International Journal of Modern Nonlinear Theory and Application
Vol.06 No.03(2017), Article ID:78124,6 pages
10.4236/ijmnta.2017.63009
On Decomposition of New Kinds of Continuity in Bitopological Space
H. Al-Malki1, S. Al-Blowi2
1Department of Mathematics, University College in Adam, Umm Al Qura University, Makkah, Saudi Arabia
2Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: May 25, 2017; Accepted: July 30, 2017; Published: August 2, 2017
ABSTRACT
In many papers, new classes of sets had been studied in topological space, then the notion of continuity between any two topological spaces (a function from X to Y is continuous if the inverse image of each open set of Y is open in X) is studied via this new classes of sets. Here the authors also introduce new classes of sets called pj-b-preopen, pj-b-B set, pj-b-t set, pj-b-semi-open and pj-sb-generalized closed set in bitopological space [1] which is a set with two topologies defined on it, then they study the notion of continuity via this set and introduce some of the theories which are studying the decomposition of continuity via this set in bitopological space.
Keywords:
pj-b-Preopen, pj-b-Semiopen, pj-b-t Set, pj-b-B Set, pj-sb-Generalized Closed, Bitopological Space
1. Introduction and Preliminaries
In topological space, there are many classes of generalized open sets given by [2] [3] [4] [5] . Tong [6] introduced the concept of t-set and B-set in topological space. [7] [8] gave some decomposition of continuity. Decomposition of pair- wise continuity was given by Jelice [9] and [10] [11] [12] . In this paper, we introduce decomposition of continuity in bitopological space via new classes of sets called pj-b-preopen, pj-b-B set, pj-b-t set, pj-b-semi-open and pj-sb-genera- lized closed set with some theories, examples and results.
Definition 1.1. Let be a subset of a space
, then
is said to be:
1) b-t-set [7] if.
2) b-B-set [7] if, where
and
is a b-t-set.
3) Locally b-closed [7] if, where
and
is a b-closed set.
4) b-preopen [7] if.
5) b-semiopen [7] if.
Definition 1.2. Let be a subset of a bitopological space
then
called pairwise p-open (or p-open) [11] if
. p-closed is the com- plement of p-open set. p-interior of
(or
) is the union of all p-open sets of a bitopological space
which contained in a subset
of
. Also, the p-closure of
(or
) is the intersection of all p-closed sets which containing
.
Definition 1.3. A subset of a bitopological space
is said to be:
1) pj-b-open [10] if.
2) pj-b-closed [10] if.
3) pj-semiopen [11] if.
4) pj-preopen [11] if.
5) pj-t-set [12] if.
6) pj-B-set [12] if, where
is p-open and
is a pj-t-set.
7) jp-regular open [12] if.
2. pj-b-t-Set, pj-b-B-Set pj-b-Semiopen, pj-b-Preopen and pj-sb-Generalized Closed
In this section, we investigated our new classes of sets pj-b-preopen, pj-b- semiopen, pj-b-t set, pj-b-B set and pj-sb-generalized closed set and study some of its fundamental properties and examples also we introduce some of important theories which is useful to study the decomposition of continuity via our new classes of sets.
Definition 2.1. A subset of a bitopological space
is said to be:
1) pj-b-t-set if.
2) pj-b-B-set if, where
is p-open and
is a pj-b-t-set.
3) pj-b-semiopen if.
4) pj-b-preopen if.
Example 2.2. Let,
and
then
is a p2-b-t-set.
Example 2.3. Let and
and
then
is a p1-b-B-set.
Example 2.4. Let and
and
then
it is p1-b-preopen.
Proposition 2.5. If and
are a subsets of a bitopological space
, then
1) is a pj-b-t set if and only if
is pj-b-semiclosed.
2) If is pj-b-closed, then it is a pj-b-t-set.
3) If and
are pj-b-t-sets, then
is a pj-b-t-set.
proof. 1) Let be pj-b-t set, then
that implies
is pj-b-semiclosed. conversely, Let
be pj-b-semiclosed set, then
. Also,
and
. Hence,
is a pj-b-t set.
2) Let be pj-b-closed, then
.
3) Let and
be pj-b-t-sets, then we have:
, Hence
is a pj-b-t-set.
The following example shows that the converse of (2) is not true in general.
Example 2.6. From example 2.2 it is clear that is a p2-b-t-set but it is not p2-b-closed.
Lemma 2.7. Let be p-open subset of a bitopological space
, then
and
.
proof. Let be p-open subset of
, then
Proposition 2.8. Let be a subsets of a bitopological space
, then
1) If is pj-t-set then it is pj-b-t-set.
2) If is pj-b-t-set then it is pj-b-B-set.
3) If is pj-B-set then it is pj-b-B-set.
proof. 1) Let be pj-t-set,then
from lem- ma 2.1
. Hence
is pj-b-t-set.
2) Let be pj-b-t-set.
and
is p-open set, then
is pj-b- B-set.
3) Let be pj-B-set i.e.
, where
is p-open and
is a pj-t- set i.e.
from lemma 2.1
. Hence
is pj-b-B-set.
Theorem 2.9. Let be a subset of a bitopological space
, then the following are equivalent:
1) is p-open set.
2) is pj-b-preopen and pj-b-B-set.
proof. (1) Þ (2) Let be p-open
but
then
is pj-b-preopen. Also,
and
is p-open and
is pj- b-B-set.
(2) Þ (1) be pj-b-preopen and pj-b-B-set. i.e.
, where
is p-open and
, then we have
Hence,
Therefore and
is p-open.
The following examples show that pj-b-preopen sets and pj-b-B-sets are independent.
Example 2.10. From example 2.3 it is clear that is a p1-b-B -set but it is not p1-b-preopen.
Example 2.11. From example 2.4 it is clear that it is p1-b-preopen but it is not a p1-b-B-set.
Corollary 2.12. A subset of a bitopological space
is p-open if and only if it is pj-α-open and pj-b-B-set.
Proposition 2.13. Let be a subsets of a bitopological space
, then the following are equivalent:
1) is jp-regular set.
2)
3) is pj-b-preopen and pj-b-t-set.
proof. (1) Þ (2) Let be jp-regular set.since
then
. Since
is pj-b-open
. Hence,
(2) Þ (3) This is obvious.
(3) Þ (1) Let be pj-b-preopen and pj-b-t-set.Then
and
is p-open by lemma 2.1
Hence,
is jp-regular set.
Definition 2.14. A subset of a bitopological space
is called pj-sb- generalized closed if pj-
, whenever
and
is pj-b- preopen.
Definition 2.15. pj- is the intersection of all pj-semiclosed sets which containing
.
Theorem 2.16. Let be a subset of a bitopological space
, the following properties are equivalent:
1) is jp-regular open set.
2) is pj-b-preopen and pj-sb-generalized closed set.
proof. (1) Þ (2) Let be jp-regular open.Then
is pj-b-open.
. Moreover, by Lemma 2.1 pj-
. Hence,
is pj-sb-generalized closed.
(2) Þ (1) Let be pj-b-preopen and pj-sb-generalized closed.
is pj-b-semiclosed. Then
. Therefore by Proposition 2.3 A is jp-regular open.
Corollary 2.17. A subset of a bitopological space
is jp-regular open if and only if it is pj-α-open and pj-b-t-set.
3. Decompositions of New Kinds of Continuity
After we had been defined and studied the propriety of our new classes of sets we are ready to study the concept of continuity between any two bitopological spaces via our new classes of sets.
Definition 3.1. A function is called pj-b-conti- nuous [10] (resp. pj-Locally b-closed continuous [10] , pj-D(c,b)-continuous [10] , pj-α-continuous [11] pj-semi continuous [11] , jp-semi continuous [11] , pj-B- continuous [12] , pj-Locally closed continuous [12] , jp-regular continuous [13] ) if
is pj-b-set (resp. pj-Locally b-closed set, pj-D(c,b)-set, pj-α-open, pj-semiopen, jp-semiopen, pj-B-set, pj-Locally closed,, jp-rgular) in
for each p-open set V of Y.
Theorem 3.2. A function is called pj-B-conti- nuous if and only if it is locally pj-b-closed-continuous and pj-semi-continuous.
proof. It is following from lemma 3.4 in [10]
Definition 3.3. Afunction is called pj-b-pre-con- tinuous (resp. pj-b-B-continuous, pj-b-t-continuous, pj-b-semi-continuous) if
is pj-b-preopen (resp. pj-b-B-set, pj-b-t-set, pj-b-semiopen) in
for each p-open set
of
.
Theorem 3.4. A function is called p-continuous if and only if it is pj-α-continuous and pj-b-B-continuous.
proof. It is follows from theorem 2.1.
Theorem 3.5. A function is called p-continuous if and only if it is pj-b-pre-continuous and pj-b-B-continuous.
proof. It is follows from corollary 2.1.
Definition 3.6. Afunction is called contra pj-sb- continuous if
is pj-sb-generalized closed in
for each p-open set
of
.
Theorem 3.7. A function is called completely p- continuous if and only if it is pj-b-pre-continuous and pj-b-t-continuous.
proof. It is follows from proposition 2.3.
Theorem 3.8 A function is called completely p-continuous if and only if it is pj-b-pre-continuous and contra pj-sb-con- tinuous.
proof. It is follows from theorem 2.2.
Theorem 3.9 A function is called completely p- continuous if and only if it is pj-α-continuous and pj-b-t-continuous.
proof. It is follows from corollary 2.2.
Cite this paper
Al-Malki, H. and Al-Blowi, S. (2017) On Decomposition of New Kinds of Continuity in Bitopological Space. International Journal of Modern Non- linear Theory and Application, 6, 98-103. https://doi.org/10.4236/ijmnta.2017.63009
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