-Regular Sets in Topology and Generalized Topology
We define and study a new class of regular sets called -regular sets. Properties of these sets are investigated for topological spaces and generalized topological spaces. Decompositions of regular open sets and regular closed sets are provided using -regular sets. Semiconnectedness is characterized by using -regular sets. -continuity and almost -continuity are introduced and investigated.
In general topology, repeated applications of interior and closure operators give rise to several different new classes of sets. Some of them are generalized form of open sets while few others are the so-called regular sets. These classes are found to have applications not only in mathematics but even in diverse fields outside the realm of mathematics [1–3].
Due to this, investigations of these sets have gained momentum in the recent days. Császár has already provided an umbrella study for generalized open sets in his latest papers [4–7]. In this paper, we introduce and study a new class of sets, called -regular sets, using semi-interior and semiclosure operators. Initially, we define them for a broader class, that is, for generalized topological spaces and discuss their various properties. Interrelationship of -regular sets with other existing classes such as semiopen sets, regular open sets, -sets, -sets, -sets, and -sets has been studied. A characterization of semiconnectedness is also provided using -regular sets. Moreover, -regular sets, where , of a generalized topological space are studied using -regular sets. In the last two sections, -regularity is studied in the domain of general topological spaces. Here several decompositions of regular open sets and regular closed sets are provided using -regular sets. In the last section, -continuity and almost -continuity are defined and interrelationship of almost -continuity with other existing mappings such as -map, graph mapping, almost precontinuity, and almost -continuity is investigated.
First we recall some definitions and results to be used in the paper.
Definition 1 (see ). Let be a nonempty set. A collection of subsets of is called a generalized topology (in brief, ) on if it is closed under arbitrary unions. The ordered pair is called generalized topological space (in brief, ).
Since an empty union amounts to the empty set, always belongs to . However, need not be a member of . The members of are called -open while the complements of -open sets are called -closed. The largest -open set contained in a set is called the interior of and is denoted by , whereas the smallest -closed set containing is called the closure of and is denoted by . For , we have, , and whenever , we have . These properties will be used in the text without any further mention.
Remark 2. Although Császár provided the definition of a generalized topology, similar notions existed prior to Császár’s work also. Peleg  in 1984 defined a similar structure which he named “semitopology” on . The corresponding “semitopological closure” is found to be monotone, enlarging, and idempotent. A subfamily of the power set of which is closed under nonempty intersection has been studied in the literature under the name of “intersection structure” ; the corresponding topped intersection structure is known as closure system.
Definition 3. Let be a . Then a subset of is called(i)semiopen  if ;(ii)--open  if ;(iii)-preopen  if ;(iv)--open  if ;(v)-set  if ;(vi)regular open (resp., regular closed)  if (resp., ).
The complement of a -semiopen (resp., --open, -preopen, and --open) set is called -semiclosed (resp., --closed, -preclosed, and --closed).
The families of regular open, regular closed, and -sets on are denoted by and , respectively.
The intersection of all -semiclosed sets containing a set is called the semiclosure of and is denoted by . Dually, the semi-interior of is defined to be the union of all -semiopen sets contained in and is denoted by .
Theorem 4 (see ). In a with , one has (i),(ii),where and denote the semi-interior and semiclosure of , respectively.
3. -Regular Sets in Generalized Topological Spaces
Although the following discussion is provided for generalized topological space, it is also valid for topological spaces as every topological space is a generalized topological space as well.
Definition 5. A subset of a generalized topological space is said to be -regular if .
The class of all -regular sets in is denoted by .
Lemma 6. For a generalized topological space , and both are -regular sets.
Proof. Let be a . Then . Now, consider . If , then and we are done. If not, then let , the largest -open set in . If then again and is -regular. Let, if possible, , and then is a -open set in and hence is again a -open set containing , which leads to a contradiction. Hence is -regular. Similarly, it can be shown that is -regular.
Thus one can say that the family of -regular sets forms an -structure . However, the family is not closed under finite union as well as finite intersection.
We have the following example.
Example 7. Let be the set of all real numbers with the usual topology . Take and ; both are -regular sets. But is not a -regular set because .
Similarly, take and ; both are -regular sets. But is not a -regular set because .
Theorem 8. Every -regular set is -semiopen.
Proof. Let be a -regular set. Then . Therefore . Hence is -semiopen.
But converse of this result is not true in general. We have the following example.
Example 9. Let with . Then is a -semiopen set because is -open. But is not -regular because .
Remark 10. The notions of -regular sets and -open sets are independent of each other. Similarly, -regular sets are different from --open, -preopen, and --open sets as well.
In Example 9 mentioned above, it is clear that is a -open set; thus it is --open, -preopen, and --open. But is not -regular.
Example 11. Let be the set of all real numbers with its usual topology . Then is a -preopen set in as . But is not -regular because .
Thus -regular sets are independent of -preopen sets.
Example 12. Let with . Then is a -regular set. But is neither -open nor -preopen or --open in because and .
Hence from the above examples, it is clear that -regular sets are independent of -open sets, -preopen sets, and --open sets as well.
Theorem 13. Let be a generalized topological space. Then a subset is -regular if it is both -semiopen and -semiclosed.
Proof. Let be -semiopen as well as -semiclosed. Then and . Therefore . Hence is -regular.
Theorem 14. Every regular open set is -regular.
Proof. Let be a regular open set. Therefore is -open and hence is -semiopen. Again as is regular open, we have . Thus . Hence is -semiclosed. Therefore by Theorem 13, it follows that is -regular.
But the converse of the above result is not true in general. We have the following example.
Example 15. In Example 12, is -regular. But is not regular open as .
Theorem 16. Every regular closed set is -regular.
Proof. Let be regular closed. Therefore is -closed and -semiclosed. As is regular closed, and hence . Thus is both -semiopen and -semiclosed. Hence is -regular in .
But the converse of the above theorem is not true in general. We have the following example.
Example 17. Let be the set of all real numbers with the usual topology . Then is not a regular closed set. But it is -regular.
In this section, we provide two interesting characterizations of -regular sets.
Theorem 18. Let be a generalized topological space. Then a subset is -regular if and only if .
Proof. For a generalized topological space, we know that Therefore, we have Consider . But ; thus . Therefore . Thus, , as (shown above). Thus is -regular if and only if .
Corollary 19. A set is -regular if and only if it is both -semiopen and -semiclosed.
Corollary 20. If a set is -regular then its complement is also -regular.
The above implications may be represented diagrammatically in Figure 1.
We complete this section by providing few interesting decompositions and results using -regular sets.
Definition 21 (see ). Let be a generalized topological space and . is said to be -regular (resp., -regular, -regular, and -regular) if (resp., , , and ).
The following theorems provide the relationships between -regular sets where and -regular sets. Here it may be mentioned that -regular sets and -regular sets are nothing but regular open sets and regular closed sets defined in Definition 3.
Theorem 22. Let be a GTS and . Then every -regular set, where , is -regular.
Proof. Let be -regular. Then . Hence by Theorem 14, it is -regular. Similarly if is -regular, then . Hence by Theorem 16, is -regular as well. Further, we know that a set is -regular if and only if it is an -regular  and a set is -regular if and only if it is -regular . Therefore every -regular set, where , is -regular.
Theorem 23. Let be a GTS and . Then the following hold. (i) is -regular if and only if is -regular and -closed if and only if is -regular and -closed.(ii) is -regular if and only if is -regular and -closed if and only if is -regular and -closed.
Proof. (i) If is -regular then it is clear that is -regular and -closed which implies that is -closed as well. Suppose is -regular and -closed. Then ; that is, and . Hence which implies that is -regular.
(ii) It is similar to (i).
Definition 24. A subset of a generalized topological space is called -closed  if whenever and is -open in , and the finite unions of regular open sets are said to be -open.
Theorem 25. Let be a GTS and . Then the following hold. (i)Every -regular set is -closed set.(ii) is -regular if and only if is -open and -regular.
Proof. (i) Let be a -regular set. Let be any -open set in containing , . Since is -regular and every -regular set is -semiclosed in , therefore, . Hence is -closed.
(ii) If is -regular then clearly is -open and -regular set. Conversely, let be -open and -regular set. Hence is -semiclosed; that is, ; that is, . Since is -open set, therefore is -preopen and hence . Thus . Hence is a -regular set.
An application of -regular set is the characterization of semiconnectedness .
Definition 26. A generalized topological space is said to be semidisconnected if there exist two -semiopen sets and such that and ; otherwise it is called semiconnected.
Theorem 27. A generalized topological space is semiconnected if and only if it does not contain any proper -regular set.
Proof. Suppose is a proper -regular set in . Then is also a -regular set. Therefore and both are -semiopen sets such that and . Hence is semidisconnected.
Conversely, let be a semidisconnected space. Then there exist two -semiopen sets and such that and . Then , a -semiclosed set as well. Hence is -semiopen and -semiclosed both. Therefore is a proper -regular set in .
Our investigations on -regular sets so far have been in the domain of generalized topological spaces. In the remaining part of our paper, we study interrelationships of the notion of -regularity with other existing topological notions. Hence, from now onwards, we confine our investigation to the domain of topological spaces only. Also, following the usual convention of topology, we denote interior and closure of a set by and , respectively, in our discussion. Since a topological space is also a generalized topological space, therefore all the results of this section so far are also valid for topological spaces.
4. Some Decompositions Using -Regular Sets
Apart from semiopen and semiclosed sets, there are several other important generalized forms of open sets and closed sets in topology such as -set, -set, -set, -set, and set. In this section, we study -regular sets in the light of these sets. We also provide some interesting decompositions of regular open and regular closed sets using the notion of -regular sets.
First of all, we provide the following definitions for topological spaces.
Definition 28. Let () be a topological space. A subset is said to be (i)a -set  if ;(ii)a -set  if there is an open set and a -set in such that ;(iii)a -set  if there exists and , such that ;(iv)a -set  if .
Theorem 29. Every -regular set is -set and -set.
Proof. Let be a -regular set. Then . Therefore because .
Thus is -set. Also is an -set as well because .
But the converse is not always true. We have the following examples.
Example 30. Let with topology . Then is a -set because . But is not -regular because is not semiopen as .
Example 31. Let with topology . Then is an -set as . But is not a -regular set because .
Corollary 32. Every -regular set is a -set, -set, and -set.
Proof. It is because every -regular set is -set and -set.
Now, we provide some decompositions of regular closed sets using -regular sets.
Theorem 33. For a topological space, the following are equivalent: (i) is regular closed;(ii) is closed and -regular;(iii) is -closed and -regular;(iv) is preclosed and -regular.
Proof. ii: they are proved in Theorem 16.
iiiii: let be a closed set; then it is -closed because every closed set is -closed.
iiiiv: they are obvious because every -closed set is preclosed as well.
iv: let be a preclosed and -regular set. Then is semiopen. Thus . Since is preclosed, therefore as well. Hence and is hence regular closed.
Now, we proceed to provide decompositions of regular open sets using -regular sets.
Theorem 34. For a topological space, the following are equivalent: (i) is regular open;(ii) is open and -regular;(iii) is preopen and -regular;(iv) is preopen and semiclosed.
Proof. ii: they are proved earlier in Theorem 14.
iiiii: let be an open set. Then it is preopen because every open set is preopen.
iiiiv: as every -regular set is semiclosed therefore is semiclosed.
iv: let be a preopen and semiclosed set. Thus and ; that is, . Hence . Thus is regular open.
Theorem 35. A set is regular open if and only if it is -open and -regular.
Proof. We have already proved in Theorem 34 that every regular open set is -regular and open and hence -open. Conversely, let be -regular; then it is an -set by Theorem 29. Hence is regular open because a set is regular open if and only if it is -open and -set .
5. -Continuity and Almost -Continuity
We first recall the following definitions.
Definition 36. A function is said to be -map  (resp., almost continuous , almost -continuous , almost semicontinuous , and almost precontinuous ) if is regular open, (resp., open, -set, semiopen, and preopen) for every regular open set in .
Now we define -continuity and almost -continuity in the following way.
Definition 37. (a) A mapping is said to be -continuous at a point if for every neighbourhood of there exists a -regular neighbourhood of such that .
(b) A mapping is said to be almost -continuous at a point if for every neighbourhood of there exists a -regular neighbourhood of such that .
A mapping is said to be -continuous (resp., almost -continuous) if it is -continuous (resp., almost -continuous) at each point of .
Since every regular open set is open, therefore every -continuous mapping is almost -continuous.
Theorem 38. For a mapping , the following are equivalent: (i) is -continuous;(ii)inverse image of every open subset of is -regular;(iii)inverse image of every closed subset of is -regular.
Proof. ii: let be any open subset of and let . Then . Therefore there exists a -regular subset in such that and . Thus ; therefore is a -regular neighbourhood of . Hence is -regular.
iiiii: let be any closed subset of . Then is open and therefore is -regular; that is, is -regular. Hence is -regular.
iii: let be open neighbourhood of ; therefore is closed, and consequently is -regular. Thus is also -regular and hence (say). Then is a -regular neighbourhood of such that .
Theorem 39. For a mapping , the following are equivalent: (i) is almost -continuous at ;(ii)for every regular open neighbourhood of , there is a -regular neighbourhood of such that .
Proof. ii: if is almost -continuous at and is a regular open neighbourhood of , then there is a -regular neighbourhood of such that .
ii: it is obvious.
Theorem 40. For a mapping , the following are equivalent: (i) is almost -continuous;(ii)inverse image of every regular open subset of is -regular;(iii)inverse image of every regular closed subset of is -regular;(iv)for each point of and for each regular open neighbourhood of , there is a -regular neighbourhood of such that .
Proof. The proof is the same as Theorem 38.
Theorem 41. If is a -continuous (resp., almost -continuous) map then it is semicontinuous (resp., almost semicontinuous).
Proof. Let be any open (resp., regular open) subset in . Then is a -regular and hence a semiopen set because is -continuous (resp., almost -continuous). Hence is semicontinuous (resp., almost semicontinuous).
Theorem 42. If is an almost -continuous map then the following hold: (i) is -map if and only if it is almost precontinuous;(ii) is -map if and only if it is almost -continuous.
Proof. (i) Let be a -map and let be any regular open subset in . Then is regular open and hence preopen. Thus is almost precontinuous.
Conversely, let be almost precontinuous and almost -regular continuous map. Let be any regular open subset of . Then is preopen as well as -regular. Hence is regular open. Therefore is a -map.
(ii) It is the same as i.
Remark 43. From the above theorem we can conclude that a map is almost continuous (a.c.s) if it is almost precontinuous and almost -continuous.
Theorem 44. Let be a function and let be the graph function defined by , for every . Then is almost -continuous if is almost -continuous.
Proof. Let and containing . Then, we have , . Since is almost -continuous, there exists a -regular set of containing such that . Therefore we obtain and hence is almost -continuous.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors express sincere thanks to the referees of the paper. The suggestions provided by one of the referees have greatly improved the presentation of the paper. This work is a part of a research work financed by the University Grant Commission (India).
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