Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

Al-shami, T. M.

doi:https://doi.org/10.1155/2021/5517702

Journal of Applied Mathematics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5517702 | https://doi.org/10.1155/2021/5517702

Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

T. M. Al-shami¹

Academic Editor: Dan Huang

Received29 Jan 2021

Revised03 Apr 2021

Accepted29 Oct 2021

Published11 Nov 2021

Abstract

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra spaces compared with their counterparts on general topology.

1. Introduction and Preliminaries

Topological space has been generalized in many manners. They can be classified into three main types; the first one was obtained by strengthening or weakening the conditions of a topology such as Alexendroff topology [1], supra topology [2], and generalized topology [3]. The second one was given by adding newly mathematical structures to the topology such as ordered topology [4], ideal topology [5], and bitopology [6]. The third one was established by defining a topology using one of the generalizations of crisp sets such as fuzzy topology [7] and soft topology [8]. Later on, wide generalizations were constructed by combining two or more types of the previous ones such as fuzzy bitopology [9], ideal ordered bitopology [10], and soft ordered topology [11].

Mashhour et al. [2] introduced the supra topology concept by deleting only the intersection condition. In analogy with topology, they studied the concepts of interior and closure operators, continuity, and separation axioms on supra topology. One of the concrete merits of supra topology is obtaining different examples that give meaning to the concepts and properties defined on a finite set; for example, the only topology defined on a finite set which is a space is the discrete topology, whereas there are several sorts of supra topologies that are a space. Additionally, it is a hard work to find counterexamples which illustrate that spaces do not imply in the cases of ; see for example, [12, 13]. After the emergence of supra topology, many authors have explored various topological notions in the spaces of supra topology such as compact [14, 15] and paracompact spaces [16], neighborhood system [17], separation axioms [18]), operators, and generalized open sets [19, 20]). Some published articles demonstrated that many topological findings are still true on supra topologies and illustrated that some of them are invalid like the distributive property of the closure (resp., interior) operator for the union (resp., intersection) between two sets and a compact set in a Hausdorff space is closed. In general, one can note that all topological properties and characterizations which are related to the intersection operator do not remain valid in supra topological spaces. In conclusion, we point to the existence of several studied extensions of supra topology such as supra ordered topology [21], supra fuzzy topology [22], supra soft topology [23], supra ideal topology [24], supra ordered bitopology [25], and supra soft ordered topology [26].

In the field of applications, the supra topology represents a framework that is general enough to model phenomena and handle practical issues. In [27], the authors elucidated how supra topological frames induced by semiopen sets handled some digital problems. It is well known that the classes of regular sets and semiopen sets define supra topology structures. In [28], these classes were examined to fix or eliminate obstacles on the digital scope.

Our goal in this article is to complete separation axioms reported in [19] by introducing the concepts of supra completely Hausdorff and supra completely regular spaces. We elaborate their master properties and demonstrate the relationships between them by some illustrative examples. Also, we investigate their behaviours with respect to subspaces, homeomorphism maps, and the finite product of spaces.

Now, we mention some definitions and results in which we need to illustrate the obtained findings.

Definition 1 (see [2]). We call as a supra topology on provided that is a subcollection of such that and the arbitrary unions of members of are a member of . A pair is called a supra topological space. Every subset of that belongs to is called supra open, and a set is called supra closed if its complement belongs to .

To refer supra topological spaces, we use the the pairs and .

Definition 2 (see [19]). For a subset of , the union of all supra open sets that is contained in is called the supra interior of (denoted by ); the intersection of all supra closed sets that contains is called the supra closure of (denoted by ).

Definition 3 (see [2]). We call a map continuous if the inverse image of every supra open subset is a supra open subset.

Proposition 4 (see [2]). A map from to is an continuous iff for every .

Definition 5 (see [19]). A supra relative topology on a subset of is given by . A pair is said to be a supra subspace of .

Lemma 6 (see[19]). A subset of , which is a supra subspace of , is supra closed iff there is a supra closed subset of such that .

Definition 7 (see [2, 19]). is said to be: (1) if for any , there is a supra open set containing (only) one of the points or (2) if for any , there are two supra open sets and such that and . If and are disjoint, then, we call supra Hausdorff (or ).(3)Supra regular if for any supra closed set and every element , there are disjoint supra open sets and satisfying that and (4)Supra normal if for any disjoint supra closed sets and , there are disjoint supra open sets and satisfying that and

which is and supra regular (resp., supra normal) is called (resp., ).

Theorem 8 (see [2]). is iff every singleton set is supra closed.

Theorem 9 (see [19]). In , the next statements are identical. (1) is supra regular(2)For each and , there is such that (3)Every supra open set can be written as follows: .

Proposition 10. .

Definition 11. Let be a family of supra topological spaces. We call a base for a supra topology on . A pair is called a finite product supra space.

Proposition 12. Let and be subsets of and , respectively. Then, .

2. Supra Completely Hausdorff Spaces

In this portion, we define a concept of supra completely Hausdorff spaces and investigate its master properties. A number of examples which clarify relationships between this concept and some separation axioms that we consider is provided.

Definition 13. is said to be supra completely Hausdorff (or supra ); if for any , there are two supra open sets and containing and , respectively, such that .

Proposition 14. Every supra space is supra , for .

Proof. We prove the proposition in case of and the other case is obvious. Let . A set is supra closed because is supra . The supra regularity of implies that there are two disjoint supra open sets and containing and , respectively. From Theorem 9, there are supra open sets and such that and . The disjointness of the supra closures of and ends the proof.

Corollary 15. When is supra regular, the next concepts are identical. (1) is supra (2) is supra (3) is supra (4) is supra

The converse of the above result is not true in general as the two examples clarify as follows.

Example 16. We construct a supra topology on as follows: It can be checked that is supra . On the other hand, is a supra closed set and . is not supra because there do not exist two disjoint supra open sets such that one of them includes and the other includes .

Example 17. We construct a supra topology on as follows: Obviously, is supra . In contrast, and the supra closure of any supra open set containing and the supra closure of any supra open set containing have a nonempty intersection, so that is not supra .

Theorem 18. The two concepts are identical if as follows. (1) is supra (2) is supra

Proof. The implication is obvious.
To prove that , let . Then, there are two disjoint supra open subsets and of containing and , respectively. It is well known that and . Now, we have the following two cases: (1) or . Then, is a supra clopen set, so that (2). Then, or . If , then, and are supra clopen sets because they are disjoint, so that . If , then, is a supra clopen set. Thus, The above two cases end the proof that is supra .

Theorem 19. If for every distinct points in there is an continuous function of into the usual topology such that ; then is supra completely Hausdorff.

Proof. Let . By hypothesis, there exists such that and . Since is supra completely Hausdorff, there are two disjoint supra open subsets and of containing and , respectively, such that . Since is an continuous map, and are supra open subsets of containing and , respectively, such that . It follows from Proposition 4 that and , so that . Hence, is supra completely Hausdorff, as required.

Definition 20. A map is said to be open (resp., closed) if the image of any supra open (resp., supra closed) set is supra open (resp., supra closed). If a bijective map is open and continuous, then it is called an homeomorphism.

Proposition 21. A map is closed iff for every .

Proof. Straightforward.

Definition 22. (1)A property that passes from a supra topological space to every supra subspace is called a supra hereditary property(2)A property that is preserved by any homeomorphism map is called a supra topological property

Theorem 23. Let be a bijective open map. If is supra completely Hausdorff, then, is supra completely Hausdorff.

Proof. Let . Then, it follows from the bijective of that there are such that and . Since is supra completely Hausdorff, there are two disjoint supra open subsets and of containing and , respectively, such that . Now, and are supra open subsets of containing and , respectively, such that . Since is a bijective open map, it is also an closed map, so that and . Thus, . Hence, is a supra completely Hausdorff, as required.

Corollary 24. The property of being a supra completely Hausdorff space is a supra topological property.

Lemma 25. Consider that is a subspace of . Then, for each , where is the supra closure operator in .

Proof.

Theorem 26. The property of being a supra completely Hausdorff space is a supra hereditary property.

Proof. Suppose that is a subspace of a supra completely Hausdorff space . Let . Then, there are two disjoint supra open subsets and of containing and , respectively, such that . Now, and are two disjoint supra open subsets of containing and , respectively. From Lemma 25, we obtain . Similarly, . The disjointness of and shows that is a supra completely Hausdorff.

Theorem 27. The finite product of supra completely Hausdorff spaces is a supra completely Hausdorff.

Proof. Consider and as two supra topological spaces.
Suppose that . This means that either or . Say, . Therefore, there are two disjoint supra open subsets and of containing and , respectively, such that . Now, and are two disjoint supra open subsets of containing and , respectively. It follows from Proposition 12 and Proposition 10 that . Hence, is a supra completely Hausdorff.

3. Supra Completely Regular Spaces

In this portion, we first investigate some properties of supra regular and supra normal spaces. Then, we define a concept of supra completely regular spaces. Compared with completely regular on topological spaces, the continuous functions are replaced by continuous functions; however, the codomain remains which is a subspace of the usual topology. We discuss some rudiments of a supra completely regular space with the help of examples and characterize it in many ways.

Theorem 28. If , then, every supra regular space is supra normal.

Proof. Let and be two disjoint nonempty supra closed subsets of a supra regular space . Now, we have the following four cases: (1). Then, is the empty set. Hence, the proof is trivial(2). Since and are disjoint, it must be that is a singleton, so that and are disjoint supra open sets as well(3). Then, or . If , say . Then, . It follows by the regularity of that there are two disjoint supra open subsets and of containing and , respectively. If , then, and are disjoint supra open sets as well(4), say . Then, . It follows by the regularity of that there are two disjoint supra open subsets and of containing and , respectively(5). Then, and are two disjoint supra open subsets of containing and , respectivelyThe above five cases end the proof that is supra normal.

Corollary 29. The next two concepts are identical if . (1) is supra (2) is supra

By the next examples, we illustrate that the converse of Theorem 28 fails and elucidate that the concepts of supra normal and supra regular spaces are independent of each other when .

Example 30. Consider that is a supra topology on . Obviously, is supra normal. In contrast, and is a supra closed set. is not supra regular because there do not exist two disjoint supra open sets separate and .

Example 31. Consider that is a supra topology on . Now, and are supra closed subsets of . One can check that is supra regular. In contrast, is not supra normal because there do not exist two disjoint supra open sets separate and .

The following result is the version of Urysohn’s lemma on supra topology and has a similar proof to Urysohn’s lemma on general topology, so that the proof is omitted.

Theorem 32. is supra normal iff for each disjoint supra-closed sets there exists an continuous map such that for each and for each .

Take into consideration that a supra topology which is defined on is a subspace of the usual topological space .

Definition 33. is said to be (1)Supra completely regular if for each and supra closed set such that if , there exists an continuous map such that and (2)Supra if it is supra completely regular and supra

Proposition 34. The following three statements are identical: (1) is supra completely regular(2)For each and supra closed set such that , there exists an continuous map such that and (3)For each and supra open set containing , there exists an continuous map such that and

Proof. . Since is supra completely regular, then for each and supra closed set such that , there exists an continuous map such that and . Therefore, a map of into is continuous. Hence, and , as required.
, obviously.
. Let be a supra closed set and such that . Then, is a supra open set. By hypothesis, there exists an continuous map such that and . Therefore, a map of into is continuous. Since and , then, is supra completely regular.

Theorem 35. For every distinct points in a supra space , there is an continuous function such that .

Proof. Let . Then, is a supra closed set because is supra . Since , it follows from the supra complete regularity of that there is an continuous map such that and . Hence, .

Definition 36. For an continuous map of into , we define (1) of into by . It is obvious that is continuous(2)The zero set by

Theorem 37. is supra completely regular iff for every and supra closed set such that , there are continuous maps and of into such that , and .

Proof. (1)Necessity. Suppose that is a supra closed set and such that . Then, there is an continuous map such that and . Let and . Obviously, the two sets and are disjoint. Now, and . By the continuity of , it follows that and are supra open subsets of . Hence, the necessary part is proved(2)Sufficiency. Let be a supra closed set and such that . Take two maps and as described in the theorem. Then, we have for each . Therefore, is an continuous map such that and . Hence, is supra completely regular

Theorem 38. Every supra completely regular space is supra regular.

Proof. Let be a supra closed subset of and let . Since is supra completely regular, there is an continuous map such that and . Since is a subspace of the usual topological space , it is supra . So that there are two disjoint supra open sets and containing and , respectively. It follows from the continuity of that and are two disjoint supra open sets containing and , respectively. Hence, is supra regular.

Corollary 39. Every supra space is supra .

Theorem 40. A supra normal space is supra completely regular iff it is supra regular.

Proof. (1)Necessity. It follows from the above theorem(2)Sufficiency. Let be a supra closed subset of and let . By hypothesis, there are two disjoint supra open sets and such that and . Now, and are two disjoint supra closed sets. By using Theorem 32, we can find an continuous map such that for each and for each . This means that and for each . Hence, is supra completely regular.

Corollary 41. Every supra space is supra .

Lemma 42. The restricted map of an continuous map is continuous.

Proof. Let be an continuous map and let . To prove that is continuous, let be a supra open subset of . Then, is a supra open subset of . Now, is a supra open subset of . Thus, is continuous.

Theorem 43. The property of being a supra completely regular space is a supra hereditary property.

Proof. Consider that is a supra subspace of a supra completely regular space . Let be a supra closed subset of and such that . Then, it follows from Lemma 6 that there is a supra closed subset of such that . Obviously, . By the supra complete regularity of , there is an continuous map such that and . Thus, is an continuous map such that and . This ends the proof that a supra completely regular space is a supra hereditary property.

Corollary 44. A supra space is a supra hereditary property.

Theorem 45. A supra completely regular space is a supra topological property.

Proof. Consider as an homeomorphism and let be a supra completely regular space. Suppose that is a supra closed subset of and such that . Then, is a supra closed subset of and . Since is supra completely regular, there is an continuous map such that and . Since is an homeomorphism, then, is continuous. Therefore, is continuous such that and . Hence, is a supra completely regular space.

Corollary 46. A supra space is a supra topological property.

Theorem 47. The finite product of supra completely regular spaces is supra completely regular.

Proof. Let be the product spaces of two supra completely regular spaces and . Suppose that and be a supra open subset of such that . Now, there exists such that . Then, for each and supra open set containing , there exists an continuous map such that and , and for each and supra open set containing , there exists an continuous map such that and . Now, we define a map by . It can be checked that is continuous, we have and . Since , then, . Hence, is supra completely regular.

Corollary 48. The property of being a supra space is preserved under a finite product.

4. Conclusion

In this paper, we have highlighted two concepts on supra topological spaces, namely, supra completely Hausdorff and supra completely regular spaces. We have investigated some of their characterizations and provided some examples to show the relationships between them. Also, we have proved that they are hereditary and topological properties as well as they are closed under the finite product of spaces.

In future works, we plan to study an application of supra topology on the information system and apply the concepts given herein and those in [19] to categorize the approximation spaces. Also, we will study functionally separation axioms [29] in the frame of supra topologies and elaborate their relationships with the concepts presented herein.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no competing interests.

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Copyright

Copyright © 2021 T. M. Al-shami. 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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