Some Properties of (1,2)^∗-Locally Closed Sets

Abstract

A new kind of generalization of (1, 2)*-closed set, namely, (1, 2)*-locally closed set, is introduced and using (1, 2)*-locally closed sets we study the concept of (1, 2)*-LC-continuity in bitopological space. Also we study (1, 2)*-contracontinuity and lastly investigate its relationship with (1, 2)*-LC-continuity.

1. Introduction and Preliminaries

It is established that generalization of closed set plays an important role in developing the various concepts in both topological and bitopological spaces. The difference of two closed subsets of an -dimensional Euclidean space was considered by Kuratowski and Sierpinski [1] in 1921 and the fundamental tool in their work is the notion of a locally closed subset of a topological space . In 1963 Kelly [2] initiated the systematic study of bitopological spaces. Then in 1989 Ganster and Reilly [3] used locally closed sets to define LC-continuity in a topological space. According to them a function is said to be LC-continuous if the inverse image of every open set in is locally closed set in . In 1990  Jelić [4] introduced (1, 2)-locally closed sets and (1, 2) LC-continuity in bitopological space. In 1991  Lellis Thivagar introduced the open set in bitopological space which is called (1, 2) open set [5]. In general we know that a (1, 2)*--open set [6] may not be a -open set in , but in this present paper we have a necessary and sufficient condition for the requirement that an (1, 2)*--open set is a -open set. Bourbaki [7] defined that a subset of a topological space is said to be locally closed if it is the intersection of an open and a closed subsets of in the year 1966. In 2004 M. Lellis Thivagar and O. Ravi introduced a generalized concept of (1, 2) open sets which is called (1, 2)*-open sets [8] in bitopological space. Using the (1, 2)*-open set and its complement in this paper we introduce (1, 2)*-locally closed set and (1, 2)*-separated set that is defined to obtain an improved result which gives that union of any two (1, 2)*-locally closed sets is again a (1, 2)*-locally closed set. We also established a relationship of (1, 2)*-regular open set [9] with (1, 2)*-Locally closed set. As an application of (1, 2)*-locally closed set we study (1, 2)*-LC-continuity. Lastly we introduce (1, 2)*-contracontinuous function and we investigate its relationship with (1, 2)*-LC-continuity.

Throughout this paper , , and denote the bitopological spaces , , and , respectively, on which no separation axioms are assumed. The concept of (1, 2)*-continuous function from a bitopological space into another bitopological space was defined as follows.

A function is said to be (1, 2)*-continuous if and only if inverse image of every -open set in is -open set in . Now we study the following definitions for ready references.

Definition 1. A subset of a bitopological space is called a (1, 2)-locally closed [4] if where is a -open set in and is -closed in .

Definition 2. Let be a subset of . Then is called -open [8] if where and .
The complement of -open set is called -closed set.

Definition 3. Let be a subset of a bitopological space . Then(i)(1, 2)*--open [6] if ,(ii)-closure of denoted by [8] is defined as the intersection of all -closed sets containing ,(iii)-interior of denoted by -int() [8] is defined as the union of all -open sets contained in ,(iv)(1, 2)*-preopen [8] if -int(-cl()),(v)(1, 2)*-regular open [9] if -int(-cl()).
The family of all -open (resp., -closed, (1, 2)*-regular open, (1, 2)*-preopen, and (1, 2)*--open) set of is denoted by (1, 2)*- (resp., (1, 2)*-, (1, 2)*-RO(), (1, 2)*-PO(), and (1, 2)*--).

Remark 4. Note that the collection of all -open subsets of need not necessarily form a topology.

Now using -open set and its complement in Section 2 we define (1, 2)*-locally closed sets and study their properties. Then we study a new form of (1, 2)*-continuity, namely, (1, 2)*-LC-continuity, and study the interrelationship between (1, 2)*-contracontinuity and (1, 2)*-LC-continuity.

2. -Locally Closed Sets

Definition 5. A subset of a bitopological space is called (1, 2)*-locally closed if where - and -, or equivalently if , for some .

Note 1. The collection of all (1, 2)*-locally closed set of is denoted by (1, 2)*-LC.

Example 6. Let and . Then (1, 2)*- and (1, 2)*-.
Hence (1, 2)*-.

Remark 7. Every (1, 2)*-open set and (1, 2)*-closed set is (1, 2)*-locally closed, but the converse may not be true as seen in the following example.

Example 8. Let , , and . Then (1, 2)*- and (1, 2)*-. Now is neither -open set nor a -closed set even though is (1, 2)*-locally closed set as .

Proposition 9. Every -open (resp., -closed) set and -open (resp., -closed) sets is (1, 2)*-locally closed set.

Proof. Since every -open (resp., -closed) set and -open (resp., -closed) set is (1, 2)*-open (resp., (1, 2)*-closed) set, hence they are (1, 2)*-locally closed set.
We can note that intersection of a (1, 2)*-locally closed set and -open (or -closed or -open or -closed) set is again a (1, 2)*-locally closed set.

Remark 10. Converse of Proposition 9 may not be true in general as shown in the following example.

Example 11. Let , , and .
We get (1, 2)*- and (1, 2)*-.
Here is a (1, 2)*-locally closed set, but is not a -open (resp., -closed) sets. Again is a (1, 2)*-locally closed set, but is not a -open (resp., -closed) sets.

Theorem 12. For a subset A of the following conditions are equivalent:(i)(1, 2)*-,(ii) for some -open set ,(iii) is -closed,(iv) is -open.

Proof. (i) (ii). Let (1, 2)*-LC. Then there exist a -open set and a -closed set of such that . Clearly . Since , .
Thus
(ii) (iii). Let -
Now,
(Where, is denoted by complement of )   -open set. Hence is a -closed set.
(iii) (iv). Since is -closed,
we get .
Thus is -open.
(iv) (i). Since is -open, this implies is -open set. Thus is -open. Hence is (1, 2)*-locally closed set (Since every -open set is (1, 2)*-locally closed set).

Proposition 13. Every (1, 2) locally closed set is (1, 2)*-locally closed.

Proof. Proof is obvious.

Remark 14. Converse of the above proposition may not be true as seen in the following example.

Example 15. Let , , and . We get (1, 2)*-, (1, 2)-, , and .
Here is (1, 2)*-locally closed but not (1, 2)-locally closed set.

Theorem 16. A subset of is (1, 2)*-locally closed if and only if is the union of a -open set and a -closed set.

Proof. Proof follows from the definition.

Remark 17. The union of any two (1, 2)*-locally closed sets may not be a (1, 2)*-locally closed set as shown in the following example.

Example 18. Let , , and .
We get (1, 2)*- and (1, 2)*-.
Here and are two (1, 2)*-locally closed sets, but is not a (1, 2)*-locally closed set.

Definition 19. Any two subsets and in a bitopological space are said to be (1, 2)*-separated sets if .

Proposition 20. Let and be any two subsets of . Suppose that the collection of all (1, 2)*-open sets is closed under finite intersection. Let -LC(), and if and are (1, 2)*-separated set in , then -LC().

Proof. By Theorem 12 there exist -open sets and of such that -cl() and -cl(). Put and - as -open subsets of . Then -cl, -cl, -cl, and -cl.
Consequently , as is -open set in , showing that .

Proposition 21. In any bitopological space intersection of any two (1, 2)*- locally closed is a (1, 2)*-locally closed set.

Proof. The proof is straightforward.

Studying the properties of (1, 2)*-locally closed set we have the idea to define a new concept in a bitopological space , namely, infra bitopology, and we also introduce (1, 2)*-LC-continuity using (1, 2)*-locally closed sets.

Definition 22. Let be a bitopological space. Any collection of subsets of (strongly related with -open set) is called an infra bitopology on , if (i), ;(ii) is closed under finite intersection.
Then the space is called an infra bitopological space.

Theorem 23. Let be a bitopological space. Then the collections of all (1, 2)*-locally closed sets of denoted by (1, 2)*-LC() are forms an infra bitopology on .

Proof. Proposition 21 serves the purpose.

Definition 24. The infra bitopology defined above is called (1, 2)*-LC infra bitopology and the space (, (1, 2)*-LC()) is called LC-infra bitopological space.

Example 25. Let , , and . We get (1, 2)*-, (1, 2)*-, and (1, 2)*-LC, .
Here (1, 2)*-LC() and collection of all (1, 2)*-locally closed set is closed under finite intersection. Hence the collection of all (1, 2)*-locally closed forms an LC-infra bitopological space on .

Theorem 26. If (1, 2)*-LC() and (1, 2)*-, then (1, 2)*-LC.

Proof. It is obvious because every (1, 2)*-closed set is (1, 2)*-locally closed set and intersection of two (1, 2)*-locally closed sets is also a (1, 2)*-locally closed.

Definition 27. A subset of is called (1, 2)*-dense (resp., (1, 2)*-nowhere dense) set if and only if -cl (resp., -.

Theorem 28. Any (1, 2)*-dense subset of is (1, 2)*-locally closed if and only if it is -open.

Proof. Let us assume that be any (1, 2)*-dense set in .
Now by the given hypothesis -cl() for some , this implies that is a -open set.
Conversely let be any -open set in . Now since every -open set is (1, 2)*-locally closed set, therefore is (1, 2)*-locally closed set.

Theorem 29. Any (1, 2)*-preopen set is (1, 2)*-locally closed if and only if it is -open.

Proof. Suppose that is a (1, 2)*-preopen set.
Now by the given hypothesis -cl() for some (1, 2)*-, since is a (1, 2)*-preopen set, therefore -int-()-cl-open set.
Converse the part that directly follows from Theorem 28.

Proposition 30. Any (1, 2)*--open set is (1, 2)*-locally closed if and only if it is -open.

Proof. Since every (1, 2)*--open set is (1, 2)*-preopen set and hence the proof is followed from Theorem 29.

Theorem 31. If any (1, 2)*-dense subset of is -open, then every subset of is (1, 2)*-locally closed.

Proof. Let be any (1, 2)*-dense subset in . Therefore .
Again let be the -cl(); that is, -cl().
Now .
This implies that is (1, 2)*-locally closed set.

Theorem 32. Any (1, 2)*-regular open set in is a (1, 2)*-locally closed set.

Proof. Let be a (1, 2)*-regular open set in . Then -int(-cl()).
Therefore -int(-cl() -; is a -open set.
Hence -LC(X).

The converse is not true in general as seen in the following example.

Example 33. Let , and .
We get (1, 2)*-,  ,  , and .

Definition 34. A mapping is said to be (1, 2)*-LC continuous if and only if the inverse image of every -open set in is (1, 2)*-locally closed in .

Proposition 35. If is (1, 2)*-LC continuous and is (1, 2)*-continuous, then (gof) is (1, 2)*-LC continuous.

Proof. Proof is obvious.

Theorem 36. A function is (1, 2)*-continuous if it is (1, 2)*-LC continuous and the preimage of every -open set is (1, 2)*-dense in .

Proof. Let be any -open set in . Since is (1, 2)*-LC continuous, then () is (1, 2)*-locally closed in . By hypothesis -cl. Then is -open in . Hence is (1, 2)*-continuous (by Theorem 28).

Definition 37. A function is called (1, 2)*-LC- irresolute if and only if inverse image of every (1, 2)*-locally closed set in is (1, 2)*-locally closed set in .

Theorem 38. Let and be any two functions; then(i)(gof) is (1, 2)*-LC-irresolute if is (1, 2)*-LC-irresolute and f is (1, 2)*-LC-irresolute;(ii)(gof) is (1, 2)*-LC continuous if is (1, 2)*-LC- continuous and is (1, 2)*-LC-irresolute.

Proof. (i) Let be (1, 2)*-locally closed set in . Since is (1, 2)*-LC-irresolute, then is (1, 2)*-locally closed set in (). As is (1, 2)*-LC-irresolute so is (1, 2)*-locally closed set in . Hence (gof) is (1, 2)*-LC-irresolute.
(ii) Let be -open set in . Since is (1, 2)*-LC-continuous, so is (1, 2)*-locally closed set in . Also is (1, 2)*-LC-irresolute so is (1, 2)*-locally closed set in . Therefore (gof) is (1, 2)*-LC- continuous.

Remark 39. Every (1, 2)*-LC-continuity may not be (1, 2)*-LC-irresolute as seen in the following example.

Example 40. Let , , and .
Then (1, 2)*-.
Let and .
Then (1, 2)*-, , and .
Define as an identity map. Then is a (1, 2)*-LC-continuity but not a (1, 2)*-LC-irresolute since (1, 2)*-LC().

3. -Contracontinuity

Definition 41. A mapping is said to be (1, 2)*-contracontinuous if the inverse image of every -open set in is -closed set in .

Theorem 42. For a function the following conditions are equivalent:(i) is (1, 2)*-contracontinuous,(ii)for each and each -closed set in with there exists a -open set in such that and ,(iii)the inverse image of each -closed set in is -open set in .

Proof. Proof is obvious.

Proposition 43. Every (1, 2)*-contracontinuous function is (1, 2)*-LC-continuous.

Proof. Proof is obvious from the definitions.

Remark 44. Converse of the above proposition may not be true in general as shown in the following example.

Example 45. Let and .
Let .
Also let and .
We get (1, 2)*-, , and .
If : is defined by , , and , then any one can prove that is (1, 2)*-LC-continuous, but it is not a (1, 2)*-contracontinuous function as is (1, 2)*-locally closed but it is not -closed set.
We know that in a topological space the concept of locally indiscrete space is defined as follows.
A topological space is called locally indiscrete space if every open set in is closed set in . Following the above definition we now define the (1, 2)*-locally indiscrete space in a bitopological space as follows.
A bitopological space is said to be (1, 2)*- locally indiscrete space if every -open set in is -closed set in .

Theorem 46. If a function is (1, 2)*-continuous and is (1, 2)*-locally indiscrete space, then is (1, 2)*-contracontinuous.

Proof. Let us consider an arbitrary -open set in ; then is -open set in as is (1, 2)*-continuous function. Again is (1, 2)*-locally indiscrete so is -closed set in . Hence is (1, 2)*-contracontinuous.

Theorem 47. Let and be two functions; then (i)(gof) is (1, 2)*-contracontinuous if is (1, 2)*-continuous and is (1, 2)*-contracontinuous,(ii)(gof) is (1, 2)*-contracontinuous if is (1, 2)*-contracontinuous and f is (1, 2)*-continuous,(iii)(gof) is (1, 2)*-contracontinuous if and is (1, 2)*-continuous and is (1, 2)*-locally indiscrete.

Proof. (i) Let be any -open set in . Since is (1, 2)*-continuous, then is -open set in and since is (1, 2)*-contracontinuous, so is -closed in . Thus is -closed set in for an arbitrary -open set in implying that (gof) is (1, 2)*-contracontinuous.
(ii) Proof is the same as (i).
(iii) Let be a -open set in . Since is (1, 2)*-continuous so is -open set in and since is (1, 2)*-locally indiscrete, so is -closed set in . Again since is (1, 2)*-continuous, so is -closed set in . Thus is -closed set in for an arbitrary -open set in which is implying that (gof) is (1, 2)*-contracontinuous.

Theorem 48. Let and be two (1, 2)*-contracontinuous functions; then(i)(gof) is (1, 2)*-continuous function;(ii)(gof) is (1, 2)*-contracontinuous function if is (1, 2)*-locally indiscrete.

Proof. (i) Let be an arbitrary -open set in . Since is (1, 2)*-contracontinuous so is -closed set in . Again is (1, 2)*-contracontinuous; thus is -open set in . Hence is -open in for any -open set in . So (gof) is (1, 2)*-continuous function.
(ii) From the above proof (i) we see that is -open in for any -open set in . But is (1, 2)*-locally indiscrete and consequently is -closed set in for any -open set in . Hence (gof) is (1, 2)*-contracontinuous function.

4. Conclusion

From the above study it is clear that in general (1, 2)*-dense set may not be a (1, 2)*-locally closed set, but a (1, 2)*-dense set is a (1, 2)*-locally closed set if and only if it contains the neighbourhood of all its points. Also it is proved that every (1, 2)*-locally closed set is -closed set if it is (1, 2)*-nowhere dense set in . There is a scope to study the concept of strongly and perfectly continuous mapping in a bitopological space and interrelationship between stronger form and weaker form of (1, 2)*-continuity and (1, 2)*-LC-continuous mapping.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to the learned reviewer for his appreciation and valuable comments.