Soft Separation Axioms and Fixed Soft Points Using Soft Semiopen Sets

Abstract

The importance of soft separation axioms comes from their vital role in classifications of soft spaces, and their interesting properties are studied. This article is devoted to introducing the concepts of -soft semi- and -soft semiregular spaces with respect to ordinary points. We formulate them by utilizing the relations of total belong and total nonbelong. The advantages behind using these relations are, first, generalization of existing comparable properties on general topology and, second, eliminating the stability shape of soft open and closed subsets of soft semiregular spaces. By some examples, we show the relationships between them as well as with soft semi- and soft semiregular spaces. Also, we explore under what conditions they are kept between soft topology and its parametric topologies. We characterize a -soft semiregular space and demonstrate that it guarantees the equivalence of -soft semi-. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces, and sum of soft topological spaces. Finally, we define a concept of semifixed soft point and study its main properties.

1. Introduction

In daily life, human beings face different kinds of uncertainties in fields such as economics, environmental and social sciences, engineering, medicine etc. To tackle such uncertainties, different approaches were proposed like fuzzy and rough set theories. However, they have their own limitations. Molodtsov [1] introduced an effective tool that is free from these limitations to solve uncertainties, namely soft set. This tool does not require specific type of parameters. Instead, it accommodates all types of parameters such as words, numbers, sentences, and functions. After that, Maji et al. [2] established some fundamental operations between two soft sets and then tackled one of their applications in decision-making problems [3]. But the authors of [4] demonstrated some shortcomings of the operations defined in [2]. They also explored novel operations between soft sets which became a goal of many studies on algebraic soft structures. Al-shami and El-Shafei [5] defined new types of soft subset and equality relations and then applied them on soft linear equations.

In 2011, Shabir and Naz [6] defined a soft topological space over a family of soft sets. They constructed the fundamental notions of soft topological spaces such as the operators of soft closure and interior, soft subspace, and soft separation axioms. Min [7] continued studying soft separation axioms and corrected some alleged results in [6]. Then, Aygünolu and Aygün [8] scrutinized the concept of compactness on soft setting. By exploiting one of the diverges between soft sets and crisp sets, Hida [9] studied a strong type of soft compactness. Al-shami [10, 11] revised many results given in many studies of soft separation axioms. Then, Al-shami et al. [12] introduced a weak type of soft compactness, namely almost soft compactness. The authors of [13] presented soft maps by using two crisp maps, one of them between the sets of parameters and the second one between the universal sets. However, the authors of [14] introduced soft maps by using the concept of soft points. Some applications of different types of soft maps were the goal of some articles (see [13–15]).

Until 2018, the belong and nonbelong relations that utilized in these studies are those given by [6]. In 2018, the authors of [16] came up new relations of belong and nonbelong between an ordinary points and soft set, namely partial belong and total nonbelong relations. In fact, these relations widely open the door to study and redefine many soft topological notions. This leads to obtain many fruitful properties and changes which can be seen significantly on the study of soft separation axioms as it was shown in [16–18].

Das and Samanta [19] studied the concept of a soft metric based on the soft real set and soft real numbers given in [20]. Wardowski [14] tackled the fixed point in the setup of soft topological spaces. Abbas et al. [21] presented soft contraction mappings and established a soft Banach fixed point theorem in the framework of soft metric spaces. Recently, many researchers explored fixed point findings in soft metric type spaces (see, for example, [22–25]). Some results related to intuitionistic fuzzy soft sets were investigated in [26, 27]. Recently, Alcantud [28] has discussed the properties of topological separability on soft setting.

One of the significant ideas that helps to prove some properties and removes some problems on soft topology is the concept of a soft point. It was first defined by Zorlutuna and Çakir [29] in order to study the interior points of a soft set and soft neighborhood systems. Then, [19, 30] simultaneously redefined soft points to discuss soft metric spaces. In fact, the recent definition of a soft point makes similarity between many set-theoretic properties and their counterparts on soft setting. Two types of soft topologies, namely enriched soft topology and extended soft topology, were studied in [8, 30], respectively. Al-shami and Koinac [31] discussed the equivalence between these two topologies and obtained interesting results. Recently, they [32] have introduced the concept of nearly soft Menger spaces and investigated main characteristics.

We organized this paper as follows: After this introduction, Section 2 addresses the basic principles about the soft sets and soft topologies. In Section (, we introduce the concepts of -soft semi- and -soft semiregular spaces with respect to ordinary points by using the relations of total belong and total nonbelong. The relationships between them and their main properties are discussed with the help of interesting examples. In Section 4, we explore a semifixed soft point theorem and study some main properties. In particular, we conclude under what conditions semifixed soft points are preserved between a soft topology and its parametric topologies. Finally, conclusions of the paper in Section 5.

2. Preliminaries

To well understand the results obtained in this study, we shall mind some essential concepts, definitions, and properties from the literature.

2.1. Soft Sets

Definition 1. [1] A mapofinto the power setof a nonempty setis called a soft set over, whereis the universal set andis a set of parameters.

In this study, we use a symbol to refer a soft set, and we identify it as ordered pairs .

A class of all soft sets defined over with is symbolized by .

Definition 2. [33] is said to be a subset of, symbolized by, iffor each.

We write provided that and .

Definition 3. [6, 16] For a soft setand, we say that(i), it is read as follows: totally belongs to , if for each (ii), it is read as follows: does not partially belong to , if for some (iii), it is read as follows: partially belongs to , if for some (iv), it is read as follows: does not totally belong to , if for each .

Definition 4. [4] The relative complement ofis another soft set, where a mapis defined byfor all.

Definition 5. [2, 16, 19, 30] A soft setoveris said to be(i)a null soft set (resp., an absolute soft set), symbolized by (resp., ), if is the empty (resp., universal) set for each (ii)a soft point if there are and so that and for each . We write that if (iii)a stable soft set if all p-approximate are equal A and it is symbolized by . In particular, we symbolize by if (iv)a countable (resp., finite) soft set if is countable (resp., finite) for each . Otherwise, it is said to be uncountable (resp., infinite).

Definition 6. [2, 4] The intersection and union ofandare defined as follows:(i)Their intersection, symbolized by , is a soft set , where a mapping is given by (ii)Their union, symbolized by , is a soft set , where a mapping is given by .

The union and intersection operators were generalized for any number of soft sets in a similar way.

Definition 7. [34] The Cartesian product ofand, symbolized by, is defined asfor each.

Definition 8. [13] A soft mapping ofinto, symbolized by, is defined by two crisp mapsandso that the image ofis a set inand pre-image ofis a set inand they are defined as follows.(i) is a set in , where for each (ii) is a set in , where for each .

Definition 9. [29] is called asoft map provided thatandaremaps, where.

2.2. Soft Topology

Definition 10. [6] A familyof soft sets overunder a fixed set of parametersis said to be a soft topology onif it satisfies the following:(i) and are elements of (ii)The intersection of a finite number of is an element of (iii)The union of an arbitrary number of is an element of .

We will call a term of soft topological space on the triple , where a terminology of soft open is given for every member in and a terminology of soft closed is given for a relative complement of a member of .

Proposition 11. [6] In, a familyis a classical topology onfor each.

We will call a term of parametric topology on and a term of parametric topological space on .

Definition 12. [6] is called a soft subspace of, whereand.

Definition 13. [35] is called a soft semiopen subset ofif.

Theorem 14. [35] Every soft open set is soft semiopen, and the union of an arbitrary number of soft semiopen sets is soft semiopen.

Definition 15. [35] The intersection of all soft semiclosed sets incontainingis the soft preclosure of. It is symbolized by.

It is clear that iff for each soft semiopen set totally containing we have ; and iff for each soft semiopen set totally containing we have .

Definition 16. [36] which does not contain disjoint soft open sets is called soft hyperconnected.

The following result will help us establish some properties of soft semiseparation axioms and soft semicompact spaces (see, for example, Theorem 42 and Proposition 49). It implies that the family of soft semiopen subsets of a soft hyperconnected space forms a new soft topology over that is finer than .

Theorem 17. [37] A class of soft semiopen subsets of a soft hyperconnected space forms a soft topology.

Proposition 18. Letbe soft open set in. Then(1)If is soft semiopen and is soft open set in , then is a soft semiopen set in (2)If is soft open in and is a soft semiopen in , then is a soft semiopen set in .

Definition 19. [35] is said to be(i)soft semi- if there is a soft semiopen set for each satisfying and ; or and (ii)soft semi- if there are two soft semiopen sets and for each satisfying and ; and and (iii)soft semi- if there are two disjoint soft semiopen sets and for each satisfying and (iv)soft semiregular if every soft semiclosed set so that , contains two disjoint soft semiopen sets and satisfying and (v)soft seminormal if we separate every two disjoint soft semiclosed sets by two disjoint soft semiopen sets(vi)soft semi- (resp., soft semi-) if it is both soft semiregular (resp., soft seminormal) and soft semi--space

Definition 20. [38] A soft semiopen cover ofis a family of soft semiopen setsso that.

Definition 21. is said to be(i)soft semi- [37] if for every , there are twodisjoint soft semiopen sets and containing and , respectively(ii)soft semicompact [38] if every soft semiopen cover of has afinite subcover.

Proposition 22. [37] A soft semicompact (resp., stable soft semicompact) set in a soft semi--space (resp., soft semi--space) is soft semiclosed.

To study the properties that preserved under soft -homeomorphism maps, the concept of a soft semi-irresolute map will be presented in this work under the name of a soft -continuous map.

Definition 23. [37] is called softif the inverse image of each soft semiopen set is soft semiopen.

Proposition 24. [38] The softimage of a soft semicompact set is soft semicompact.

The definitions of soft semicontinuous, soft semiopen, soft semiclosed, and soft semihomeomorphism maps were given in [35] in a similar way of their counterparts in classical topology.

Definition 25. A soft topologyonis said to be(i)an enriched soft topology [8] if all soft sets so that or are members of (ii)an extended soft topology [30] if , where is a parametric topology on .

Al-shami and Kočinac [31] proved that extended and enriched soft topologies are identical and obtained some useful results that help to discover the interrelations between soft topology and its parametric topologies.

Theorem 26. [31] A subsetof an extended soft topological spaceis soft semiopen if and only if each p-approximate element ofis semiopen.

Proposition 27. [39] Letbe a class of mutually disjoint soft topological spaces and. Then the class
forms a soft topology on with .

Definition 28. [39] given in the above proposition is said to be the sum of soft topological spaces and is symbolized by.

Theorem 29. [39] A soft setis soft semiopen (resp., soft semiclosed) inif and only if allare soft semiopen (resp., soft semiclosed) in.

Proposition 30. [14] Letbe a soft map so thatis a soft point. Then,is a unique fixed point of.

Theorem 31. [29] Let, whereandare two soft topological spaces. A class of all arbitrary union of members ofdefines a soft topology overunder a fixed set of parameters.

Lemma 32. Letandbe two subsets ofand, respectively. Then(i)(ii).

3. Semisoft Separation Axioms

In this section, the concepts of -soft semi- and -soft semiregular spaces are introduced, where denotes the total belong and total nonbelong relations that are utilized in the definitions of these concepts. The relationships between them are shown, and their main features are studied. In addition, their behaviours with the concepts of hereditary, topological, and additive properties are investigated. Some examples are provided to elucidate the obtained results.

First of all, we see that it is necessary to classify containment into several categories as it is shown in remark below. Factually, this classification will play a vital role in redefining many soft theoretic-set and soft topological concepts, in particular, the concepts of soft interior and closure operators, soft compactness, and soft separation axioms.

Remark 33. For a soft setand, we have the following notes:(i) totally contains if (ii) does not partially contain if (iii) partially contains if (iv) does not totally contain if .

Definition 34. is said to be(i)-soft semi- if there exists a soft semiopen set for every satisfying and or and (ii)-soft semi- if there exist soft semiopen sets and for every satisfying and ; and and (iii)-soft semi- if there exist two disjoint soft semiopen sets and for every satisfying and ; and and (iv)-soft semiregular if for every soft semiclosed set so that , there exist disjoint soft semiopen sets and so that and (v)-soft semi- (resp., -soft semi-) if it is both -soft semiregular (resp., soft seminormal) and -soft semi-.

Remark 35. It can be noted that ifandare disjoint soft set, theniff. This implies thatis a-soft semi--space iff is a soft semi--space. That is, the concepts of a-soft semi--space and a soft semi--space are equivalent.

We can say that is -soft semi- if there exist two disjoint soft semiopen sets and for every so that and totally contain and , respectively.

Remark 36. The soft semiregular spaces imply a strict condition on the shape of soft semiopen and soft semiclosed subsets. To explain this matter, letbe a soft semiclosed set so that. Then, we have two cases:(i)There are so that and . This case is impossible because there do not exist two disjoint soft sets and containing and , respectively(ii)For each , . This implies that must be stable.

As a direct consequence, we infer that every soft semiclosed and soft semiopen subsets of a soft semiregular space must be stable. However, this matter does not hold on the -soft semiregular spaces because we replace a partial nonbelong relation by a total nonbelong relation. Therefore a -soft semiregular space needs not be stable.

Proposition 37. (i)A -soft semi--space is always soft semi- for (ii)A soft semiregular space is always -soft semiregular(iii)A soft semi--space is always -soft semi-.

Proof. Since implies , then the proofs of (i) and (ii) follow.
To prove (iii), it suffices to prove that a soft semi--space is -soft semi- when is soft semiregular. Suppose . Then, there exist two soft semiopen sets and so that and ; and and . Since and are soft semiopen subsets of a soft semiregular space, then they are stable. So and . Thus, is -soft semi-. Hence, we obtain the desired result.
The following examples clarifies that the converse of the above proposition is not always true.

Example 1. Let. A familyis a soft topology on, where

Since the soft sets and are not soft semiopen subsets of , then it is not -soft semi-. However, it is clear that is a soft semi--space. Also, it is soft seminormal. Therefore, it is soft semi-.

Example 2. Let. A familyis a soft topology on, where

It can be checked that a set in is soft open iff it is soft semiopen. Since there exists an unstable soft semiopen set in , then is not soft semiregular. Therefore, it is not soft semi-. However, is a -soft semiregular space. Also, it is a -soft semi--space. Hence, it is a -soft semi--space.

Before we show the relationship between -soft semi--spaces, we need to prove the next useful lemma.

Lemma 3. is a-soft semi--space iffis soft semiclosed for every.

Proof. Necessity: for any distinct point than , contains a soft semiopen set and totally contains so that . So and for each . Thus, is soft semiopen. Hence, is soft semiclosed.
Sufficiency: let . By hypothesis, and are soft semiclosed sets. Then and are soft semiopen sets so that and . Obviously, and . Hence, is -soft semi-.

Proposition 38. Every-soft semi--space is-soft semi-for.

Proof. We suffice by proving the cases of .
For , let in a -soft semi--space . Then, is soft semiclosed. Since and is -soft semiregular, then contains two disjoint soft semiopen sets and satisfying and . Therefore, is -soft semi-.
For , let be a soft semiclosed set so that . Since is -soft semi-, then is soft semiclosed. Since and is soft seminormal, then there are disjoint soft semiopen sets and so that and . Hence, is -soft semi-.

We explain that the converse of proposition above is not always true by presenting the next example.

Example 1. Let. A familyis a soft topology on. The following soft sets withare all soft semiopen subsets of.

Since the soft sets are a soft semiopen set in , then it is -soft semi-. However, is not a soft semiopen set in . Hence, it is not -soft semi-.

Example 2. Letbe a soft topology on the set of natural numbers, whereis a set of parameters. By a simple check, one notes that a class of soft semiopen and a class of soft open sets incoincide. For each, we haveandare soft semiopen sets so thatand; andand. Therefore,is-soft semi-. In contrast, all nonnull proper semiopen sets have nonnull soft intersection. Hence,is not-soft semi-.

Example 3. As we know, the coincidence of soft topological and classical topological spaces occurs ifis a singleton. Then, it suffices to consider examples that satisfy a semi--space but not semi-and satisfy a semi--space but not semi-.

In what follows, we found some properties of -soft semi- and -soft semiregular.

Lemma 39. Letbe a subset ofand. Theniff there is a soft semiopen settotally containingso that.

Proof. Let . Then, . So, . Conversely, if there exists a soft semiopen set totally containing so that , then . Therefore, . Since , then .

Proposition 40. Ifis a-soft semi--space, thenfor every.

Proof. Let in a -soft semi--space. Then, there is a soft semiopen set so that and or and . Say, and . Now, . So, by the above lemma, . But . Hence, we obtain the desired result.

Corollary 41. Ifis a-soft semi--space, thenfor alland.

Theorem 42. Letbe a finite set andbe soft hyperconnected. Thenis a-soft semi--space if and only iffor each.

Proof. To prove the “if” part, let . Then for each , we have a soft semiopen set so that and . Therefore . Since is chosen arbitrary, then the desired result is proved.
To prove the “only if” part, let the given conditions be satisfied and let . Let . Since , then for each there is a soft semiopen set so that and . Since is soft hyperconnected, then it follows from Theorem 17 that is a soft semiopen set so that and . Similarly, we can get a soft semiopen set so that and . Thus, is a -soft semi--space.

Theorem 43. Ifis an extended-soft semi--space, thenis soft semiclosed for all.

Proof. By Lemma 3, we have is a soft semiopen set. Since is extended, then a soft set , where and for each , is a soft semiopen set. Therefore, is soft semiopen. Thus, is soft semiclosed.

Corollary 44. Ifis an extended-soft semi--space, then the intersection of all soft semiopen sets containingis exactlyfor each.

Proof. Let be a soft subset of . Since is a soft semiclosed set for every , then is a soft semiopen set containing . Therefore, , as required.

Theorem 45. Letbe finite soft hyperconnected. Then,is-soft semi-if and only if it is-soft semi-.

Proof. Necessity: it is obtained from Proposition 38.
Sufficiency: for each , we have and are soft semiclosed sets. Since is finite, then and are soft semiclosed sets. Since is soft hyperconnected, then and are soft semiopen sets. The disjointness of and ends the proof that is -soft semi-.

Remark 46. In Example2, note thatis not a soft semiopen set for each. This clarifies that a soft setin a-soft semi--space need not be soft semiopen if the universal set is infinite.

Theorem 47. is-soft semiregular iff for every soft semiopen subsetoftotally containing, there is a soft semiopen setso that.

Proof. Let be a soft semiopen set totally containing . Then is semisoft closed and . So contains two disjoint soft semiopen sets and satisfying and . Thus, . Hence, . Conversely, let be a soft semiclosed set. Then, for each , we have . By hypothesis, there is a soft semiopen set totally containing so that . Therefore, and . Thus, is -soft semiregular, as required.

Theorem 48. The three concepts below are identical ifis a-soft semiregular space.(i)a -soft semi--space(ii)a -soft semi--space(iii)a -soft semi--space.

Proof. The directions and are obvious.
To prove , let in a -soft semi--space . Then, there exists a soft semiopen set so that and , or and . Say, and . Obviously, and . Since is -soft semiregular, then there exist two disjoint soft semiopen sets and so that and . Hence, is -soft semi-.

Proposition 49. Letbe finite soft hyperconnected. Ifis a-soft semi--space, then it i-soft semiregular.

Proof. Let be a soft semiclosed set and so that . Then for each . By hypothesis, there are two disjoint soft semiopen sets and so that and . Since is a finite set, then there is a finite number of soft semiopen sets so that . Since is soft hyperconnected, then it follows from Theorem 17 that is a soft semiopen set containing . Since , then is -soft semiregular.

Corollary 50. The following concepts are identical ifis finite soft hyperconnected.(i)a -soft semi--space(ii)a -soft semi--space(iii)a -soft semi--space.

Proof. We obtain the directions and from Proposition 38.
We obtain the direction from Theorem 45.
We obtain the direction from Proposition 49.

Theorem 51. The property of being a-soft semi--spaceis a soft open hereditary.

Proof. We suffice by proving the cases of .
Let be a soft open subspace of a -soft semi--space . To prove that is -soft semi-, let . Since is a -soft semi--space, then there exist two soft semiopen sets and so that and ; and and . Therefore, and so that and . It follows from Proposition 18 that and are soft semiopen subsets of , so that is -soft semi-.
To prove that is -soft semiregular, let and be a soft semiclosed subset of so that . Then, is a soft semiclosed subset of so that . Therefore, there exist disjoint soft semiopen subsets and of so that and . Now, and are disjoint soft semiopen subsets of so that and . Thus, is -soft semiregular.
Hence, is -soft semi-, as required.

Theorem 52. Letbe extended and. Then,is-soft semi-iffis semi-for each.

Proof. We prove the theorem in the case of , and one can similarly prove the other cases.
Necessity: let in . Then, there exist two soft semiopen sets and so that and ; and and . Obviously, and ; and and . Since is extended, then it follows from Theorem 26 that and are semiopen subsets of for each . Thus, is a semi--space. To prove that is seminormal, let and be two disjoint semiclosed subsets of . Let and be two soft sets given by , , and for each . It follows from Theorem 26 that and are two disjoint soft semiclosed subsets of . By hypothesis, there exist two disjoint soft semiopen sets and so that and . This implies that and . Since is extended, then it follows from Theorem 26 that and are semiopen subsets of . Thus, is a seminormal space. Hence, it is a semi--space.
Sufficiency: let in . Then there exists two semiopen subsets and of so that and ; and and . Let and be two soft sets given by and for each . Since is extended, then it follows from Theorem 26 that and are soft semiopen subsets of so that and ; and and . Thus, is a -soft semi--space. To prove that is soft seminormal, let and be two disjoint soft semiclosed subsets of . Since is extended, then it follows from Theorem 26 that and are two disjoint semiclosed subsets of . By hypothesis, there exist two disjoint semiopen subsets and of so that and . Let and be two soft sets given by and for each . Since is extended, then it follows from Theorem 26 that and are two disjoint soft semiopen subsets of so that and . Thus, is soft seminormal. Hence, it is a -soft semi--space.

In the following examples, we show that a condition of an extended soft topology given in the above theorem is not superfluous.

Example 1. Letbe the same as in Example1. We showed thatis not-soft semi-. On the other hand,andare the discrete topology on. Hence, the two parametric topological spacesandare semi-.

Theorem 53. The property of being a-soft semi--spaceis preserved under a finite product soft spaces.

Proof. We prove the theorem in case of . The other cases follow similar lines.
Let and be two -soft semi--spaces and let in . Then, or . Without loss of generality, let . Then there exist two disjoint soft semiopen subsets and of so that and ; and and . Obviously, and are two disjoint soft semiopen subsets so that and ; and and . Hence, is a -soft semi--space.

For the sake of brevity, we omit the proof of the following result.

Theorem 54. The property of being a-soft semi--space is an additive property for.

We complete this section by discussing some interrelations between -soft semi--spaces and soft semicompact spaces.

Proposition 55. A stable soft semicompact set in a-soft semi--space is soft semiclosed.

Proof. It follows from Proposition 22 and Remark 35.

Theorem 56. Letbe a soft semicompact subset of a soft hyperconnected-soft semi--space. If, then there are disjoint soft semiopen setsandso thatand.

Proof. Let . Then for each . Since is a -soft semi--space, then there exist disjoint soft semiopen sets and so that and . Therefore, forms a soft semiopen cover of . Since is soft semicompact, then . By the soft hyperconnectedness of , we obtain is a soft semiopen set. Hence, we obtain the desired result.

Theorem 57. Every soft hyperconnected, soft semicompact, and-soft semi--space is-soft semiregular.

Proof. Let be a soft semiclosed subset of soft semicompact and -soft semi--space so that . Then, is soft semicompact. By Theorem 56, there exist disjoint soft semiopen sets and so that and . Thus, is -soft semiregular.

Corollary 58. Every soft hyperconnected, soft semicompact, and-soft semi--space is-soft semi-.

Lemma 59. Letbe a soft semiopen subset of a soft semiregular space. Then, for each, there exists a soft semiopen setso that.

Proof. Let be a soft semiopen set so that . Then, . Since is soft semiregular, then there exist two disjoint soft semiopen sets and totally containing and , respectively. Thus, . Hence, .

Theorem 60. Letbe a soft semicompact subset of a soft semiregular space andbe a soft semiopen set containin. Then, there exists a soft semiopen setso that.

Proof. Let the given conditions be satisfied. Then, for each , we have . Therefore, there is a soft semiopen set so that . Now, is a soft semiopen cover of . Since is soft semicompact, then . Putting . Thus, .

Corollary 61. Ifis soft semicompact and soft semi-, then it is-soft semi-.

Proof. Suppose that and are two disjoint soft semiclosed sets. Then, . Since is soft semicompact, then is soft semicompact and since is soft semiregular, then there is a soft semiopen set so that . Obviously, and . Thus, is soft seminormal. Since is soft semi-, then it is -soft semi-. Hence, it is -soft semi-.

4. Semifixed Soft Points of Soft Mappings

In this section, we introduce a semifixed soft point property and investigate some main features, in particular, those related to parametric topological spaces.

Theorem 62. Letbe a collection of soft subsets of a soft semicompact spacesatisfying the following:(i) for each (ii) is a soft semiclosed set for each (iii) for each .

Then, .

Proof. Suppose that . Then, . It follows from (ii) that is a soft semiopen cover of . By hypothesis of soft semicompactness, there exist , so that . It follows from (iii) that . This yields a contradiction. Thus, we obtain the proof that .

Proposition 63. Letbe a soft semicompact and soft semi--space andbe a soft-continuous map. Then there exists a unique soft pointof.

Proof. Let be a family of soft subsets of . It is clear that for each . Since is soft -continuous, then is a soft semicompact set for each and since is soft semi-, then is also a soft semiclosed set for each . It follows from Theorem 62 that is a nonnull soft set. Note that . To show that , suppose that there is a so that . Let . Obviously, and for each . Now, is a soft compact subset of a soft semi--space, then is soft closed; therefore, is a soft semiclosed set for each . By Theorem 62, there exists a soft point so that . Therefore, . This is a contradiction. Thus, . Hence, the proof is complete.

Definition 64. (i) is said to have a semifixed soft point property if every soft -continuous map has a fixed soft point(ii)A property is said to be a -soft topological property if the property is preserved by soft -homeomorphism maps.

Proposition 65. The property of being a semifixed soft point is a-soft topological property.

Proof. Let and be a soft -homeomorphic. Then, there is a bijective soft map so that and are soft -continuous. Since has a semifixed soft point property, then every soft semi-continuous map has a semifixed soft point. Now, let be a soft semi-continuous. Obviously, is a soft semi-continuous. Also, is a soft semi-continuous. Since has a semifixed soft point property, then for some . consequently, . This implies that . Thus, is a semifixed soft point of . Hence, has a semifixed soft point property, as required.

Before we investigate a relationship between soft topological space and their parametric topological spaces in terms of possessing a fixed (soft) point, we need to prove the following result.

Theorem 66. Ifis extended, thenis soft-continuous iffis-continuous.

Proof. Necessity: let be a semiopen subset of . Then contains a soft semiopen set satisfying . Since is a soft -continuous map, then is a soft semiopen set. Definition 8 implies that a soft subset of is given by for each . By hypothesis, is extended; we obtain from Theorem 26 that a subset of is semiopen. Hence, a map is -continuous.
Sufficiency: let be a soft semiopen subset of . According to Definition 8, a soft subset of is given by for each . Since a map is -continuous, then a subset of is semiopen. By hypothesis, is extended; we obtain from Theorem 26 that is a soft semiopen subset of . Hence, a soft map is soft semi-continuous.

Definition 67. is said to have a semifixed point property if every semi-continuous maphas a fixed point.

Proposition 68. has the property of a semifixed soft point iffhas the property of a semifixed point for each.

Proof. Necessity: let has the property of a semifixed soft point. Then every soft -continuous map has a fixed soft point. Say, . It follows from the above theorem that is -continuous. Since is a fixed soft point of , then it must be that . Thus, has a fixed point. Hence, we obtain the desired result.
Sufficiency: let has the property of a semifixed point for each . Then, every semi-continuous map has a fixed point. Say, . It follows from the above theorem that is soft semi-continuous. Since is a fixed point of , then it must be that . Thus, has a fixed soft point. Hence, we obtain the desired result.

5. Conclusion

This work is a contribution of studying new classes to soft topology. First, we have introduced new soft separation axioms with respect to ordinary points by using total belong and total nonbelong relations. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft semiregular spaces. In general, we have studied their main properties and showed the interrelations between them with help of interesting examples. Second, we have defined semifixed soft point and investigated its basic properties.

As future works, we shall study them on the contents of supra soft topological spaces, minimal soft topological spaces, and soft weak structures. Finally, we hope that the concepts initiated herein will find their applications in many fields soon.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no competing interests.

Acknowledgments

We are highly grateful to the anonymous referees for their helpful comments and suggestions for improving the paper.