# Maximal Soft Compact and Maximal Soft Connected Topologies

Authors: Samer Al Ghour; Zanyar A. Ameen
Journal: Applied Computational Intelligence and Soft Computing
 (2022)

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## 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 softT1, while a maximal soft connected topology is soft T0. Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal.


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## Body

## 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: Sections1 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 Γ 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

LetZ be an initial universe, PZ be all subsets of Z and E be a set of parameters. An ordered pair Y,E=e,Ye:e∈E is said to be a soft set over Z, where Y:E⟶PZ is a set-valued mapping. The family of all soft sets on Z is represented by PZ,E. A soft element [16] is a soft set Y,E over Z in which Y,E=z for all e∈E, where z∈Z, and is denoted by z,E. A soft point [17], denoted by ze, is a soft set Y,E over Z in which Ye=z and Ye′=∅ for each e′≠e, e′∈E, where e∈E and z∈Z. A statement ze∈˜Y,E means that z∈Ye. The singleton soft set ze is referred to ze=e,z,e′,∅,…:∀e′∈E,e′≠e. The soft set Z,E\Y,E (or simply Yc,E) is the complement of Y,E, where Yc:E⟶PZ is given by Yce=Z\Ye for all e∈E. A soft subset Y,E over Z is called null, denoted by Φ˜, if Ye=∅ for any e∈E and called absolute, denoted by Z˜, if Ye=Z for any e∈E. Notice that Z˜c=Φ˜ and Φ˜c=Z˜. It is said that X,E1 is a soft subset of Y,E2 (written by X,E1⊆˜Y,E2, [18]) if E1⊆E2 and Xe⊆Ye for any e∈E1. We say X,E1=Y,E2 if X,E1⊆˜Y,E2 and Y,E2⊆˜X,E1.Definition 1.
(see [19, 20]). Let Yi,E:i∈I be a family of soft sets over Z, where I is any index set.(i)
The intersection ofYi,E, for i∈I, is a soft set Y,E such that Ye=∩i∈IYie for each e∈E and denoted by Y,E=∩˜i∈IYi,E(ii)
The union ofYi,E, for i∈I, is a soft set Y,E such that Ye=∪i∈IYie for each e∈E and denoted by Y,E=∪˜i∈IYi,EDefinition 2.
(see [7]). A collection ℑ of PZ,E is said to be a soft topology on Z if the following conditions are satisfied:(i)
Φ˜,Z˜∈˜ℑ(ii)
IfY1,E,Y2,E∈˜ℑ, then Y1,E∩˜Y2,E∈˜ℑ(iii)
If anyYi,E:i∈I⊆˜ℑ, then ∪˜i∈IYi,E∈˜ℑ
Terminologically, we callZ,ℑ,E a soft topological space on Z. The elements of ℑ are called soft ℑ-open sets (or simply soft open sets when no confusion arise), and their complements are called soft ℑ-closed sets (or soft closed sets).
In what follows, byZ,ℑ,E we mean a soft topological space, by two distinct soft points ue,ve′ we mean either u≠v or e≠e′, and by two disjoint soft sets Y,E,X,E over Z, we mean Y,E∩˜X,E=Φ˜.Definition 3.
(see [21]). A subcollection ℬ⊆ℑ is called a soft base for the soft topology ℑ if each element of ℑ is a union of elements of ℬ.Definition 4.
(see [7]). Let Y,E≠Φ˜ be a soft subset of Z,ℑ,E. Then, ℑY:=G,E∩˜Y,E:G,E∈˜ℑ is called a soft relative topology over Y and Y,ℑY,E is a soft subspace of Z,ℑ,E.Definition 5.
(see [7]). Let Y,E be a soft subset of Z,ℑ,E. The soft interior of Y,E, denoted by IntℑY,E, is the largest soft open set contained in Y,E. The soft closure of Y,E, denoted by ClℑY,E, is the smallest soft closed set which contains Y,E. The soft closure and interior of a soft subset Y,E in the soft subspace X,ℑX,E are, respectively, denoted by ClXY,E and IntXY,E.Lemma 1 (see [22]).
For a soft subsetY,E of Z,ℑ,E,(1)IntℑY,Ec=ClℑY,Ec,ClℑY,Ec=IntℑY,Ec.Definition 6.
(see [23, 24]). Let Z,E,Y,E′ be soft sets, and let p:Z⟶Y,q:E⟶E′ be functions. The image of a soft set A,E⊆˜Z,E under f:Z,E⟶Y,E′ is a soft subset fA,E=fA,qE of Y,E′ which is given by(2)fAe′=∪e∈q−1e′∩EpAe,q−1e′∩E≠∅,∅,otherwise,for each e′∈E′.
The inverse image of a soft setB,E′⊆˜Y,E′ under f is a soft subset f−1B,E′=f−1B,q−1E′ such that(3)f−1Be=p−1Bqe,qe∈E′,∅,otherwise,for each e∈E.
The soft functionf is bijective if both p and q are bijective.Lemma 2.
LetZ,E,Y,E′ be soft sets. If f:Z,E⟶Y,E′ is bijection, then fA,Ec=fA,Ec for each A,E⊆˜Z,E.Proof. It follows from Theorem 3.14 in [24].Definition 7.
(see [16]). Let Z,ℑ,E and Y,T,E′ be soft topological spaces. A soft function f:Z,ℑ,E⟶Y,T,E′ is said to be(i)
Soft continuous if the inverse image of each soft open subset ofY,T,E′ is a soft open subset of Z,ℑ,E(ii)
Soft open if the image of each soft open subset ofZ,ℑ,E is a soft open subset of Y,T,E′(iii)
Soft homeomorphism if it is a soft open and soft continuous bijection fromZ,ℑ,E onto Y,T,E′


## 3. Maximal Soft Topologies

Definition 8.
Letℑ1,ℑ2 be two soft topologies on Z. It is said that ℑ1 is coarser than ℑ2 if ℑ1⊂˜ℑ2. And ℑ1 is finer than ℑ2 if ℑ2⊂˜ℑ1.Lemma 3.
Letℱ=ℑα:α∈Λ be a family of soft topologies on Z, where Λ is any index set. Then ℑ=∩˜α∈Λℑα is a soft topology on Z.Proof. Evidently,Φ˜,Z˜ belong to ℑ as they belong to ℑα for all αs. Let Y1,E,Y2,E∈˜ℑ. Then Y1,E,Y2,E∈˜ℑα for all α∈Λ. Since all ℑα are soft topologies, so Y1,E∩˜Y2,E∈˜ℑα for all α∈Λ. Therefore Y1,E∩˜Y2,E∈˜∩˜α∈Λℑα=ℑ. Let Yλ:λ∈Δ be a collection of sets in ℑ. Then for each α∈Λ, Yλ∈˜ℑα for all λ∈Δ. But for each α, ℑα is a soft topology on Z, so ∪˜λ∈ΔYλ∈˜ℑα for all α∈Λ. Hence ∪˜λ∈ΔYλ∈˜∩˜α∈Λℑα=ℑ.The above result is an extension of Proposition 6 in [7].Lemma 4.
LetC be a collection of soft subsets over Z. There exists a unique soft topology ℑ on Z including C and if ℑ0 is any other soft topology on Z that includes C, then ℑ⊂˜ℑ0.Proof. Notice that such a soft topology always exists becausePZ,E is the soft topology on Z which includes C. Consider ℑ, the intersection of all those soft topologies on Z which include C. Then it follows from Lemma 3 that ℑ is the required soft topology.Definition 9.
LetC be a collection of soft subsets over Z. The unique soft topology obtained in the above lemma is called the soft topology on Z generated by the collection C and is denoted by ℑC, which is the smallest soft topology on Z including C.
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.
Letℑ1,ℑ2 be two soft topologies on Z. The generating soft topology ℑℑ1∪˜ℑ2 is identical to the soft topology ℑℱ generated by ℱ=G1,E∩˜G2,E:G1,E∈˜ℑ1,G2,E∈˜ℑ2.Proof. Sinceℑ1,ℑ2 soft topologies, so they include Z˜. By taking Z˜=Gi,E, for i=1,2, then ℱ will exactly contain ℑ1,ℑ2. By the uniqueness of the generating soft topology, ℑℱ=ℑℑ1∪˜ℑ2, see Definition 9.Theorem 1.
The setΣ of all soft topologies over a common universe Z forms a complete lattice under soft set inclusion “⊂˜”.Proof. One can easily show that⊂˜ is a partially ordered set on Σ. It remains to prove that every subset of Σ has the greatest lower bound and the least upper bound. Let Σ0 be a subset of Σ. By Lemma 3, ∧Σ0=∩˜ℑ0:ℑ0∈˜Σ0 is the greatest lower bound of Σ0. By Lemma 6, ∨Σ0=ℑ∪˜ℑ0:ℑ0∈˜Σ0 is the least upper bound of Σ0.Remark 1.
Note that the indiscrete soft topologyℑI on Z is the minimal (smallest) element in Σ and the discrete soft topology ℑD on Z is the maximal (largest) element in Σ. It is worth remarking that ℑD is the maximal soft Hausdorff topology and ℑI 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.
LetZ,ℑ,E be a soft topological space with the property Γ. Then ℑ is called Γ-maximal if any soft topology finer than ℑ does not have the property Γ.


### 3.1. Maximal Soft Compact Topologies

Recall that a spaceZ,ℑ,E is called soft compact [8] if each soft open cover of Z˜ has a finite subcover. If we replace Γ by soft compactness, Definition 10 will be as follows.Definition 11.
LetZ,ℑ,E be a soft compact space. Then ℑ is called maximal soft compact if any soft topology finer than ℑ is not soft compact.
The following example shows the structure of maximal soft compact spaces.Example 1.
Consider the set of naturalsℕ and E=e. Define a soft topology ℑ=YE⊆˜ℕ˜:YE=neforn≠1or1e∈˜YEifYEcis finite∪˜Φ˜. Then ℕ,ℑ,E is maximal soft compact.Lemma 6 (see [25], Proposition 5.1).
LetZ,ℑ,E be a soft compact space and let Y,E⊆˜Z˜. If Y,E is soft closed, then Y,E is soft compact.Definition 12.
(see [26]). Let Z,ℑ,E be any soft topological space and let Y,E be any soft non-open subset over Z. The soft topology ℑ∗ on Z generated by ℑ∪˜Y,E is said to be an s-extension of ℑ and it is denoted by ℑ∗=ℑY,E (or shortly, ℑ∗=ℑY).Lemma 7 (see [26], Theorem 3.2).
LetZ,ℑ,E be a soft compact space. Then Z,ℑY,E is soft compact if and only if Yc,E is soft compact in Z,ℑ,E.Proposition 1.
A soft topological spaceZ,ℑ,E is maximal soft compact if and only if the family of all soft ℑ-closed sets is equal to the family of all soft ℑ-compact sets.Proof. Assume thatℑ is maximal soft compact. If Y,E is a soft compact set but not soft closed, then Yc,E is not soft open. By Lemma 7, ℑYc is soft compact. But ℑ⊂˜ℑYc, so this violates the maximality of ℑ. Hence Y,E must be closed.Conversely, suppose the family of all softℑ-closed sets is equal to the family of all soft ℑ-compact sets. If ℑ is not maximal soft compact, there exists a soft compact topology ℑ∗ such that ℑ⊂˜ℑ∗. Therefore there is a set Y,E⊂˜Z˜ which is soft ℑ∗-closed but not soft ℑ-closed. By assumption, Y,E is not ℑ-compact. Therefore, there exists a soft ℑ-open cover U of Y,E which has no finite subcover. Since ℑ⊂˜ℑ∗, so U is also a soft ℑ∗-open cover of Y,E. This means that Y,E is not soft ℑ∗-compact. But this is a contradiction, because Y,E is soft ℑ∗-closed and, by Lemma 6, it must be soft ℑ∗-compact. This proves that ℑ has to be maximal soft compact.Proposition 2.
A soft topological spaceZ,ℑ,E is maximal soft compact if and only if each soft continuous bijection from a soft compact space onto Z,ℑ,E is a soft homeomorphism.Proof. AssumeZ,ℑ,E is maximal soft compact. Let f:Y,ℑ′,E′⟶Z,ℑ,E be a soft continuous bijection, where Y,ℑ′,E′ is soft compact. Take T=fU,E′:U,E′∈˜ℑ′. Evidently Z,ℑ,E is a soft topology and f is a soft homeomorphism. Since Y,ℑ′,E′ is soft compact, Z,ℑ,E is also soft compact under f. But ℑ⊆˜T and ℑ is maximal, thus T=ℑ.Conversely, ifZ,ℑ,E is not maximal soft compact, then there exists a soft compact topology ℑ∗ on Z such that ℑ⊆˜ℑ∗. Then the identity soft function h:Z,ℑ∗,E⟶Z,ℑ,E is a soft continuous bijection but not a soft homeomorphism. This completes the proof.Theorem 2.
For a soft topological spaceZ,ℑ,E, the following are equivalent:(1)
ℑ is maximal soft compact(2)
The family of all softℑ-closed sets is equal to the family of all soft ℑ-compact sets(3)
Each soft continuous bijection from a soft compact space ontoZ,ℑ,E is a soft homeomorphismProof.(1)⇒ (2) Proposition 1(2)⇒ (3) Suppose f is a soft continuous bijection from a soft compact space Y,ℑ′,E′ onto Z,ℑ,E. It remains to check that f is soft open. Let G,E′ be soft ℑ′-open. Then Gc,E′ is soft ℑ′-closed. Since Y,ℑ′,E′ is soft compact space, by Lemma, 3.11, Gc,E′ is soft ℑ′-compact. From the soft continuity of f, fGc,E′ is soft ℑ-compact and by (2) fGc,E′ is soft ℑ-closed. Since f is bijective, fGc,E′=fG,E′c and so fG,E′ is soft open. Hence f is a soft open function.(3)⇒ (1) Proposition 2.Lemma 8 (see [11], Theorem 4.1).
LetZ,ℑ,E be a soft topological space. If each singleton soft set is soft closed, then Z,ℑ,E is soft T1.The above is true in various soft point theories, see [27].Theorem 3.
IfZ,ℑ,E is a maximal soft compact space, then Z,ℑ,E is soft T1.Proof. One can easily show that each singleton soft set is soft compact. SinceZ,ℑ,E 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 A,E from Z,ℑ,E is called stable if there exists a subset Y of Z such that Ae=Y for each e∈E.Definition 14.
We call a soft topological spaceZ,ℑ,E stable if each soft open is stable.Lemma 9.
IfZ,ℑ,E is a stable soft T2-space, then each soft compact is soft closed.Proof. It follows from Lemma 7 and Theorem 8 in [15].Theorem 4.
IfZ,ℑ,E is a stable soft compact T2-space, then Z,ℑ,E is maximal soft compact.Proof. Ifℑ is not maximal soft compact, there exists a soft compact topology ℑ∗ on Z such that ℑ⊂˜ℑ∗. Pick a set Y,E to be soft ℑ∗-open but not soft ℑ-open. Let ℑ0=ℑY. Then ℑ⊂˜ℑ0⊂˜ℑ∗ and so ℑ0 is soft compact. By Lemma 7, Yc,E is a soft ℑ-compact set and by Lemma 9, it is soft ℑ-closed. This implies Y,E is soft ℑ-open, which contradicts to the choice of Y,E. Hence Z,ℑ,E is maximal soft compact.


### 3.2. Maximal Soft Connected Topologies

Definition 15.
(see [9]). A soft topological space Z,ℑ,E 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 Z,ℑ,E be a soft connected space. Then ℑ is called maximal soft connected if any soft topology finer than ℑ is not soft connected.We start by giving some examples of maximal soft connected spaces.Example 2.
Letℝ be the set of reals and let E=e1,e2. The soft topological space ℝ,ℑ,E is maximal soft connected, where ℑ=YE⊆˜ℝ˜:0e1∈˜YE∪˜Φ˜˜.Example 3.
(see [26], Example 3.3). The soft topological space ℝ,ℑ,E is maximal soft connected, where ℑ=YE⊆˜ℝ˜:0e1∉˜YE∪˜ℝ˜, ℝ is the set of reals, and E=e1,e2.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 e1 is maximal, while the crisp topology of e2 is not maximal.Theorem 5.
LetY,E be a soft subset of a soft topological space Z,ℑ,E. If Y,E,Yc,E are soft connected (as soft subspaces) and either of them is soft open, then ℑ is maximal soft connected.Proof. Assume Y,E≠Φ˜≠Yc,E, otherwise the result trivially holds. Suppose that Y,E,Yc,E are not soft ℑ-open. Then, by taking ℑ∗=ℑY, we obtain a disconnected soft topology ℑ∗ such that ℑ⊂˜ℑ∗. Therefore, there are disjoint soft ℑ∗-open sets G,E,H,E such that Z˜=G,E∪˜H,E. W.L.O.G assume that G,E∩˜Yc,E≠Φ˜≠H,E∩˜Yc,E. Then G,E∩˜Yc,E,H,E∩˜Yc,E are disjoint soft ℑYc∗-open and G,E∩˜Yc,E∪˜H,E∩˜Yc,E=Yc,E. By Remark 2.2 (vii) in [26], G,E∩˜Yc,E,H,E∩˜Yc,E are disjoint soft ℑYc-open which means that Yc,ℑYc,E is not soft connected, a contradiction. Therefore, either G,E⊆˜Y,E or H,E⊆˜Y,E. Suppose G,E⊆˜Y,E. If G,E=Y,E, then H,E=Yc,E, which is a soft ℑ-open set, but that is not possible (as it is soft ℑ∗-open). Therefore, we must have H,E∩˜Y,E≠Φ˜. Hence G,E∪˜H,E∩˜Y,E=Y,E, where G,E,H,E∩˜Y,E are soft ℑY∗-open. But, by Remark 2.2 (vii) in [26] soft ℑY∗-open and soft ℑY-open are similar, thus Y,ℑY,E is soft disconnected, again contradiction. The result is proved.Theorem 5. Let Z,ℑ,E be a soft connected space. If ℑ is maximal soft connected, then Z,ℑ,E is soft T0.Proof. SupposeZ,ℑ,E is not soft T0. Then there are soft points ye,ze′ with ye≠ze′ such that ye,ze′∈˜G,E for all soft ℑ-open sets G,E. Therefore, ye∈˜Clze′ and ze′∈˜Clye. Let ℑ∗=ℑze′. Then ℑ⊂˜ℑ∗ and so ℑ∗ is not soft connected as ℑ is maximal soft connected. Therefore, there exist disjoint soft ℑ∗-open sets G,E,H,E such that G,E∪˜H,E=Z˜. Thus either ye,ze′∈˜G,E or ye,ze′∈˜H,E. If ye,ze′∈˜G,E, by Remark 2.2 (ii) in [26], each soft ℑ∗-open set containing ye is also soft ℑ-open, so G,E is soft ℑ-open. Therefore, there is a soft ℑ-open U,E such that ye∈˜U,E⊆˜G,E. But U,E is a soft ℑ-open containing ze′ such that U,E⊆˜G,E. Thus U,E is a soft ℑ-open containing each of its points, and similarly for H,E. This implies that Z,ℑ,E is soft disconnected, which is impossible.Proposition 3.
IfY,E is a soft open connected subset of a maximal soft connected space Z,ℑ,E, then Y,ℑY,E is maximal soft connected.Proof. IfY,ℑY,E is not maximal soft connected, then there exists a soft connected topology T on Y such that TY⊂˜T. Let X,E⊆˜Y,E be a soft ℑ-open set but not soft TY-open. If T∗=ℑYX, then ℑY⊂˜T∗ and so T∗ is soft connected. If ℑ∗=ℑX, then ℑ⊂˜ℑ∗ but ℑ∗ cannot be soft connected. Therefore, there are disjoint soft ℑ∗-open sets G,E,H,E such that G,E∪˜H,E=Z˜. Then either Y,E⊆˜G,E or Y,E⊆˜H,E, differently Y,T∗,E=Y,ℑY∗,E will be soft disconnected by the disjoint soft T∗-open sets G,E∩˜Y,E,H,E∩˜Y,E (impossible). Hence, we assume Y,E⊆˜G,E. By Remark 2.2 (ii), for all soft points ze that belongs to some soft ℑ∗-open set but does not belong to any soft ℑ-open, we have(4)ze∈˜X,E⊆˜Y,E⊆˜G,E.This means thatG,E is soft ℑ-open as Y,E is soft ℑ-open. Similarly, one can show that H,E is soft ℑ-open. Hence G,E,H,E are disjoint soft ℑ-open and G,E∪˜H,E=Z˜, which proves that Z,ℑ,E is not soft connected, a contradiction.Proposition 4.
IfY,E is a soft closed connected subset of a maximal soft connected space Z,ℑ,E, then Y,ℑY,E is maximal soft connected.Proof. AssumeY,ℑY,E is not maximal soft connected. Take a set X,E⊂˜Y,E which is not soft ℑY-open. If T=ℑYX, then ℑY⊂˜T and so T is soft connected. Let W,E=Z˜\Y,E\X,E. Then, we have W,E∩˜Y,E=X,E. Therefore, ℑYX=ℑY∗, where ℑ∗ is an s-extension of ℑ with respect to W,E (i.e. ℑ∗=ℑW). Since ℑ is maximal soft connected and ℑ⊂˜ℑ∗, then ℑ∗ is soft disconnected. Thus there exist disjoint soft ℑ∗-open sets G,E,H,E such that G,E∪˜H,E=Z˜. Since ℑY∗ is soft connected, then either Y,E⊂˜G,E or Y,E⊂˜H,E. Suppose Y,E⊂˜G,E. We consider two cases: (i) suppose ze∈˜G,E. Since G,E is soft ℑ∗-open, there are soft ℑ-open sets U,E,V,E such that ze∈˜U,E∪˜V,E∩˜W,E⊂˜G,E. If ze∈˜U,E⊂˜G,E and so ze∈˜IntℑG,E. If ze∈˜V,E∩˜W,E⊂˜G,E, then ze∈˜V,E=V,E∩˜Z˜\W,E∪˜W,E⊂˜G,E as V,E∩˜Z˜\W,E⊂˜Z˜\W,E=Y,E\X,E⊆˜Y,E⊆˜G,E. Again ze∈˜IntℑG,E. Since ze was arbitrarily taken, so G,E is soft ℑ-open. (ii) suppose ze∈˜H,E. Since H,E is soft ℑ∗-open, there are soft ℑ-open sets U,E,V,E such that ze∈˜U,E∪˜V,E∩˜W,E⊂˜H,E. If ze∈˜U,E⊂˜H,E and so ze∈˜IntℑH,E. If ze∈˜V,E∩˜W,E, then, since ze∈˜H,E⊆˜Z˜\Y,E⊂˜W,E, so ze∈˜V,E∩˜Z˜\Y,E⊂˜V,E∩˜W,E. Since Z˜\Y,E is soft ℑ-open, therefore ze∈˜IntℑH,E. Thus H,E is soft ℑ-open. This means that Z,ℑ,E is not soft connected, which is impossible. Hence Y,ℑY,E must be maximal soft connected.Definition 16. (see [29]). A soft topological space Z,ℑ,E is called soft submaximal if each soft ℑ-dense set is soft ℑ-open.Lemma 10.
LetZ,ℑ,E be a soft connected space and let ℑ∗=ℑY be an s-extension of ℑ over Z. If Y,E is soft ℑ-dense, then Z,ℑ∗,E is soft connectedProof. It follows from Theorem 3.17 [26].Lemma 11.
IfZ,ℑ,E is a maximal soft connected space, then Z,ℑ,E is soft submaximal.Proof. Let (D, E) be a soft ℑ-dense set over Z. By Lemma 10, Z,ℑ∗,E is soft connected, where ℑ∗=ℑD, but ℑ is soft maximal, hence ℑ=ℑ∗. Thus (D, E) must be a soft ℑ-open.Theorem 5. If Y,E is a soft connected subset of a maximal soft connected space Z,ℑ,E, then Y,ℑY,E is maximal soft connected.Proof. SinceY,E is soft connected, then ClℑY,E is also soft connected and, by Proposition 4, ClℑY,E is maximal. By Lemma 11, Y,E is soft open in ClℑY,E because Y,E is soft dense in ClℑY,E. By Proposition 3, Y,E is maximal soft connected.


## 3.1. Maximal Soft Compact Topologies

Recall that a spaceZ,ℑ,E is called soft compact [8] if each soft open cover of Z˜ has a finite subcover. If we replace Γ by soft compactness, Definition 10 will be as follows.Definition 11.
LetZ,ℑ,E be a soft compact space. Then ℑ is called maximal soft compact if any soft topology finer than ℑ is not soft compact.
The following example shows the structure of maximal soft compact spaces.Example 1.
Consider the set of naturalsℕ and E=e. Define a soft topology ℑ=YE⊆˜ℕ˜:YE=neforn≠1or1e∈˜YEifYEcis finite∪˜Φ˜. Then ℕ,ℑ,E is maximal soft compact.Lemma 6 (see [25], Proposition 5.1).
LetZ,ℑ,E be a soft compact space and let Y,E⊆˜Z˜. If Y,E is soft closed, then Y,E is soft compact.Definition 12.
(see [26]). Let Z,ℑ,E be any soft topological space and let Y,E be any soft non-open subset over Z. The soft topology ℑ∗ on Z generated by ℑ∪˜Y,E is said to be an s-extension of ℑ and it is denoted by ℑ∗=ℑY,E (or shortly, ℑ∗=ℑY).Lemma 7 (see [26], Theorem 3.2).
LetZ,ℑ,E be a soft compact space. Then Z,ℑY,E is soft compact if and only if Yc,E is soft compact in Z,ℑ,E.Proposition 1.
A soft topological spaceZ,ℑ,E is maximal soft compact if and only if the family of all soft ℑ-closed sets is equal to the family of all soft ℑ-compact sets.Proof. Assume thatℑ is maximal soft compact. If Y,E is a soft compact set but not soft closed, then Yc,E is not soft open. By Lemma 7, ℑYc is soft compact. But ℑ⊂˜ℑYc, so this violates the maximality of ℑ. Hence Y,E must be closed.Conversely, suppose the family of all softℑ-closed sets is equal to the family of all soft ℑ-compact sets. If ℑ is not maximal soft compact, there exists a soft compact topology ℑ∗ such that ℑ⊂˜ℑ∗. Therefore there is a set Y,E⊂˜Z˜ which is soft ℑ∗-closed but not soft ℑ-closed. By assumption, Y,E is not ℑ-compact. Therefore, there exists a soft ℑ-open cover U of Y,E which has no finite subcover. Since ℑ⊂˜ℑ∗, so U is also a soft ℑ∗-open cover of Y,E. This means that Y,E is not soft ℑ∗-compact. But this is a contradiction, because Y,E is soft ℑ∗-closed and, by Lemma 6, it must be soft ℑ∗-compact. This proves that ℑ has to be maximal soft compact.Proposition 2.
A soft topological spaceZ,ℑ,E is maximal soft compact if and only if each soft continuous bijection from a soft compact space onto Z,ℑ,E is a soft homeomorphism.Proof. AssumeZ,ℑ,E is maximal soft compact. Let f:Y,ℑ′,E′⟶Z,ℑ,E be a soft continuous bijection, where Y,ℑ′,E′ is soft compact. Take T=fU,E′:U,E′∈˜ℑ′. Evidently Z,ℑ,E is a soft topology and f is a soft homeomorphism. Since Y,ℑ′,E′ is soft compact, Z,ℑ,E is also soft compact under f. But ℑ⊆˜T and ℑ is maximal, thus T=ℑ.Conversely, ifZ,ℑ,E is not maximal soft compact, then there exists a soft compact topology ℑ∗ on Z such that ℑ⊆˜ℑ∗. Then the identity soft function h:Z,ℑ∗,E⟶Z,ℑ,E is a soft continuous bijection but not a soft homeomorphism. This completes the proof.Theorem 2.
For a soft topological spaceZ,ℑ,E, the following are equivalent:(1)
ℑ is maximal soft compact(2)
The family of all softℑ-closed sets is equal to the family of all soft ℑ-compact sets(3)
Each soft continuous bijection from a soft compact space ontoZ,ℑ,E is a soft homeomorphismProof.(1)⇒ (2) Proposition 1(2)⇒ (3) Suppose f is a soft continuous bijection from a soft compact space Y,ℑ′,E′ onto Z,ℑ,E. It remains to check that f is soft open. Let G,E′ be soft ℑ′-open. Then Gc,E′ is soft ℑ′-closed. Since Y,ℑ′,E′ is soft compact space, by Lemma, 3.11, Gc,E′ is soft ℑ′-compact. From the soft continuity of f, fGc,E′ is soft ℑ-compact and by (2) fGc,E′ is soft ℑ-closed. Since f is bijective, fGc,E′=fG,E′c and so fG,E′ is soft open. Hence f is a soft open function.(3)⇒ (1) Proposition 2.Lemma 8 (see [11], Theorem 4.1).
LetZ,ℑ,E be a soft topological space. If each singleton soft set is soft closed, then Z,ℑ,E is soft T1.The above is true in various soft point theories, see [27].Theorem 3.
IfZ,ℑ,E is a maximal soft compact space, then Z,ℑ,E is soft T1.Proof. One can easily show that each singleton soft set is soft compact. SinceZ,ℑ,E 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 A,E from Z,ℑ,E is called stable if there exists a subset Y of Z such that Ae=Y for each e∈E.Definition 14.
We call a soft topological spaceZ,ℑ,E stable if each soft open is stable.Lemma 9.
IfZ,ℑ,E is a stable soft T2-space, then each soft compact is soft closed.Proof. It follows from Lemma 7 and Theorem 8 in [15].Theorem 4.
IfZ,ℑ,E is a stable soft compact T2-space, then Z,ℑ,E is maximal soft compact.Proof. Ifℑ is not maximal soft compact, there exists a soft compact topology ℑ∗ on Z such that ℑ⊂˜ℑ∗. Pick a set Y,E to be soft ℑ∗-open but not soft ℑ-open. Let ℑ0=ℑY. Then ℑ⊂˜ℑ0⊂˜ℑ∗ and so ℑ0 is soft compact. By Lemma 7, Yc,E is a soft ℑ-compact set and by Lemma 9, it is soft ℑ-closed. This implies Y,E is soft ℑ-open, which contradicts to the choice of Y,E. Hence Z,ℑ,E is maximal soft compact.


## 3.2. Maximal Soft Connected Topologies

Definition 15.
(see [9]). A soft topological space Z,ℑ,E 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 Z,ℑ,E be a soft connected space. Then ℑ is called maximal soft connected if any soft topology finer than ℑ is not soft connected.We start by giving some examples of maximal soft connected spaces.Example 2.
Letℝ be the set of reals and let E=e1,e2. The soft topological space ℝ,ℑ,E is maximal soft connected, where ℑ=YE⊆˜ℝ˜:0e1∈˜YE∪˜Φ˜˜.Example 3.
(see [26], Example 3.3). The soft topological space ℝ,ℑ,E is maximal soft connected, where ℑ=YE⊆˜ℝ˜:0e1∉˜YE∪˜ℝ˜, ℝ is the set of reals, and E=e1,e2.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 e1 is maximal, while the crisp topology of e2 is not maximal.Theorem 5.
LetY,E be a soft subset of a soft topological space Z,ℑ,E. If Y,E,Yc,E are soft connected (as soft subspaces) and either of them is soft open, then ℑ is maximal soft connected.Proof. Assume Y,E≠Φ˜≠Yc,E, otherwise the result trivially holds. Suppose that Y,E,Yc,E are not soft ℑ-open. Then, by taking ℑ∗=ℑY, we obtain a disconnected soft topology ℑ∗ such that ℑ⊂˜ℑ∗. Therefore, there are disjoint soft ℑ∗-open sets G,E,H,E such that Z˜=G,E∪˜H,E. W.L.O.G assume that G,E∩˜Yc,E≠Φ˜≠H,E∩˜Yc,E. Then G,E∩˜Yc,E,H,E∩˜Yc,E are disjoint soft ℑYc∗-open and G,E∩˜Yc,E∪˜H,E∩˜Yc,E=Yc,E. By Remark 2.2 (vii) in [26], G,E∩˜Yc,E,H,E∩˜Yc,E are disjoint soft ℑYc-open which means that Yc,ℑYc,E is not soft connected, a contradiction. Therefore, either G,E⊆˜Y,E or H,E⊆˜Y,E. Suppose G,E⊆˜Y,E. If G,E=Y,E, then H,E=Yc,E, which is a soft ℑ-open set, but that is not possible (as it is soft ℑ∗-open). Therefore, we must have H,E∩˜Y,E≠Φ˜. Hence G,E∪˜H,E∩˜Y,E=Y,E, where G,E,H,E∩˜Y,E are soft ℑY∗-open. But, by Remark 2.2 (vii) in [26] soft ℑY∗-open and soft ℑY-open are similar, thus Y,ℑY,E is soft disconnected, again contradiction. The result is proved.Theorem 5. Let Z,ℑ,E be a soft connected space. If ℑ is maximal soft connected, then Z,ℑ,E is soft T0.Proof. SupposeZ,ℑ,E is not soft T0. Then there are soft points ye,ze′ with ye≠ze′ such that ye,ze′∈˜G,E for all soft ℑ-open sets G,E. Therefore, ye∈˜Clze′ and ze′∈˜Clye. Let ℑ∗=ℑze′. Then ℑ⊂˜ℑ∗ and so ℑ∗ is not soft connected as ℑ is maximal soft connected. Therefore, there exist disjoint soft ℑ∗-open sets G,E,H,E such that G,E∪˜H,E=Z˜. Thus either ye,ze′∈˜G,E or ye,ze′∈˜H,E. If ye,ze′∈˜G,E, by Remark 2.2 (ii) in [26], each soft ℑ∗-open set containing ye is also soft ℑ-open, so G,E is soft ℑ-open. Therefore, there is a soft ℑ-open U,E such that ye∈˜U,E⊆˜G,E. But U,E is a soft ℑ-open containing ze′ such that U,E⊆˜G,E. Thus U,E is a soft ℑ-open containing each of its points, and similarly for H,E. This implies that Z,ℑ,E is soft disconnected, which is impossible.Proposition 3.
IfY,E is a soft open connected subset of a maximal soft connected space Z,ℑ,E, then Y,ℑY,E is maximal soft connected.Proof. IfY,ℑY,E is not maximal soft connected, then there exists a soft connected topology T on Y such that TY⊂˜T. Let X,E⊆˜Y,E be a soft ℑ-open set but not soft TY-open. If T∗=ℑYX, then ℑY⊂˜T∗ and so T∗ is soft connected. If ℑ∗=ℑX, then ℑ⊂˜ℑ∗ but ℑ∗ cannot be soft connected. Therefore, there are disjoint soft ℑ∗-open sets G,E,H,E such that G,E∪˜H,E=Z˜. Then either Y,E⊆˜G,E or Y,E⊆˜H,E, differently Y,T∗,E=Y,ℑY∗,E will be soft disconnected by the disjoint soft T∗-open sets G,E∩˜Y,E,H,E∩˜Y,E (impossible). Hence, we assume Y,E⊆˜G,E. By Remark 2.2 (ii), for all soft points ze that belongs to some soft ℑ∗-open set but does not belong to any soft ℑ-open, we have(4)ze∈˜X,E⊆˜Y,E⊆˜G,E.This means thatG,E is soft ℑ-open as Y,E is soft ℑ-open. Similarly, one can show that H,E is soft ℑ-open. Hence G,E,H,E are disjoint soft ℑ-open and G,E∪˜H,E=Z˜, which proves that Z,ℑ,E is not soft connected, a contradiction.Proposition 4.
IfY,E is a soft closed connected subset of a maximal soft connected space Z,ℑ,E, then Y,ℑY,E is maximal soft connected.Proof. AssumeY,ℑY,E is not maximal soft connected. Take a set X,E⊂˜Y,E which is not soft ℑY-open. If T=ℑYX, then ℑY⊂˜T and so T is soft connected. Let W,E=Z˜\Y,E\X,E. Then, we have W,E∩˜Y,E=X,E. Therefore, ℑYX=ℑY∗, where ℑ∗ is an s-extension of ℑ with respect to W,E (i.e. ℑ∗=ℑW). Since ℑ is maximal soft connected and ℑ⊂˜ℑ∗, then ℑ∗ is soft disconnected. Thus there exist disjoint soft ℑ∗-open sets G,E,H,E such that G,E∪˜H,E=Z˜. Since ℑY∗ is soft connected, then either Y,E⊂˜G,E or Y,E⊂˜H,E. Suppose Y,E⊂˜G,E. We consider two cases: (i) suppose ze∈˜G,E. Since G,E is soft ℑ∗-open, there are soft ℑ-open sets U,E,V,E such that ze∈˜U,E∪˜V,E∩˜W,E⊂˜G,E. If ze∈˜U,E⊂˜G,E and so ze∈˜IntℑG,E. If ze∈˜V,E∩˜W,E⊂˜G,E, then ze∈˜V,E=V,E∩˜Z˜\W,E∪˜W,E⊂˜G,E as V,E∩˜Z˜\W,E⊂˜Z˜\W,E=Y,E\X,E⊆˜Y,E⊆˜G,E. Again ze∈˜IntℑG,E. Since ze was arbitrarily taken, so G,E is soft ℑ-open. (ii) suppose ze∈˜H,E. Since H,E is soft ℑ∗-open, there are soft ℑ-open sets U,E,V,E such that ze∈˜U,E∪˜V,E∩˜W,E⊂˜H,E. If ze∈˜U,E⊂˜H,E and so ze∈˜IntℑH,E. If ze∈˜V,E∩˜W,E, then, since ze∈˜H,E⊆˜Z˜\Y,E⊂˜W,E, so ze∈˜V,E∩˜Z˜\Y,E⊂˜V,E∩˜W,E. Since Z˜\Y,E is soft ℑ-open, therefore ze∈˜IntℑH,E. Thus H,E is soft ℑ-open. This means that Z,ℑ,E is not soft connected, which is impossible. Hence Y,ℑY,E must be maximal soft connected.Definition 16. (see [29]). A soft topological space Z,ℑ,E is called soft submaximal if each soft ℑ-dense set is soft ℑ-open.Lemma 10.
LetZ,ℑ,E be a soft connected space and let ℑ∗=ℑY be an s-extension of ℑ over Z. If Y,E is soft ℑ-dense, then Z,ℑ∗,E is soft connectedProof. It follows from Theorem 3.17 [26].Lemma 11.
IfZ,ℑ,E is a maximal soft connected space, then Z,ℑ,E is soft submaximal.Proof. Let (D, E) be a soft ℑ-dense set over Z. By Lemma 10, Z,ℑ∗,E is soft connected, where ℑ∗=ℑD, but ℑ is soft maximal, hence ℑ=ℑ∗. Thus (D, E) must be a soft ℑ-open.Theorem 5. If Y,E is a soft connected subset of a maximal soft connected space Z,ℑ,E, then Y,ℑY,E is maximal soft connected.Proof. SinceY,E is soft connected, then ClℑY,E is also soft connected and, by Proposition 4, ClℑY,E is maximal. By Lemma 11, Y,E is soft open in ClℑY,E because Y,E is soft dense in ClℑY,E. By Proposition 3, Y,E 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 compactT2 topologies and is contained in the class of soft T1 topologies. Furthermore, we have seen that the class of maximal soft connected topologies is contained in the class of soft T0 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 topologyℑ on Z with the property Γ is Γ-maximal if any soft topology finer than ℑ does not have the property Γ. 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.



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