Maximal Soft Compact and Maximal Soft Connected Topologies

Al Ghour, Samer; Ameen, Zanyar A.

doi:https://doi.org/10.1155/2022/9860015

Applied Computational Intelligence and Soft Computing

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

Research Article | Open Access

Volume 2022 | Article ID 9860015 | https://doi.org/10.1155/2022/9860015

Maximal Soft Compact and Maximal Soft Connected Topologies

Samer Al Ghour¹and Zanyar A. Ameen²

Academic Editor: Anandakumar Haldorai

Received24 Nov 2021

Revised19 Dec 2021

Accepted05 Jan 2022

Published08 Feb 2022

Abstract

In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft , while a maximal soft connected topology is soft . Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal.

1. Introduction

The real world is far too complex for our instant comprehension. We construct “models” of reality that are simplified versions of reality. Unfortunately, these mathematical models are just too complex, and we are unable to obtain exact solutions. Traditional classical methods are ineffective for modeling problems in engineering, physics, computer sciences, economics, social sciences, medical sciences, and many other domains due to the unpredictability of data. This could be owing to the unpredictability of natural environmental occurrences, human knowledge of the real world, or the limitations of measurement tools. For instance, ambiguity or confusion on the border between states or between major cities, the precise population growth rate in a country’s rural areas, or making judgments in a machine-based environment using database information. As a result, classical set theory, which is predicated on the crisp and accurate case, may not be totally adequate for dealing with such uncertainty concerns. The theory of fuzzy sets [1], the theory of intuitionistic fuzzy sets [2], the theory of vague sets [3], the theory of interval mathematics [4], and the theory of rough sets [5] are some of the theories. These theories might be seen as instruments for dealing with uncertainty, but each has its own set of problems. The insufficiency of the theory’s parametrization tool, as highlighted by Molodtsov in [6], could be the cause of these difficulties. He invented the word “soft set theory” to describe a new mathematical tool that is free of the issues discussed above. He stated the core results of the new theory in his paper [6], and effectively applied it to a variety of fields, including smoothness of functions, game theory, operations research, Riemann-integration, and probability theory. A “soft set” is a collection of approximations to an object’s description.

General topology is the branch of topology that deals with the fundamental set-theoretic notions and constructions used in topology. It is the foundation of most other topics in topology, including differential topology, geometric topology, and algebraic topology. Soft topology, which combines soft set theory and topology, is another field of topology. It is concerned with a structure on the set of all soft sets and is motivated by the standard axioms of classical topological space. The work of Shabir and Nazs [7], in particular, was crucial in establishing the field of soft topology. After that various classes of soft topological spaces have been proposed, such as: soft compact [8], soft connected [9], soft paracompact [9], soft extremely disconnected [10], and soft separable spaces [11], soft J-spaces [12], soft Menger spaces [13] and soft separation axioms [11, 14]. At this point, it is worth remarking that not all classical results in topology are true in soft topology, see Theorem 4 in [15]. Introducing all the above terminologies, arguments, and Remark 1 motivate us to study the structure of maximal soft compact and maximal soft connected topologies.

This paper is organized as follows: Sections 1 and 2 are dedicated to a brief introduction and preliminary concepts from soft set theory and soft topology. Section 3 starts by showing that the set of all soft topologies on a common universe forms a complete lattice. The definition of a maximal soft topology with property is given, followed by two subsections. The first one defines the concept of a maximal soft compact topology. Some properties and a characterization of maximal soft compact topologies are established. The second subsection concerns the fundamental properties of maximal soft connected topologies. It also contains some examples that present the structure of maximal soft connected topologies. Finally, the main result on maximal soft connected topologies is demonstrated. Section 4 concludes the summary of our findings and proposes possible lines for future work.

2. Preliminaries

Let be an initial universe, be all subsets of and be a set of parameters. An ordered pair is said to be a soft set over , where is a set-valued mapping. The family of all soft sets on is represented by . A soft element [16] is a soft set over in which for all , where , and is denoted by . A soft point [17], denoted by , is a soft set over in which and for each , , where and . A statement means that . The singleton soft set is referred to . The soft set (or simply ) is the complement of , where is given by for all . A soft subset over is called null, denoted by , if for any and called absolute, denoted by , if for any . Notice that and . It is said that is a soft subset of (written by , [18]) if and for any . We say if and .

Definition 1. (see [19, 20]). Let be a family of soft sets over , where is any index set.(i)The intersection of , for , is a soft set such that for each and denoted by (ii)The union of , for , is a soft set such that for each and denoted by

Definition 2. (see [7]). A collection of is said to be a soft topology on if the following conditions are satisfied:(i)(ii)If , then (iii)If any , then Terminologically, we call a soft topological space on . The elements of are called soft -open sets (or simply soft open sets when no confusion arise), and their complements are called soft -closed sets (or soft closed sets).
In what follows, by we mean a soft topological space, by two distinct soft points we mean either or , and by two disjoint soft sets over , we mean .

Definition 3. (see [21]). A subcollection is called a soft base for the soft topology if each element of is a union of elements of .

Definition 4. (see [7]). Let be a soft subset of . Then, is called a soft relative topology over and is a soft subspace of .

Definition 5. (see [7]). Let be a soft subset of . The soft interior of , denoted by , is the largest soft open set contained in . The soft closure of , denoted by , is the smallest soft closed set which contains . The soft closure and interior of a soft subset in the soft subspace are, respectively, denoted by and .

Lemma 1 (see [22]). For a soft subset of ,

Definition 6. (see [23, 24]). Let be soft sets, and let be functions. The image of a soft set under is a soft subset of which is given byfor each .
The inverse image of a soft set under is a soft subset such thatfor each .
The soft function is bijective if both and are bijective.

Lemma 2. Let be soft sets. If is bijection, then for each .

Proof. It follows from Theorem 3.14 in [24].

Definition 7. (see [16]). Let and be soft topological spaces. A soft function is said to be(i)Soft continuous if the inverse image of each soft open subset of is a soft open subset of (ii)Soft open if the image of each soft open subset of is a soft open subset of (iii)Soft homeomorphism if it is a soft open and soft continuous bijection from onto

3. Maximal Soft Topologies

Definition 8. Let be two soft topologies on . It is said that is coarser than if . And is finer than if .

Lemma 3. Let be a family of soft topologies on , where is any index set. Then is a soft topology on .

Proof. Evidently, belong to as they belong to for all s. Let . Then for all . Since all are soft topologies, so for all . Therefore . Let be a collection of sets in . Then for each , for all . But for each , is a soft topology on , so for all . Hence .

The above result is an extension of Proposition 6 in [7].

Lemma 4. Let be a collection of soft subsets over . There exists a unique soft topology on including and if is any other soft topology on that includes , then .

Proof. Notice that such a soft topology always exists because is the soft topology on which includes . Consider , the intersection of all those soft topologies on which include . Then it follows from Lemma 3 that is the required soft topology.

Definition 9. Let be a collection of soft subsets over . The unique soft topology obtained in the above lemma is called the soft topology on generated by the collection and is denoted by , which is the smallest soft topology on including .
The union of two soft topologies is not a soft topology, see example 3 in [7], but we can generate a unique soft topology that includes both of them.

Lemma 5. Let be two soft topologies on . The generating soft topology is identical to the soft topology generated by .

Proof. Since soft topologies, so they include . By taking , for , then will exactly contain . By the uniqueness of the generating soft topology, , see Definition 9.

Theorem 1. The set of all soft topologies over a common universe forms a complete lattice under soft set inclusion “”.

Proof. One can easily show that is a partially ordered set on . It remains to prove that every subset of has the greatest lower bound and the least upper bound. Let be a subset of . By Lemma 3, is the greatest lower bound of . By Lemma 6, is the least upper bound of .

Remark 1. Note that the indiscrete soft topology on is the minimal (smallest) element in and the discrete soft topology on is the maximal (largest) element in . It is worth remarking that is the maximal soft Hausdorff topology and is the minimal soft compact (and minimal soft connected) topology. From the latter statement, we understand that maximal covering and connectedness properties are more interesting to study. On the other hand, minimal separation axioms are more interesting. Hence, we focus on considering maximal soft compact and maximal soft connected spaces.

Definition 10. Let be a soft topological space with the property . Then is called -maximal if any soft topology finer than does not have the property .

3.1. Maximal Soft Compact Topologies

Recall that a space is called soft compact [8] if each soft open cover of has a finite subcover. If we replace by soft compactness, Definition 10 will be as follows.

Definition 11. Let be a soft compact space. Then is called maximal soft compact if any soft topology finer than is not soft compact.
The following example shows the structure of maximal soft compact spaces.

Example 1. Consider the set of naturals and . Define a soft topology . Then is maximal soft compact.

Lemma 6 (see [25], Proposition 5.1). Let be a soft compact space and let . If is soft closed, then is soft compact.

Definition 12. (see [26]). Let be any soft topological space and let be any soft non-open subset over . The soft topology on generated by is said to be an s-extension of and it is denoted by (or shortly, ).

Lemma 7 (see [26], Theorem 3.2). Let be a soft compact space. Then is soft compact if and only if is soft compact in .

Proposition 1. A soft topological space is maximal soft compact if and only if the family of all soft -closed sets is equal to the family of all soft -compact sets.

Proof. Assume that is maximal soft compact. If is a soft compact set but not soft closed, then is not soft open. By Lemma 7, is soft compact. But , so this violates the maximality of . Hence must be closed.

Conversely, suppose the family of all soft -closed sets is equal to the family of all soft -compact sets. If is not maximal soft compact, there exists a soft compact topology such that . Therefore there is a set which is soft -closed but not soft -closed. By assumption, is not -compact. Therefore, there exists a soft -open cover of which has no finite subcover. Since , so is also a soft -open cover of . This means that is not soft -compact. But this is a contradiction, because is soft -closed and, by Lemma 6, it must be soft -compact. This proves that has to be maximal soft compact.

Proposition 2. A soft topological space is maximal soft compact if and only if each soft continuous bijection from a soft compact space onto is a soft homeomorphism.

Proof. Assume is maximal soft compact. Let be a soft continuous bijection, where is soft compact. Take . Evidently is a soft topology and is a soft homeomorphism. Since is soft compact, is also soft compact under . But and is maximal, thus .

Conversely, if is not maximal soft compact, then there exists a soft compact topology on such that . Then the identity soft function is a soft continuous bijection but not a soft homeomorphism. This completes the proof.

Theorem 2. For a soft topological space , the following are equivalent:(1) is maximal soft compact(2)The family of all soft -closed sets is equal to the family of all soft -compact sets(3)Each soft continuous bijection from a soft compact space onto is a soft homeomorphism

Proof. (1) (2) Proposition 1 (2) (3) Suppose is a soft continuous bijection from a soft compact space onto . It remains to check that is soft open. Let be soft -open. Then is soft -closed. Since is soft compact space, by Lemma, 3.11, is soft -compact. From the soft continuity of , is soft -compact and by (2) is soft -closed. Since is bijective, and so is soft open. Hence is a soft open function. (3) (1) Proposition 2.

Lemma 8 (see [11], Theorem 4.1). Let be a soft topological space. If each singleton soft set is soft closed, then is soft .

The above is true in various soft point theories, see [27].

Theorem 3. If is a maximal soft compact space, then is soft .

Proof. One can easily show that each singleton soft set is soft compact. Since is maximal soft compact, by Proposition 1, each singleton soft set is soft closed and Lemma 8 finishes the proof.

Definition 13. (see [15]). A soft set from is called stable if there exists a subset of such that for each .

Definition 14. We call a soft topological space stable if each soft open is stable.

Lemma 9. If is a stable soft -space, then each soft compact is soft closed.

Proof. It follows from Lemma 7 and Theorem 8 in [15].

Theorem 4. If is a stable soft compact -space, then is maximal soft compact.

Proof. If is not maximal soft compact, there exists a soft compact topology on such that . Pick a set to be soft -open but not soft -open. Let . Then and so is soft compact. By Lemma 7, is a soft -compact set and by Lemma 9, it is soft -closed. This implies is soft -open, which contradicts to the choice of . Hence is maximal soft compact.

3.2. Maximal Soft Connected Topologies

Definition 15. (see [9]). A soft topological space is called soft connected if it cannot be written as a union of two disjoint soft open sets. Otherwise, it called soft disconnected.

Definition 16. Let be a soft connected space. Then is called maximal soft connected if any soft topology finer than is not soft connected.

We start by giving some examples of maximal soft connected spaces.

Example 2. Let be the set of reals and let . The soft topological space is maximal soft connected, where .

Example 3. (see [26], Example 3.3). The soft topological space is maximal soft connected, where , is the set of reals, and .

Remark 2. From the above example, we shall remark that the maximal soft topology dealt with in this note does not have a nice connection with maximal crisp topology [28], in general, due to the concept of soft point we select. The soft topology given in Example 3 is maximal and the crisp topology of is maximal, while the crisp topology of is not maximal.

Theorem 5. Let be a soft subset of a soft topological space . If are soft connected (as soft subspaces) and either of them is soft open, then is maximal soft connected.

Proof. Assume , otherwise the result trivially holds. Suppose that are not soft -open. Then, by taking , we obtain a disconnected soft topology such that . Therefore, there are disjoint soft -open sets such that . W.L.O.G assume that . Then are disjoint soft -open and . By Remark 2.2 (vii) in [26], are disjoint soft -open which means that is not soft connected, a contradiction. Therefore, either or . Suppose . If , then , which is a soft -open set, but that is not possible (as it is soft -open). Therefore, we must have . Hence , where are soft -open. But, by Remark 2.2 (vii) in [26] soft -open and soft -open are similar, thus is soft disconnected, again contradiction. The result is proved.

Theorem 5. Let be a soft connected space. If is maximal soft connected, then is soft .

Proof. Suppose is not soft . Then there are soft points with such that for all soft -open sets . Therefore, and . Let . Then and so is not soft connected as is maximal soft connected. Therefore, there exist disjoint soft -open sets such that . Thus either or . If , by Remark 2.2 (ii) in [26], each soft -open set containing is also soft -open, so is soft -open. Therefore, there is a soft -open such that . But is a soft -open containing such that . Thus is a soft -open containing each of its points, and similarly for . This implies that is soft disconnected, which is impossible.

Proposition 3. If is a soft open connected subset of a maximal soft connected space , then is maximal soft connected.

Proof. If is not maximal soft connected, then there exists a soft connected topology on such that . Let be a soft -open set but not soft -open. If , then and so is soft connected. If , then but cannot be soft connected. Therefore, there are disjoint soft -open sets such that . Then either or , differently will be soft disconnected by the disjoint soft -open sets (impossible). Hence, we assume . By Remark 2.2 (ii), for all soft points that belongs to some soft -open set but does not belong to any soft -open, we have

This means that is soft -open as is soft -open. Similarly, one can show that is soft -open. Hence are disjoint soft -open and , which proves that is not soft connected, a contradiction.

Proposition 4. If is a soft closed connected subset of a maximal soft connected space , then is maximal soft connected.

Proof. Assume is not maximal soft connected. Take a set which is not soft -open. If , then and so is soft connected. Let . Then, we have . Therefore, , where is an s-extension of with respect to (i.e. ). Since is maximal soft connected and , then is soft disconnected. Thus there exist disjoint soft -open sets such that . Since is soft connected, then either or . Suppose . We consider two cases: (i) suppose . Since is soft -open, there are soft -open sets such that . If and so . If , then as . Again . Since was arbitrarily taken, so is soft -open. (ii) suppose . Since is soft -open, there are soft -open sets such that . If and so . If , then, since , so . Since is soft -open, therefore . Thus is soft -open. This means that is not soft connected, which is impossible. Hence must be maximal soft connected.

Definition 16. (see [29]). A soft topological space is called soft submaximal if each soft -dense set is soft -open.

Lemma 10. Let be a soft connected space and let be an s-extension of over . If is soft -dense, then is soft connected

Proof. It follows from Theorem 3.17 [26].

Lemma 11. If is a maximal soft connected space, then is soft submaximal.

Proof. Let (D, E) be a soft -dense set over Z. By Lemma 10, is soft connected, where , but is soft maximal, hence . Thus (D, E) must be a soft -open.

Theorem 5. If is a soft connected subset of a maximal soft connected space , then is maximal soft connected.

Proof. Since is soft connected, then is also soft connected and, by Proposition 4, is maximal. By Lemma 11, is soft open in because is soft dense in . By Proposition 3, is maximal soft connected.

4. Conclusion

The growth of topology has been supported by the continual supply of classes of topological spaces, examples, and their properties and relations. As a result, extending the area of soft topological spaces in the same way is significant. We have shown that the collection of all soft topologies on a non-empty set is a complete lattice. The minimal element in this lattice is the soft indiscrete topology, which is both soft compact and soft connected. One might ask the question of what be will be the structure of maximal soft compact and maximal soft connected topologies. The third section answers this question and gives some more properties of these topologies. We have characterized maximal soft compact topologies in terms of soft closed-compact subsets and soft homeomorphisms. It is shown that the class of maximal soft compact topologies contains the class of stable soft compact topologies and is contained in the class of soft topologies. Furthermore, we have seen that the class of maximal soft connected topologies is contained in the class of soft topologies. With the help of a simple extension of a soft topology, we have found that any soft connected topology relativised with a maximal soft connected is also maximal.

As part of future work, the following tasks are expected to be completed:(i)Different soft point theories can be applied to all the results presented in this paper.(ii)Recall that a soft topology on with the property is -maximal if any soft topology finer than does not have the property . One can examine different soft topological properties in place of soft compact or soft connected, namely: soft countably compact, soft sequentially compact, soft paracompact, soft Menger, soft path-connected, soft J-topologies, and so on.(iii)This work is done on soft topologies, one can work on different topological structures, like fuzzy soft topologies, supra topologies, infra topologies, etc.

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 © 2022 Samer Al Ghour and Zanyar A. Ameen. 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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