Some Results of - Semiconnectedness and Compactness in Bitopological Spaces

Abstract

We are going to establish some results of - semiconnectedness and compactness in a bitopological space. Besides, we will investigate several results in - semiconnectedness for subsets in bitopological spaces. In particular, we will discuss the relationship related to semiconnectedness between the topological spaces and bitopological space. That is, if a bitopological space is - semiconnected, then the topological spaces and are -semiconnected. In addition, we introduce the result which states that a bitopological space is - semiconnected if and only if and are the only subsets of which are - semiclopen sets. Moreover, we have proved some results in compactness also. Altogether, several results of - semiconnectedness and compactness in a bitopological space have been discussed.

1. Introduction

The concept “bitopological space” was established by Kelly in [1]. He introduced this concept in his journal of London Mathematical Society in 1963. He initiated his study about bitopological space by using quasimetric and its conjugate. A quasimetric on a set is a nonnegative real valued function on the Cartesian product of that satisfies the following three axioms:(1),  (2),  (3) if and only if ,  

However, the quasimetric cannot be a metric. Because the symmetric property does not hold for quasimetric. Moreover, every metric space is a topological space. But this is not true for bitopological space in general. Anyhow, bitopological spaces exist for quasimetric spaces. Maheshwari and Prasad [2] introduced semiopen sets in bitopological spaces in 1977. In 1987, the notion -open sets in bitopological spaces was introduced by Banerjee [3]. After that, Khedr [4] introduced and studied about - open sets. Later, Fukutake [5] defined one kind of semiopen sets in 1989. Recently, Edward Samuel and Balan [6] established the concept - semiopen sets in bitopological spaces. We have already published some properties of - semiopen/closed sets in bitopological spaces in [7]. Moreover, we have presented some results of - semiconnectedness in bitopological spaces in [8]. In this paper, we are going to discuss the following results:(1)Let be a family of - semiconnected subsets of a bitopological space with . Then is also - semiconnected.(2)If a bitopological space is - semiconnected, then is - semiconnected.(3)If a bitopological space is - semiconnected, then the topological spaces and are -semiconnected.(4)A bitopological space is - semiconnected if and only if and are the only subsets of which are - semiclopen sets.(5)In a bitopological space , if is - semiconnected then cannot be expressed as the union of two sets and with such that is - semiopen and is - semiopen.(6)In a bitopological space , if cannot be expressed as the union of two nonempty sets and with such that is - semiopen and is - semiopen, then does not contain any nonempty proper subset which is both - semiopen and - semiclosed.(7)The union of any family of - semiconnected sets with a nonempty intersection is - semiconnected.(8)Every - semicompact space is - compact.(9)If is - semiclosed subset of a - semicompact space , then is - semicompact.(10)If is - closed subset of a - semicompact space , then is - semicompact.

2. Materials and Methods

Let -, -, - , -, and - be the interior, closure, -interior, -closure, and -semiclosure of with respect to the topology , respectively, . Let -int and -cl are the -interior and -closure of with respect to the topology , respectively, , where and are semiregularization of and , respectively.

Definition 1. For a nonempty set , we define two topologies and    and may be the same or distinct). Then the triple is called a bitopological space.

Definition 2 (see [1]). Let be subset of a bitopological space . Then is called -open, if . Complement of -open set is called -closed set.

Definition 3 (see [9]). Let be subset of a bitopological space . Then is called(1)-regular open, if -int-cl;(2)-regular open, if -int-cl;(3)-semiopen, if -cl-int;(4)-semiclosed, if -int-cl

Definition 4 (see [9]). Let be subset of bitopological space . Then,(1) is said to be - open set, if, for , there exists -regular open set such that . Complement of - open set is called - closed set;(2) is said to be - open set, if for , there exists -regular open set such that . Complement of - open set is called - closed set;(3)Collection of all - open sets and - open sets are denoted by and respectively. And also and .

Definition 5 (see [6]). Let be subset of bitopological space . Then, is called - semiopen set, if there exists an - open set such that -cl().
Complement of - open set is called - closed set.

Definition 6 (see [6]). A subset is called a - semidisconnected subset of a bitopological space , if - semiopen sets such that , and . Otherwise is called a - semiconnected subset.

Definition 7 (see [6]). A bitopological space is called - semiconnected space, if cannot be expressed as the union of two disjoint sets and such that -scl-scl. Suppose can be so expressed, then is called - semidisconnected space and we write and it is called - semiseparation of .

Definition 8. A nonempty collection is called a - semiopen cover of a bitopological space , if and -- and contains at least one member of - and one member of -.

Definition 9. A cover of a bitopological space is called - open cover of , if and , and .

Definition 10. A bitopological space is called - compact, if every - open cover of has a finite subcover.

Definition 11. A bitopological space is called - semicompact, if every - semiopen cover of has a finite subcover.

3. Results

3.1. Semiconnectedness

Proposition 12 (see [8]). Let be family of - semiconnected subsets of a bitopological space such that ; then is also - semiconnected.

Proof. Let . Now, assume that is not - semiconnected. Then there exist two - semiopen sets and if , and , then let (other case is similar). Now there exist such that also (since ) and also and (since , which shows that is - semidisconnected subset and it is a contradiction. So, is - semiconnected.

Proposition 13. If a bitopological space is - semiconnected, then is - semiconnected.

Proof. Suppose is - semiconnected. Then cannot be expressed as the union of two nonempty disjoint sets and such that -- Also is - semiopen and is - semiopen. Since and , we have every - semiopen and - semiopen are - semiopen and - semiopen, respectively. Therefore, cannot be expressed as the union of two nonempty disjoint sets and such that is - semiopen and is - semiopen, respectively. Hence, is - semiconnected.

Proposition 14 (see [8]). If a bitopological space is - semiconnected, then the topological spaces and are -semiconnected.

Proof. Since every - open set and - open set are - semiopen set and - semiopen set, respectively, So if and are -semidisconnected spaces then the bitopological space becomes -semidisconnected. But this is impossible. So and are -semiconnected spaces.

Proposition 15 (see [8]). A bitopological space is - semiconnected if and only if and are the only subsets of which are - semiclopen sets (simultaneously semiopen and semiclosed.)

Proof. Consider a - semiconnected space ; let and is - semiclopen set; then is - semidisconnected in the bitopological space, which is contradiction. So and are the only subsets of which are both - semiclopen sets. Conversely, let and be the only subsets of which are both - semiclopen sets. If the bitopological space is - semidisconnected, so there exists a - semidisconnection of the bitopological space. So and ; then and both are - semiclopen sets and each of them is neither nor . This is a contradiction. Therefore, is - semiconnected.

Proposition 16. If is - semiconnected then cannot be expressed as the union of two nonempty sets with such that is - semiopen and is - semiopen.

Proof. Assume that can be expressed as the union of two nonempty disjoint sets and such that is - semiopen and is - semiopen, respectively. Since , we have . Then --. Therefore, -. Similarly, we can prove -. Hence, --. This contradicts our supposition. Thus, cannot be expressed as the union of two nonempty disjoint sets and such that is - semiopen and is - semiopen.

Proposition 17. If cannot be expressed as the union of two disjoint sets and such that is - semiopen and is - semiopen, then does not contain any nonempty proper subset which is both - semiopen and - semiclosed.

Proof. Let cannot be expressed as the union of two nonempty sets with such that is - semiopen and is - semiopen. If contains a nonempty proper subset which is both - semiopen and - semiclosed. Then , where is - semiopen, is - semiopen, and are disjoint. This is a contradiction. Thus, does not contain any nonempty proper subset which is both - semiopen and - semiclosed.

Proposition 18. The union of any family of - semiconnected sets with a nonempty intersection is - semiconnected.

Proof. Take , where each is - semiconnected with . Suppose that is not - semiconnected. Then , where and are two nonempty disjoint sets such that --. Since is - semiconnected and , we have or . Therefore, or as . Hence, or . Since , we have . Therefore, ,  . So, . Therefore, or . Suppose . Since , we have . Therefore, . Thus, . This contradicts . Thus, is - semiconnected.

3.2. Semicompactness

Proposition 19. Every - semicompact space is - compact.

Proof. Take to be - semicompact. Let be a pairwise open cover of . Then and and contains at least one member of and one member of . Since every - open set is - semiopen, we have and -- and contains at least one member of - and one member of -. Therefore, is - semiopen cover of . Since is - semicompact, has a finite subcover. Thus, is - compact.

Proposition 20. If is - semiclosed subset of a - semicompact space then is - semicompact.

Proof. Let be a - semicompact space. Let be - semiopen cover of . Since is - semiclosed, is - semiopen. And be a - semiopen cover of . Since is - semicompact, . Hence, . Thus, is - semicompact.

Proposition 21. If is - closed subset of a - semicompact space then is - semicompact.

Proof. Since every - closed set is - semiclosed, is - semiclosed. Then by applying Proposition 20, is - semicompact.

4. Discussion

In this paper, we have used the result that every - closed set is - semiclosed. Moreover, if and are two - semicompact subsets of , then is also - semicompact. Besides, every - semicompact space is - compact. The concept semiconnectedness and compactness is used in various parts of Mathematics. Simultaneously, the bitopological spaces have several applications in analysis, general topology, and theory of ordered topological spaces.

5. Concluding Remarks

In this paper, some results of - semiconnectedness and compactness in bitopological spaces have been discussed. Simultaneously, we have some important results which are related to connectedness and compactness. So we are interested to check whether those results will work for bitopological space or not. If the results fail to hold for bitopology, we are going to illustrate by examples. Further, we want to find how uniform continuity will work in bitopological spaces.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

M. Arunmaran thanks his supervisor K. Kannan, Department of Mathematics and Statistics, University of Jaffna, Sri Lanka, for providing important references from the literature.