Relative Continuity and New Decompositions of Continuity in Bitopological Spaces ()
1. Introduction
The concept of bitopological spaces has been introduced by Kelly [1] . Functions and continuous functions stand among the most important notions in mathematical science. Many different weak forms of continuity in bitopological spaces have been introduced in the literature. For instance, we have pairwise almost and pairwise weakly continuity [2] , pairwise semi-continuity [3] , pairwise pre continuity [4] , pairwise ρ-continuity [5] , pairwise α-continuity [5] and many others, see ([6] [7] ). N. Levine, in [8] introduced decomposition of continuity in topological spaces. In 2004 [9] Tong introduced twenty weak forms of continuity in topological spaces. In this paper, we generalize the results obtained by Tong to the setting of bitopological spaces.
Throughout this paper
and
(or briefly, X and
) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let
be a subset of X, by
(resp.
) we denote the closure (resp. interior) of A with respect to
(or
) and
will denote the complement of
. Here
and
.
2. Preliminaries
We recall some known definitions
Definition 1 ([3] ) A subset
of a bitopological space
is called
-semi open if there is an
- open set U in X such that
.
Definition 2 ([3] ) A function
is called
-semi continuous if
is
- semi open in X for each
-open set V of Y.
Definition 3 ([2] ) A function
is called
-weakly (resp.
-almost) continuous if for each point
and each
-open set V of Y containing
, there exists an
-open set U of X con- taining
such that
(resp.
.
Definition 4 ([5] ) A subset A of a bitopological space
is called ij-α-open if
.
Definition 5 ([5] ) A function
is called
-continuous if
is
- open in X for each i-open set V of Y.
Definition 6 ([4] ) A subset A of a bitopological space
is called ij-pre open if
.
Definition 7 ([4] ) A function
is called
pre continuous if
is
-pre open in X for each i-open set V of Y.
The relations of the above weak forms of continuity are as follows:
![]()
[Diagram 1]
3. Classification of ij-Weak Continuity
Lemma 1 For a subset
of a bitopological space
, we have
1)
;
2)
;
3)
;
4)
.
Proof (1) and (2) are obvious. (3) Since
, then
. Therefore,
. On the other hand,
, then
(4) Similar to (3).
Proposition 1 Let
be a function. Then
1)
is
-continuous if and only if
for each
-open set
in
;
2)
is
-pre continuous if and only if
for each
-open set
in
;
3) f is
-continuous if and only if
for each
-open set
in
.
It is known [2] that a function
is
-weakly continuous if and only if for each
-open set
of
,
. From this we define the following.
Definition 8 Let
be a function. Then
1)
is
-pre weakly continuous if and only if
for each
-open set V in Y;
2)
is
-weakly continuous if and only if
for each
- open set V in Y.
It is well known [2] that
is
-almost continuous if and only if
. From this we define the following group of definitions.
Definition 9 Let
be a function. Then
1)
is
-pre-almost continuous if and only if
for each
- open set V in Y;
2) f is ij-α-almost continuous if and only if
for each i-
mopen set V in Y.
Lemma 2 A function
is
-semi continuous if and only if for each
-open set
of
,
.
Proof Let
be an
-semi continuous function. Then
is
-semi open in
for each
-open set
of
. Since
is a
-semi open set in
, there exist an
-open set
such that
.
Since
we have
. Hence,
, and therefore,
.
Conversely, assume that
for each
-open set
. Now
. Put
. Then there exists an
-open set
such that
. It means
is
-semi open in
for each
-open set
of
. Hence,
is an
-semi continuous function.
In view of the above lemma we define the following:
Definition 10 Let
be a function. Then
1)
is
-weak semi continuous if and only if
for each
-open set V in Y;
2)
is
-almost semi continuous if and only if
for each
- open set V in Y.
Definition 11 Let
be a function. Then:
1)
is
-pre semi continuous if and only if
for each
-open set V
in Y;
2)
is
-pre weak semi continuous if and only if
for each i- open set V in Y;
3)
is
-pre almost semi continuous if and only if
for each
-open set V in Y.
The following diagram gives the relations between all the weak forms of continuity
![]()
[Diagram 2]
Proof (Proof of some relations in Diagram 2).
1)
-weak continuity
-weak continuity
Let
be an
-weak continuous function. Then ![]()
for each
-open set
of
. Since
,
. Hence,
is
-weak continuous;
2)
-weak continuity
-pre weak continuity.
Let
. This implies
hence
. Assume
is
-weak continuous. Then
for each
-open set
of
.
Since
,
.
Hence, f is ij-pre weak continuous.
We could use similar ways to prove other relations in Diagram 2.
4. Classification of Relative Continuity
Let
be a function. Then
is
-continuous if and only if
is an
-open set in
for each
-open set
in
. If we change the requirement on
from being
-open in
to being
-open in a subspace, then we can obtain many new weak forms of continuity.
Definition 12 Let
be a function. Then
1)
is
-continuous if and only if
is an
-open set in the subspace
for each
-open set
in
;
2) f is ij-pre
continuous if and only if
is an
-open set in the subspace
for each i- open set
in
;
3) f is
-continuous if and only if
is an
-open set in the subspace
for each
-open set
in
.
Proposition 2 Any function f is an i#-continuous function.
Proof Let
be a function. For each i-open set V in Y we have
, then
is an i-open set in the subspace
. Hence,
is
-continuous function.
Definition 13 Let
be a function. Then
1) f is
-weak
continuous if and only if
is an i-open set in the subspace
for each i-open set V in Y;
2) f is ij-pre weak
continuous if and only if
is an i-open set in the subspace
for each
-open set V in Y;
3)
is
-weak
continuous if and only if
is an i-open set in the subspace
for each
-open set V in Y.
Definition 14 Let
be a function. Then
1) f is
-almost#continuous if and only if
is an i-open set in the subspace
for each
-open set
in
;
2)
is
-pre almost
continuous if and only if
is an
-open set in the subspace
for each
-open set
in
;
3)
is
-almost
continuous if and only if
is an
-open set in the subspace
for each
-open set
in
.
Definition 15 Let
be a function. Then
1)
is a
-pre-semi
continuous if and only if
is an
-open set in the subspace
for each
-open set
in
;
2)
is a
-pre weak semi
continuous if and only if
is an
-open set in the subspace
for each
-open set
in
;
3)
is a
-pre-almost semi
continuous if and only if
is an
-open set in the subspace
for each
-open set
in
.
Lemma 3 Let
and
be a bitopological space. Then
for
.
Proof Let
. Then there exists an i-open set V in the subspace
such that
. We
can write
, where
is an
-open set in
. Therefore,
. Hence,
is an
-open set in the subspace Z.
Conversely, assume that
. Then there exists an
-open
in
such that
. Since
,
where
is an
-open set in the subspace
. Hence
.
Lemma 4 If
and
is an
-open set in
then
is also
-open relative to
for
.
Proof The proof follows immediately from
where
is an
-open in
.
The following diagram gives the relations between all the weak forms of continuity
![]()
[Diagram 3]
Proof (Proof of some relations in Diagram 2).
1)
-pre weak semi
continuity
-pre almost semi
weak continuity;
Let
be an
-pre weak semi# continuous. Then
is an i-open set in the
subspace
. Now
. By Lemmas 4.6 and 4.7, we
obtain
is an
-open in the subspace
. Hence,
is
pre almost semi continuous.
2)
-pre almost semi
continuity
-pre semi
continuity;
Let
be
-pre almost semi
continuous. Then
is an
-open set in
. Since
,
. Therefore,
. By Lemmas 4.6 and 4.7, we obtain
is an i-open in the subspace
. Hence,
is
-pre semi# continuous.
3)
-pre#continuity
pre semi# continuity;
Let
be
-pre
continuous function. Then
is
-open set in the
subspace
. Since
, then by using Lemma 4.6
and Lemma 4.7, we obtain
is an
-open in the subspace
. So
is
-pre semi# continuous.
4)
-pre almost# continuity
-pre# continuity;
Let
be
-pre almost# continuous function. Then
is
-open set in
. Since
,
. So
, by using Lemmas 4.6 and 4.7, we obtain
is
-open
in the subspace
. Then
is an
-pre
continuous.
5)
-pre almost#continuity
-pre semi#continuity;
Let
be
pre almost
continuous function. Then
is
-open set in
. Since
, then
, therefore
. This implies
, so
. Then
. By using Lemmas 4.6 and 4.7, we obtain
is
-open in the subspace
. So
is an
-pre semi
continuous.
We could also use the similar ways to prove other relations in Diagram 3.
The following examples show that the reverse implications of Diagram 3 is not true.
Example 1 Let
,
,
,
,
and
Define a map
by
,
,
. The map
is 12-pre weak
continuous but not 12-
-weak
continuous because
which is not 1-open in the subspace
.
Example 2 Let
,
,
,
,
and
. Define a map
by
, ![]()
Then the map f is 12-pre weak# continuous but not 12-pre weak semi# continuous, because ![]()
which is not 1-open set in the subspace
.
Example 3 Let
,
,
, ![]()
and
Define a map
by
,
,
and
. The map f is 21-pre almost# continuous but not 21-pre almost semi# continuous because
is not 2-open in the subspace
.
5. Decompositions of i-Continuity and Pairwise Continuity
For a property
of a function
, we say that
is pairwise
if
is 12-
and 21-
. For example,
is called pairwise weakly continuous if it is 12-weakly continuous and 21-weakly continuous.
is pairwise continuous if
and
are continuous.
In this section we will give eight decompositions of
-continuity and pairwaise continuity.
Lemma 5 Let
be a mapping with
and let
be another
mapping with
for each
-open set
of
. Let
be a function such that for each
- open set V in Y,
1)
;
2) There is an
-open set G of X such that
.
Then
is
-continuous.
Proof Since
, then
. Therefore,
. We have
proved that
is an
-open set and hence
is
-continuous.
Now we turn to the decomposition of
-continuity and pairwise continuity.
Theorem 1 Let
be a function. Then each of the following conditions implies that
is
-continuous.
1)
is
-pre continuous and
-pre
-continuous;
2)
is
-continuous and
-continuous;
3)
is
-weakly continuous and
-weak
-continuous;
4)
is
-pre weakly continuous and
-pre weak
-continuous;
5)
is
-weakly continuous and
-weak
-continuous;
6)
is
-almost continuous and
-almost
-continuous;
7)
is
-it pre-almost continuous and
-pre-almost
-continuous;
8)
is
-almost continuous and
-almost
-continuous.
Proof
1) Since f is ij-pre continuous,
. Since f is ij-pre#-continuous,
, where O is i-open set in X. By Lemma 5.1, f is continuous, where
and;
2) Since
is
-continuous,
. Since
is
-continuous,
, where
is
-open set in
. By Lemma 5.1,
is continuous, where
and;
3) Since
is
-weakly continuous,
. Since
is
-weak
-continuous,
, where
is
-open set in
. By Lemma 5.1,
is continuous, where
and
;
4) Since
is
-pre weakly continuous,
. Since
is
-pre weak
-
continuous,
where
is
-open set in
. By Lemma 5.1,
is con- tinuous, where
and
;
5) Since
is
-weakly continuous,
. Since
is
-
weak#-continuous,
, where
is
-open set in
. By Lemma 5.1,
is continuous, where
and
;
6) Since
is
-almost continuous,
. Since
is
-almost
- continuous,
, where
is
-open set in
. By Lemma 5.1,
is
continuous, where
and
;
7) Since
is
-pre-almost continuous,
. Since
is
-pre- almost
-continuous,
, where
is
-open set in
. By Lemma
5.1,
is continuous, where
and
;
8) Since
is
-almost continuous,
. Since
is ij-α-almost
-continuous, so
, where
is
-open set in X.
By Lemma 5.1,
is continuous, where
and
.
Corollary 1 Let
be a function. Then each of the following conditions implies that
is pairwise continuous.
1)
is pairwise pre continuous and pairwise pre
-continuous;
2)
is pairwise
-continuous and pairwise
-continuous;
3)
is pairwise weakly continuous and pairwise weakly
-continuous;
4)
is pairwise pre weakly continuous and pairwise pre weak
-continuous;
5)
is pairwise
-weakly continuous and pairwise
-weak
-continuous;
6)
is pairwise almost continuous and pairwise almost
-continuous;
7)
is pairwise pre-almost continuous and pairwise pre-almost
-continuous;
8)
is pairwise
-almost continuous and pairwise
-almost
-continuous.
Proof The proof follows immediately from Theorem 5.3.
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), king Abdulaziz University, Jeddah, under grat No. (363-006-D1433). The author, therefore, acknowledge with thanks DSR technical and financial support.