every countable-B-open cover of X has finite subcover.

Definition 12. A subset S of a bi-operator topological space (X, τ, T1, T2) is said to be
B-dense, if BCl(S) = X.

Remark 3. If T1(S) = Int(Cl(S)), T2(S) = Cl(Int(S)), then B-dense will be b-dense and
BCl(S) will be bCl(S) such that b-dense is a set in X if bCl(S) = X.

Definition 13. A bi-operator topological space (X, τ, T1, T2) is called a B-connected pro-
vided X is not a union of two nonempty B-open sets.

Definition 14. A topological space (X, τ) is said to be a weakly Hausdorff space [19], if
each element of X is an intersection of regular closed sets.

Definition 15. A topological space (X, τ) is an Urysohn space [4], if for every pair of
distinct points x and y in X, there exist open sets U and V such that x ∈ U, y ∈ V and
Cl(U) ∩ Cl(V ) = ∅.



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 363

3. Some properties of B-open sets

Lemma 1. Let (X, τ, T1, T2) be a bi-operator topological space given by

T1(S) = Int(Cl(S)), T2(S) = Cl(Int(S)).

Then

(i) BInt(S) = sInt(S) ∪ pInt(S).

(ii) BCl(S) = sCl(S) ∩ pCl(S).

Proof. It is sufficient to prove only the first assertion. As we have stated in Remark 2
(iv) that BInt(S) is the union of all B-open sets contained in S, therefore

BInt(S) ⊃ Cl(Int(BInt(S))) ∪ Int(Cl(BInt(S))) ⊃ Cl(Int(S)) ∪ Int(Cl(S))).

Thus, with the help of Proposition 1, we obtain

BInt(S) = S ∩ [Cl(Int(S)) ∪ Int(Cl(S))]

= [S ∩ Cl(Int(S))] ∪ [S ∩ Int(Cl(S))]

= sInt(S) ∪ pInt(S).

The opposite direction is evident. One can prove the second statement in a similar way.

Lemma 2. Let (X, τ, T1, T2) be a bi-operator topological space, suppose that

T1(W ∩ Z) = T1(W ) ∩ T1(Z)

and
T2(W ∩ Z) = T2(W ) ∩ T2(Z),

for all W ∈ τ, Z ⊆ X then the following assertions are satisfied:

(i) The intersection of an open set with a B-open set is a B-open set.

(ii) The union of any family of B-open sets is a B-open set.

Proof. (i) Assume that there exists U ∈ τ , which is an open set, and V is a B-open
set. We are going to show that U ∩ V is also a B-open set. Since U is open, then

U ⊆ T1(U), U ⊆ T2(U.

By the definition of the B-open set:

V ⊆ T1(V ) ∪ T2(V ).

Now,

U ∩ V ⊆ U ∩ [T1(V ) ∪ T2(V )]



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 364

= [U ∩ T1(V )] ∪ [U ∩ T2(V )]

⊆ [T1(U) ∩ T1(V )] ∪ [T2(U) ∩ T2(V )]

= [T1(U ∩ V )] ∪ [T2(U ∩ V )],

as wanted to be shown.
(ii) Suppose that F = {Va| a ∈ λ} is a family of B-open set,

Va ⊆ T1(Va) ∪ T2(Va).

Then we have, ⋃
a

Va ⊆
⋃
a

(T1(Va) ∪ T2(Va))

=
⋃
a

T1(Va) ∪
⋃
a

T2(Va).

It is clear that
⋃

a T1(Va) = T1(
⋃

a Va) and
⋃

a T2(Va) = T2(
⋃

a Va), therefore⋃
a

Va ⊆ T1(
⋃
a

Va) ∪ T2(
⋃
a

Va).

Thus,
⋃

a Va is a B-open set, which completes the proof.

Proposition 2. Let (X, τ, T1, T2) be a bi-operator topological space. If the function
f : (X, τ, T1, T2)→ (Y, σ) has a contra-B-closed graph, then the inverse image of a contra-
compact set S of Y is B-closed in X.

Proof. Assume that S is a contra-compact set of Y and x /∈ f−1(S), i.e. for all
a ∈ S, (x, a) /∈ G(f). Then there exist Ua which is B-closed containing x and Va which is
closed in Y containing a such that

f(Ua) ∩ Va = ∅.

On the other hand one can consider F = {S ∩ Va| a ∈ S} and F is closed cover of the
subspace S. We have that S is contra-compact, then there exists a1, a2, ..., an such that
S ⊆ ∪ni=1Vai . Now, if U = ∩ni=1Uai , then U is B-closed containing x and f(U) ∩ S = ∅,
therefore U ∩ f−1(S) = ∅. Hence x /∈ BCl(f−1(S)), this shows that f−1(S) is B-closed.

Proposition 3. Let f : (X, τ, T1, T2) → (Y, σ) from a bi-operator topological space to a
contra-compact space which has a contra-B-closed graph, then f is a contra-B-continuous
function.

Proof. Let F = {Va| a ∈ λ} be a cover of an open set U ⊂ Y by the closed subsets
Va of U for each a ∈ λ. Thus, there exists a closed set Wa of Y where Va = Wa ∩ U, i.e.
{Wa| a ∈ λ}∪{U c} is a closed cover of Y . But Y is a contra-compact space, namely, there
exist a1, a2, ..., an such that Y = ∪ni=1Wai ∪ U c. Hence U = ∪ni=1Vai , and consequently U



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 365

is contra-compact. From previous proposition f−1(U) is B-closed in X, thus f is contra-
B-continuous.

The proof of the next lemma is immediate, since g is contra-B-continuous, so f−1(U) =
g−1(X × U) is B-closed in X, then f is contra-B-continuous.

Lemma 3. Let f : (X, τ, T1, T2)→ (Y, σ) be a function and g : (X, τ, T1, T2) → (X × Y )
be a graph function of f defined by g(x) = (x, f(x)) for every x ∈ X. If g is contra-B-
continuous then f is contra-B-continuous.

Proposition 4. Let f : (X, τ, T1, T2) → (Y, σ) be contra-B-continuous and g : (X, τ) →
(Y, σ) is contra-continuous. If Y is an Urysohn space, then E = {x ∈ X| f(x) = g(x)} is
B-closed in X.

Proof. Suppose that x ∈ Ec, this implies that f(x) 6= g(x). Since Y is an Urysohn
space, then there exist open sets U and V such that f(x) ∈ U, g(x) ∈ V and Cl(U) ∩
Cl(V ) = ∅. Since the function f is contra-B-continuous, f−1(Cl(U)) is B-open in X
and g is contra-continuous, therefore g−1(Cl(V )) is open in X. If we consider W =
f−1(Cl(U)), Z = g−1(Cl(V )), then x ∈ W ∩ Z = S where S is B-open in X and
f(S) ∩ g(S) ⊆ f(W ) ∩ g(Z) ⊆ Cl(U) ∩ Cl(V ) = ∅. Hence f(S) ∩ g(S) = ∅ and S ∩ E =
∅, S ⊆ Ec where S is B-open. We conclude that x /∈ BCl(E), and so E is B-closed in X.

Corollary 1. Let f : (X, τ, T1, T2) → (Y, σ) be contra-B-continuous and let g : (X, τ) →
(Y, σ) be contra-continuous. If Y is an Urysohn space and f = g on a B-dense set S ⊆ X,
then f = g on X.

Proof. From the previous result E = {x ∈ X|f(x) = g(x)} is B-closed in X. Now we
assumed that f = g on B-dense set and S ⊆ E. Since f is contra-B-continuous and g is
contra-continuous, then X = BCl(S) ⊆ BCl(E) = S. Therefore, f = g on X.

Proposition 5. If f : (X, τ, T1, T2) → (Y, σ) is contra-B-continuous from a B-connected
space onto Y , then Y is not a discrete space.

Proof. Let Y be a discrete space and ∅ 6= S ⊂ Y, then S is a proper nonempty open
and closed subset of Y . Then f−1(S) is a proper nonempty B-open and B-closed subset
of X such that X = f−1(S)∪ (f−1(S))c which means that X is B-disconnected space and
this contradicts our assumption. Thus, Y is not discrete.

Definition 16. A function f : (X, τ, T1, T2) → (Y, σ) is called an almost contra-B-
continuous function, if f−1(V ) is a B-closed for every regular open set V in Y.

Proposition 6. Let f : (X, τ, T1, T2)→ (Y, σ) be a surjective almost contra-B-continuous
function, then:

(i) if X is B-Lindelöf, then Y is contra-R-Lindelöf.

(ii) if X is B-compact, then Y is contra-R-compact.



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 366

(iii) if X is countable-B-compact, then Y is countable contra-R-compact.

Proof. We are going to prove (i) and (ii) and one can prove (iii) in a similar way.
(i) Consider a family S = {Va| a ∈ λ} to be a regular closed cover of Y, at the same time
let S∗ = {f−1(Va)| a ∈ λ} be a B-open cover of X. But X is B-Lindelöf, then there exist
a1, a2, ..., an such that X = ∪∞i=1f

−1(Vai), we have Y = f(X) = f(∪∞i=1f
−1(Vai)). Then

Y = ∪∞i=1(Vai) is contra-R-Lindelöf.
(ii) Using the same technique as above, let F = {Ua| a ∈ λ} be a regular closed cover of
Y since f is a surjective almost contra-B-continuous function. So F∗ = {f−1(Ua)| a ∈ λ}
is a B-open cover of X but X is B-compact, then there exists a1, a2, ..., an where X =
∪ni=1f

−1(Uai). Consequently,

Y = f(X) = f(∪ni=1f
−1(Uai)) = ∪ni=1(Uai).

This clearly forces Y to be contra-R-compact.

Definition 17. [16] A function f : X → Y is called:

• almost continuous, if f−1(V ) is open in X for every regular open set V in Y .

• R-continuous, if f−1(V ) is a regular open set of X for each regular closed set V in
Y .

Lemma 4. [16] If a function f : X → Y is almost contra-b-continuous and almost
continuous, then f is a R-continuous function.

Proposition 7. Let f : (X, τ, T1, T2)→ (Y, σ) be an almost contra-B-continuous and sur-
jective almost-continuous function, suppose T1(S) = Int(Cl(S)) and T2(S) = Cl(Int(S)),
then Y is:

(i) contra-R-compact, if X is contra-R-compact.

(ii) R-compact, if X is R-compact.

(iii) R-Lindelöf, if X is R-Lindelöf.

(iv) countable-R-compact, if X is countableR-compact.

(v) countable contra-R-compact, if X is countable contra-R-compact.

(vi) contra-R-Lindelöf, if X is contra-R-Lindelöf.

Proof. It is enough to prove (i) and for the rest one can use the same methods to prove
them.
(i) T1(S) = Int(Cl(S)) and T2(S) = Cl(Int(S)) are given. So f is almost contra-B-
continuous and surjective almost-continuous, by the above lemma, f is R-continuous, that
is the inverse of each regular closed set in Y is regular in X. Assume that S = {Va| a ∈ λ}
is a regular closed cover of Y. Consequently, S∗ = {f−1(Va)| a ∈ λ} is a regular closed
cover of X, but X is contra-R-compact, therefore there exists a1, a2, ..., an such that X =
∪ni=1f

−1(Vai) and Y = f(X) = f(∪ni=1f
−1(Vai)), which shows that Y is contra-R-compact.



L. M. Alabdulsada / Eur. J. Pure Appl. Math, 12 (2) (2019), 358-369 367

Proposition 8. If f : (X, τ, T1, T2) → (Y, σ) is a contra-B-continuous function and S is
B-compact relative to X, then f(X) is contra-compact in Y.

Proof. Let F = {Va| a ∈ λ} be any cover of f(S). It follows from the closed set of the
subspace of f(S) for all a ∈ λ that there exists a closed set Sa of Y such that Sa∩f(S) = Va
and for each x ∈ S, there exists a(x) ∈ λ where f(S) ∈ Sa(x). Then there exists Ux which
is B-open, this implies that f(Ux) ∈ Sa(x) such that the family F∗ = {Ux| x ∈ S} is a cover
of S by B-open of X. But S is B-compact relative to X, so there exist x1, ..., xn ∈ S and
S ⊆ ∪ni=1f

−1Uxi . Hence f(S) ⊆ f(∪ni=1f
−1Uxi) = ∪ni=1Uxi , therefore, f(S) = ∪ni=1Va(xi).

Corollary 2. If f : (X, τ, T1, T2) → (Y, σ) is a contra-B-continuous surjective function
and X is B-compact, then Y is contra-compact.

Definition 18. A function f : (X, τ, T1, T2)→ (Y, σ) is called almost weakly-B-continuous,
if for each x ∈ X and regular set V containing f(x) there exist U which is a B-open set
in X containing x such that f(U) ⊆ Cl(V ).

Proposition 9. Let a function f : (X, τ, T1, T2)→ (Y, σ) be an almost contra-B-continuous
and Y be an Urysohn space, then G(f) is regular in X × Y.

Proof. Let (x, y) ∈ X×Y \G(f), it follows that y 6= f(x), since Y is an Urysohn, then
there exist open sets U and V containing f(x) and y, respectively where U ∩V = ∅. Then

Int(Cl(U)) ∩ Cl(Int(V )) = ∅.

Since f is almost contra-B-continuous, we have that f−1(Int(Cl(U))) is B-closed in X
containing x. If W = f−1(Int(Cl(U))), then f(W ) ⊆ Int(Cl(U)) such that f(W ) ∩
Int(Cl(V )) = ∅ and Int(Cl(V ) is regular in Y . Hence G(f) is B-regular in X × Y.

Proposition 10. Suppose that f : (X, τ, T1, T2)→ (Y, σ) has a B-regular graph. If f is a
surjective function, then Y is weakly Hausdorff.

Proof. Let y, ȳ be any two distinct points of Y . Since f is surjective, then there exists
x ∈ X where f(x) = y. Notice (x, ȳ) ∈ (X×Y )\G(f), by the definition of B-regular graph,
there exists B-closed set U of X and a regular open Fȳ in Y such that (x, ȳ) ∈ U ×Fȳ and
f(U)∩ Fȳ = ∅. Since f(x) ∈ f(U) and y /∈ Fȳ, ȳ /∈ F c

ȳ which is regular closed in Y, we get
y = ∩

ȳ 6=y
F c
ȳ . Thus, Y is weakly Hausdorff.

Proposition 11. Let f : (X, τ, T1, T2)→ (Y, σ) be a function which has a B-regular graph.
If f is an injective function, then X is B-Frechet.

Proof. Assume that x1, x2 are any two distinct points of X. Since f is injective, it
follows that (x1, f(x2)) ∈ (X×Y )\G(f), by the definition of B-regular graph. Then there
exist a B-closed set U of X and a regular open set V in Y such that (x1, f(x2)) ⊂ U × V
and f(U)∩ V = ∅, therefore U ∩ f−1(V ) = ∅ and x2 /∈ U . Thus x1 /∈ U c, x2 ∈ U c and U c

is B-open which means that {x1}c is B-open, that is {x1} is B-closed. So X is is B-Frechet.



REFERENCES 368

Proposition 12. Let f : (X, τ, T1, T2)→ (Y, σ) be a weakly-B-continuous function and Y
be an Urysohn space. Then G(f) is contra B-regular in X × Y.

Proof. Let us consider (x, y) ∈ (X × Y ) \ G(f), therefore y 6= f(x). Since Y is an
Urysohn space, then there exist two open sets U and V in Y containing y and f(x),
respectively. Consider that W is a B-open set containing x and Cl(U)∩Cl(W ) = ∅. Since
we are working under the assumption that f is weakly-B-continuous, then f(x) ⊆ Cl(V )
which implies that f(W ) ∩ Cl(U) = f(W ) ∩ Cl(Int(U)) = ∅ and Cl(Int(U)) is regular
closed containing y. Hence G(f) is a contra B-regular graph in X × Y, which completes
the proof.

Acknowledgements

I would like to express my sincere gratitude to my supervisor Dr. László Kozma, for
carefully reviewing this paper, providing beneficial suggestions.

References

[1] A. Al-Omari and M.S.M. Noorani. Some properties of contra-b-continuous and almost
contra-b-continuous functions. Eur. J. Pure Appl. Math., 2(2):213–230, 2009.

[2] L. M. Alabdulsada. On the class of weakly almost contra-T ∗-continuous functions.
to appear in Publ. Math. Debrecen, 2019.

[3] D. Andrijević. On b-open sets. Mat. Vesnik, 48(1-2):59–64, 1996.

[4] S. P. Arya and M. P. Bhamini. Some generalizations of pairwise urysohn spaces.
Indian J. Pure Appl. Math., 18(12):1088–1093, 1987.

[5] M.E. Abd El-Monsef A.S. Mashhour and S.N. El-Deeb. On precontinuous and weak
precontinuous mappings. Proc. Math. Phys. Soc. Egypt., 53:47–53, 1982.

[6] J. Dontchev. Contra-continuous functions and strongly s-closed spaces. Internat.J.
Math. Math. Sci., 19(2):303–310, 1996.

[7] E. Ekici. Almost contra-precontinuous functions. Bull. Malays. Math. Sci. Soc.(2),
27(1):53–65, 2004.

[8] E. Ekici. On the notion of (γ, s)-continuous functions. Demonstratio Math., 38(3):715–
727, 2005.

[9] E. Ekici. New forms of contra-continuity. Carpathian J. Math., 24(1):37–45, 2008.

[10] E. Ekici. On contra πg-continuous functions. Chaos Solitons Fractals., 35(1):71–81,
2008.



REFERENCES 369

[11] A. L. Moussa H. J. Mustafa and A. K. Mazal. Operator topological spaces. Journal
of the College of Education, 1:213–221, 2011.

[12] N. Levine. Semi-open sets and semi-continuity in topological spaces. Amer. Math.
Monthly., 70:36–41, 1963.

[13] S.N. El-Deeb M.E. Abd El-Monsef and R.A. Mahmoud. β-open sets and β-continuous
mapping. Bull. Fac. Sci. Assiut Univ. A, 12(1):77–90, 1983.

[14] H. J. Mustafa and L. M. Alabdulsada. On almost contra T ∗-continuous functions. J.
of Kufa for Math. and Comp., 1(6):1–6, 2012.

[15] O. Nj̊astad. On some classes of nearly open sets. Pacific J. Math., 15:961–970, 1965.

[16] T. Noiri. On almost continuous functions. Indian J. Pure Appl. Math., 20(6):571–576,
1989.

[17] J. B. Toranagatti S. S. Benchalli, P. G. Patil and S. R. Vighneshi. Contra δgb-
continuous functions in topological spaces. Eur. J. Pure Appl. Math., 10(2):312–322,
2017.

[18] M.K. Singal and A. Mathur. On nearly-compact spaces. Boll. Un. Mat. Ital. (4),
2:702–710, 1969.

[19] T. Soundararajan. Weakly hausdorff spaces and the cardinality of topological spaces.
pages 301–306, 1971.

[20] M.H. Stone. Applications of the theory of boolean rings to general topology. Trans.
Amer. Math. Soc., 41(3):375–481, 1937.

[21] A. Al-Omari T. Noiri and M.S.M. Noorani. Weakly b-open functions. Math. Balkanica
(N.S.), 23(1-2):1–13, 2009.

[22] S. Willard. General topology spaces. Addison Wesley., 1970.