Theta Omega Topological Operators and Some Product Theorems

Al Ghour, Samer; El-Issa, Salma

doi:https://doi.org/10.1155/2021/6438053

Mathematical Problems in Engineering

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Volume 2021 | Article ID 6438053 | https://doi.org/10.1155/2021/6438053

Theta Omega Topological Operators and Some Product Theorems

Samer Al Ghour¹and Salma El-Issa¹

Academic Editor: Ewa Rak

Received06 Jul 2021

Revised01 Aug 2021

Accepted27 Oct 2021

Published16 Nov 2021

Abstract

We introduce and investigate the concepts of -limit points and -interior points, and we use them to introduce two new topological operators. For a subset of a topological space , denote the set of all limit points of (resp. -limit points of , -limit points of , interior points of , -interior points of , and -interior points of ) by (resp. , , , , and ). Several results regarding the two new topological operators are given. In particular, we show that lies strictly between and and lies strictly between and . We show that (resp. and ) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.

1. Introduction

In 1943, Fomin [1] introduced the notion of -continuity. For the purpose of studying the important class of -closed spaces in terms of arbitrary filterbases, the notions of -open subsets, -closed subsets, and -closure were introduced by Velicko [2] in 1966, in which he showed that the family of -open sets in a topological space forms a topology on denoted by (see also [3]). The work of Velicko is continued by [3–26] and others. Hdeib [27] introduced the class of -closed sets by which he introduced and investigated the notion of -continuity. The family of all -open sets in is denoted by . It is known that is a topology on which is finer than . Research related to -open sets is still a hot area of research [28–36]. In 2017, Al Ghour and Irshidat [37] introduced -open subsets, -closed subsets, and -closure utilizing the topological spaces and . It is proved in [37] that forms a topology on which lies between and , and that if and only if is -regular. Also, in [37], topological spaces were characterized via -open sets. Authors in [35] introduced -connectedness and some new separation axioms. Also, research in [37] was continued by various researchers in [28–31]. The notion of interior operators is important in the axiomatization of modal logics. Judging from the importance of limit points in mathematical analysis, introducing a new limit point notion in any topological structure is still a hot area of research. The first goal of this paper is to introduce and investigate the concepts of -limit points and -interior points.

In general topology, several topological properties are not finitely productive, such as paracompactness, strong paracompactness, Lindelöfness, and metacompactness. The area of research regarding the problem “What conditions on and to insure that their product has property ”is still hot[38–45]. The second goal of this paper is to introduce several product theorems concerning metacompactness.

2. Preliminaries

From now on TS will denote topological space for simplicity. Let and be TSs and let with as nonempty. Then, is called -open set in [27] if for each , there is and a countable set such that . The relative topology on is denoted by , and the product topology on is denoted by . The closure of in (resp. , ) is denoted by (resp. , ). A point is in -closure of [2] () if for every with , . is called -closed [2] if . The complement of a -closed set is called a -open set. It is known that if and only if is regular. A TS is called -regular [37] if for each closed set in and , there exist and such that , , and . In [37], the author defined -closure operator as follows: a point is in -closure of if for any with we have . is called -closed if . The complement of a -closed set is called a -open set. A TS is called metacompact [46] if every open cover of has a point-finite open refinement.

The following sequence of definitions and theorems will be used in the sequel.

Definition 1 (see [47]). A TS is called locally countable if for each , there is such that is countable and .

Definition 2. (see [48]). A TS is called antilocally countable if each is uncountable.

Definition 3. (see [9]). Let be a TS . A point is called -limit point of if for each with , . The set of all -limit points of is called the -derived set of and is denoted by .

Definition 4. (see [9]). Let be a TS and . A point is called a -interior point of if there exists such that . The set of all -interior points of is called the -interior of and is denoted by .

Theorem 1 (see [37]). If is locally countable and , then .

Theorem 2 (see [37]). If is antilocally countable and , then .

Theorem 3 (see [37]). For any TS , .

Theorem 4 (see [2]). A TS is regular if and only if .

Theorem 5 (see [37]). Let be a TS and . Then, is -open set if and only if for each , there exists such that.

Definition 5. Let and be TSs and let . Then,(a) is called -scattered if every , there is and a compact set such that [49](b) is called strongly placed in if for every and with , there are and such that [50](c) is called scattered relative to if for each , there exists and such that is Lindelöf and strongly placed in [50]It is well known that if and are TSs and is -scattered, then is scattered relative to but not conversely.

Definition 6. (see [51]). A Hausdorff TS is called ultraparacompact if every open cover of has a locally finite clopen refinement.
Ellis [51] showed that a Hausdorff space is ultraparacompact if every open cover has a pairwise disjoint open refinement.

Theorem 6 (see [52]). Let be closed and continuous with regular. If is metacompact and is Lindelöf for each , then is metacompact.

Theorem 7 (see [50]). For any two TSs and , is strongly placed in if and only if the projection is closed.

Theorem 8 (see [35]). Let and be TSs and let . If is strongly placed in and , then is strongly placed in .

3. Theta Omega Limit Points

In this section, we explore the concept of -limit points of a set and study its fundamental properties.

Definition 7. Let be a TS and . A point is called -limit point of if for each with , .
The set of all -limit points of is called the -derived set of and is denoted by .
The following result shows that -derived set of a set contains the derived set of and contained in the -derived set of .

Theorem 9. Let be a TS . The derived set of is denoted by . Then, .

Proof. To see that , let , then there exists such that and . By Theorem 3, and so . Therefore, we have . To see that , let , then there exists such that and . By Theorem 3, and so . Therefore, we have .
The following example shows that the equality of each of the inclusions in Theorem 9 does not hold in general.

Example 1. (Example 2.26 of [37]). Let and let . It is proved in [37] that and . Let . Then, , , and .
Under the condition “regularity,” the -derived set, the derived set, and the -derived set are all equal.

Theorem 10. Let be a regular TS and . Then, .

Proof. It follows from Theorems 3, 4, and 9.
“Local countability” is a sufficient condition for the -derived set and the derived set to be equal to each other:

Theorem 11. Let be a locally countable TS and . Then, .

Proof. By Theorem 9, we have . To see that , suppose to the contrary that there is . Since , there is such that . By Theorem 1, and so . We conclude that , a contradiction.
“Antilocal countability” is a sufficient condition for the -derived set and the -derived set to be equal to each other.

Theorem 12. Let be an antilocally countable TS and . Then, .

Proof. By Theorem 9, we have . To see that , suppose to the contrary that there is . Since , there is such that . By Theorem 2, and so . We conclude that , a contradiction.
In Theorems 13–16, we give some natural properties for -derived set.

Theorem 13. Let be a TS. If , then .

Proof. Let , there exists such that and . Since , then and hence . It follows that .

Theorem 14. Let be a TS, and let be subsets of . Then, .

Proof. By Theorem 13, and . Therefore, . Now, let , then there exist -open sets such that , , and . Let . Then, andThus, .

Theorem 15. Let be a TS, and let be subsets of . Then, .

Proof. By Theorem 13, and . Then, .
The following example shows that the inclusion in Theorem 15 can not be replaced by equality in general.

Example 2. (Example 2.26 of [37]). Let and . It is proved in [37] that . Let and . Then, and , so . On the other hand, .

Theorem 16. Let be a TS and . Then, .

Proof. Let . Let with . Since , . Choose . Since and , then . Choose . Since and , then . Thus, and hence .
The following example shows that the inclusion in Theorem 16 cannot be replaced by equality in general.

Example 3. Let and . Let . It is proved in [37] that . By Example 2, . On the other hand,

4. Theta Omega Interior Points

In this section, we explore the concept of -interior points of a set and study its fundamental properties.

Definition 8. Let be a TS and . A point is called a -interior point of if there exists such that . The set of all -interior points of is called the -interior of and is denoted by .
The following result shows that the -interior of a set contains the -interior and contained in the -interior of .

Theorem 17. Let be a TS and . Then, .
The following example shows that each of the two inclusions in Theorem 17 cannot be replaced by equality in general.

Example 4. (Example 2.26 of [37]). Let and let . Let and . It is proved in [37] that and . We have but . Also, we have but .

Theorem 18. Let be a TS and . If such that , then .

Proof. It follows directly from the definition of.
-interior is always open.

Theorem 19. Let be a TS and . Then, is -open.

Proof. By the definition of and Theorem 18, for every , there exists such that . By Theorem 5, it follows that is -open.
The following is a characterization of -open via -interior.

Theorem 20. Let be a TS and . Then, is -open if and only if .

Proof. Necessity: suppose that is a -open set. By the definition, we have . To see that , let . By Theorem 5, there exists such that . Then, .
Sufficiency: suppose that . Then, by Theorem 19, is -open.
The results in the rest of this section are some natural properties of -interior.

Theorem 21. Let be a TS and . Then, .

Proof. Follows from Theorem 20.

Theorem 22. Let be a TS and . Then, .

Proof. To see that , let . Then, there is such that and , so we have . This shows that . To see that , let . Then, , and so there is such that . Therefore, we have , and so .

Theorem 23. Let be a TS and . Then, .

Proof. By Theorem 22,

Theorem 24. Let be a TS and let . Then, .

Proof. Let . Then, there exists such that . Since , then . Thus, .

Theorem 25. Let be a TS and let and be subsets of . Then, .

Proof. By Theorem 24, we have and . Thus, .

Theorem 26. Let be a TS, and let be subsets of . Then, .

Proof. By Theorem 24, we have and . Thus, . To see that , let . Then, there exist such that and . Let . Then, and . It follows that .

5. Metacompactness Product Theorems

In this section, we introduce several product theorems concerning metacompactness.

The following result will be used in the proof of Theorems 28 and 29.

Theorem 27. Let and be metacompact TSs. If for every there exists such that and is metacompact, then is metacompact.

Proof. Let be an open cover of . For every , choose such that and is metacompact. Since is an open cover of the metacompact TS , then it has a point-finite open refinement . For each , is metacompact and has as an open cover, and hence has a point-finite open refinement . It is not difficult to see that is a point-finite open refinement of . It follows that is metacompact.
The following two product theorems concerning metacompactness will be used in the proof of Theorem 31 which is the main result of this section:

Theorem 28. Let and be regular metacompact TSs. If for every there exists such that , is strongly placed in , and is Lindelöf, then is metacompact.

Proof. For each , choose such that , is strongly placed in , and is Lindelöf. For every , is strongly placed in and so by Theorem 6, the projection function is a closed function. For every , is Lindelöf and since , then is Lindelöf. For every , is metacompact and so by Theorem 6, is metacompact. Thus, by Theorem 27, we have is metacompact.

Theorem 29. Let and be metacompact TSs and let such that is closed in , is Lindelöf, and is strongly placed in . If for all there is such that is metacompact, then is metacompact.

Proof. Let be an open cover of . For each , is Lindelöf with , and so there exists such that is countable and . Since is strongly places in , then for every , there exist and such that . Since is an open cover of the metacompact TS , then it has a point-finite open refinement . For each , choose such that . Then, by Theorem 27 and the assumption, it is not difficult to see that is metacompact. Since is an open cover of , then it has a point-finite open refinement . It is not difficult to check thatis a point-finite open refinement of . Therefore, is metacompact.

Theorem 30. Let be ultraparacompact and be metacompact. Suppose there exists such that is closed in and for every there exists such that , is strongly placed in , and is Lindelöf, and for every , there is such that and is metacompact. Then, is metacompact.

Proof. By assumption there exists such that for all , is strongly placed in and is Lindelöf, and . Since is ultraparacompact, then has a pairwise disjoint open refinement . For every , choose such that . Put . Since is ultraparacompact and is an open cover of , then has a pairwise disjoint open refinement . For every , meets at most one member of . For every , let , then or ; for some, it follows thatis closed inandis Lindelöf and by Theorem 8; it is strongly placed in. By the assumption on and Theorem 29, we conclude that is metacompact. Since is metacompact, then is metacompact.
Now, we are ready to state the main result of this section.

Theorem 31. Let be ultraparacompact and be regular and metacompact such that is scattered relative to the product , then is metacompact.

Proof. Denote by and = {: there is no such that and is strongly placed in and closure is Lindelöf }. If there is an ordinal such that has been defined and , then . If is a limit ordinal, then . Since is scattered relative to , then there exists an ordinal such that .
The proof proceeds by transfinite induction on . If , then for every there exists such that and is strongly placed in and closure . And by Theorem 28, is metacompact. If , then for every point there exists such that , is strongly placed in , and is Lindelöf and if , choose a clopen set such that . Since , then is scattered relative to and is ultraparacompact, and by the inductive assumption, it follows that is metacompact.
If for the limit ordinal , then the open cover has a pairwise disjoint open refinement . For each , choose such that . Therefore, , and hence is metacompact. Since , it follows that is metacompact.

Corollary 1. The product of an ultraparacompact -scattered TS with a metacompact regular TS is again metacompact.
By the end of this paper, the authors found it is suitable to raise the following open question.

Question 1. Let and be regular and metacompact TSs such is scattered relative to the product . Is metacompact?

6. Conclusion

In this work, the research via -open sets is continued by introducing the notions of -limit points and -interior points. Several relationships regarding these two notions are introduced. Moreover, several product theorems concerning metacompactness are given. In future studies, the following topics could be considered: (1) define -border, -frontier, and -exterior of a set using -open sets and (2) try to solve Question 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Samer Al Ghour and Salma El-Issa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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