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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279

Published online September 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.3.269

© The Korean Institute of Intelligent Systems

Soft Minimal Soft Sets and Soft Prehomogeneity in Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Correspondence to :
Samer Al Ghour (algore@just.edu.jo)

Received: July 4, 2021; Revised: August 16, 2021; Accepted: August 28, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft sets and soft minimal soft preopen sets as two new classes of soft sets in soft topological spaces, and we define soft prehomogeneity as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points, and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous soft topological space that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete soft topological space in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.

Keywords: Soft minimal soft open sets, Soft preopen soft sets, Soft locally indiscrete, Prehomogeneity, Generated soft topology

This paper follows the notions and terminologies as appeared in [1] and [2]. In this paper, TS and STS will denote topological space and soft topological space, respectively. Molodtsov [3] defined soft sets in 1999. The soft set theory offers a general mathematical tool for dealing with uncertain objects. Let U be a universal set and E be a set of parameters. A soft set over U relative to E is a function G : E → ℘(U). SS (U,E) will denote the family of all soft sets over U relative to E. In this paper, the null soft set and the absolute soft set will be denoted by 0E and 1E, respectively. The structure of STSs was defined in [4] as follows: An STS is a triplet (U, τ,E), where τSS (U,E), τ contains 0E and 1E, τ is closed under finite soft intersection, and τ is closed under arbitrary soft union. Let (U, τ,E) be an STS and FSS(X,A), then F is said to be a soft open set in (U, τ,E) if Fτ and F is said to be a soft closed set in (U, τ,E) if 1AF is a soft open set in (U, τ,E). The family of all soft closed sets in (U, τ,E) will be denoted by τc. Soft topological concepts and their applications are still a hot area of research [1,2,526]. The concepts of minimal open sets in TSs were defined and investigated in [27]. Then research via this concept is continued by various researchers [2831]. Author in [32] defined and investigated the concepts of minimal sets and minimal preopen sets, in which he also introduced the notion of prehomogeneity. Soft minimal soft open sets in STSs were defined in [33], then research via them is continued in [1] and others.

In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft open sets and soft minimal soft preopen sets as two new classes of soft sets in STSs, and we define soft prehomogeneity. as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous STS that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.

Herein, we recall several related definitions and results.

Let (X,ℑ) be a TS and let SX. Then ℑc will denote the family of all closed sets in (X,ℑ), Cl(S) (resp. Int (S)) will denote the closure of S in (X,ℑ) (resp. the interior of S in (X,ℑ)), and M(x,ℑ) will denote the set ∩{V ∈ ℑ : xV }.

Definition 2.1

Let (X,ℑ) be a TS and let SX. Then S is said to be

(a) [34] a preopen set in (X,ℑ) if SInt (Cl(S)). Equivalently: S is a preopen set in (X,ℑ) if and only if there exists V ∈ ℑ such that SVCl(S).

(b) [27] a minimal open set in (X,ℑ) if for all V ∈ ℑ with VS either V = ∅︀ or V = S.

(c) [32] a minimal set in (X,ℑ) if there exists xX such that S = M(x,ℑ).

(d) [32] a minimal preopen set in (X,ℑ) if S is a preopen set in (X,ℑ) and for all preopen set V in (X,ℑ) with VS either V = ∅︀ or V = S.

The family of all preopen sets (resp. minimal open sets, minimal sets, minimal preopen sets) in (X,ℑ) will be denoted by PO(X,ℑ) (resp. min(X,ℑ), min s(X,ℑ), min(PO(X,ℑ))).

Definition 2.2 [34]

A function is said to be

(a) preirresolute if f1(V )∈PO(X,ℑ) for all .

(b) prehomeomorphism if f is a bijection and both f and f1 are preirresolute.

Definition 2.3 [27]

A TS (X,ℑ) is called prehomogeneous if for any x, yX, there exists a prehomeomorphism f : (X,ℑ) → (X,ℑ) such that f (x) = y.

Definition 2.4 [35]

Let M,NSS (X,A).

(a) M is a soft subset of N, denoted by M⊆̃N, if M(a) ⊆ N(a) for each aA.

(b) M and N are said to be soft equal, denoted by F = G, if M⊆̃N and N⊆̃M.

(c) Soft union of M and N is denoted by M∪̃N and defined to be the soft set M∪̃NSS (X,A) where (M∪̃N) (a) = M(a) ∪ N (a) for each aA.

(d) Soft intersection of M and N is denoted by M∩̃N and defined to be the soft set M∩̃NSS (X,A) where (M∩̃N) (a) = M(a) ∩ N (a) for each aA.

(e) The difference of M and N is denoted by MN and defined to be the soft set MNSS(X,A) where (MN)(a) = M(a) – N(a) for each aA.

Definition 2.5 [36]

Let Δ be an arbitrary indexed set and {Gα : α ∈ Δ} ⊆ SS (X,A).

(a) The soft union of these soft sets is the soft set denoted by αΔ˜Gα and defined by (αΔ˜Gα)(a)=αΔGα(a) for each aA.

(b) The soft intersection of these soft sets is the soft set denoted by αΔ˜Gα and defined by (αΔ˜Gα)(a)=αΔGα(a) for each aA.

Definition 2.6 [1]

Let X be a universal set and A is a set of parameters. Then GSS(X,A) defined by

G(a)={Y,if a=e,,if ae

will be denoted by eY.

Definition 2.7 [37]

Let X be a universal set and A be a set of parameters. Then GSS(X,A) defined by

G(a)={{x},if a=e,,if ae

will be denoted by ex and will be called a soft point.

Definition 2.8 [37]

Let GSS (X,A) and axSP (X,A). Then ax is said to belong to F (notation: ax∊̃G) if ax ⊆̃G or equivalently: ax∊̃G if and only if xG(a).

Theorem 2.9 [4]

Let (X, τ,A) be an STS. Then the collection {F(a) : Fτ} defines a topology on X for every aA. This topology will be denoted by τa.

Theorem 2.10 [38]

Let (X,ℑ) be a TS. Then the collection

{FSS(X,A):F(a)for all aA}

defines a soft topology on X relative to A. This soft topology will be denoted by τ (ℑ).

Let (X, τ,A) be an STS and let FSS(X,A). Then Clτ (F) (resp. Intτ (F)) will denote the soft closure of F in (X, τ,A) (resp. the soft interior of F in (X, τ,A)).

Definition 2.11

Let (X, τ,A) be a TS and let FSS(X,A). Then F is said to be

(a) [39] a soft preopen set in (X, τ,A) if FIntτ (Clτ (F)). Equivalently: F is a soft preopen set in (X, τ,A) if and only if there exists Gτ such that FGClτ (F).

(b) [33] a soft minimal soft open set in (X, τ,A) if for all G ∈ ℑ with G⊆̃F either G = 0A or G = F.

The family of all soft preopen sets (resp. soft minimal soft open sets) in (X, τ,A) will be denoted by PO(X, τ,A) (resp. min(X, τ,A)).

Definition 2.12 [40]

An STS (X, τ,A) is said to be soft locally indiscrete if τ = τc.

Definition 2.13

A soft mapping fpu : (X, τ,A) → (Y, σ,B) is said to be

(a) soft preirresolute if fpu-1(F)PO(X,τ,A) for all FPO(Y, σ,B).

(b) soft prehomeomorphism if fpu is bijective, and fpu : (X, τ,A) → (Y, σ,B) and fp1u−1: (Y, σ,B) → (X, τ,A) are soft preirresolute.

3. Soft Minimal Soft Open Sets, Soft Minimal Soft Sets and Soft Minimal Soft Preopen Sets

We start this section by the following two characterizations of soft minimal soft open sets:

Theorem 3.1

Let (X, τ,A) and let FSS(X,A)–{0A}. Then the following are equivalent:

(a) F ∈ min (X, τ,A).

(b) For every GSS(X,A)–{0A} with G⊆̃F, F ⊆̃Clτ (G).

(c) For every GSS(X,A)–{0A} with G⊆̃F, Clτ (G) = Clτ (F).

Proof

(a) =⇒ (b) Let GSS(X,A) – {0A} with G⊆̃F. Let ax∊̃F and let Mτ such that ax∊̃M, then F ∩̃M ≠ 0A and so F ⊆̃M. Thus, 0AG = G∩̃M. Therefore, ax∊̃Clτ (G).

(b) =⇒ (c) Let GSS(X,A) – {0A} with G⊆̃F. Then Clτ (G)⊆̃Clτ (F). On the other hand, by (b) we have F ⊆̃Clτ (G), and so Clτ (F)⊆̃Clτ (Clτ (G)) = Clτ (G). Thus, Clτ (G) = Clτ (F).

(c) =⇒(a) Suppose to the contrary that there exists Gτ – {0A} such that G⊆̃F and GF. Choose ax∊̃FG. Then we have ax∊̃ (1AG) ∈ τc and so Clτ (ax)⊆̃Clτ (1AG) = 1AG. On the other hand, by (c), Clτ (ax) = Clτ (F) = Clτ (G), a contradiction.

Theorem 3.2

Let (X, τ,A) and let F ∈ min (X, τ,A). Then for every GSS(X,A) – {0A} with G⊆̃F, we have GPO(X, τ,A).

Proof

Let F ∈ min (X, τ,A) and GSS(X,A) – {0A} with G⊆̃F. Then by Theorem 3.1, G⊆̃F ⊆̃Clτ (G). Hence GPO(X, τ,A).

Theorem 3.3

For an STS (X, τ,A), the following are equivalent:

• (a) (X, τ,A) is soft locally indiscrete.

• (b) PO(X, τ,A) = SS(X,A).

• (c) SP(X,A) ⊆ PO(X, τ,A).

• (d) τcPO(X, τ,A).

Proof

(a) =⇒ (b) We need only to show that SS(X,A) ⊆ PO(X, τ,A). Let FSS(X,A), then Clτ (F) ∈ τc. So by (a), Clτ (F) ∈ τ. Since F ⊆̃Clτ (F)⊆̃Clτ (F), then FPO(X, τ,A).

(b) =⇒ (c) Since SP(X,A) ⊆ SS(X,A), then by (b) SP(X,A) ⊆ PO(X, τ,A).

(c)=⇒(d) LetGτc. Let ax∊̃G. By (c), axPO(X, τ,A) and so, ax ⊆̃Intτ (Clτ (ax)) ⊆̃Intτ (Clτ (G)). Hence, ax∊̃Intτ (Clτ (G)). Therefore, G⊆̃Intτ (Clτ (G)). Thus, GPO(X, τ,A).

(d) =⇒ (a) It is sufficient to show that τcτ. Let Gτc, then by (d), G⊆̃Intτ (Clτ (G)) = Intτ (G). But Intτ (G)⊆̃G is always true. It follows that Intτ (G) = G and hence Gτ.

Let (X, τ,A) be an STS. For each axSP(X,A), denote the soft set ∩̃ {F : Fτ and ax∊̃F} by M(ax ).

Definition 3.4

Let (X, τ,A) be an STS and GSS(X,A), then G is called a soft minimal soft set in (X, τ,A) if G = M(ax ) for some axSP(X,A). The family of all soft minimal soft sets in (X, τ,A) will be denoted by minss(X, τ,A).

Theorem 3.5

Let (X, τ,A) be an STS and G ∈ min(X, τ,A), then for every ax∊̃G, G = M(ax ).

Proof

Suppose that G ∈ min(X, τ,A) and let ax∊̃G. Then by definition of M(ax ) we have M(ax ) ⊆̃G. On the other hand, to see that G⊆̃M(ax ), suppose to the contrary that there is by∊̃GM(ax ). Then by∊̃G and there exists Fτ such that ax∊̃F and by∊̃1AF. Since ax∊̃G∩̃Fτ and G ∈ min(X, τ,A), then G⊆̃F and hence by∊̃F, a contradiction.

Corollary 3.6

For any STS (X, τ,A), min(X, τ,A) ⊆ minss(X, τ,A).

The following example will show that the inclusion in Corollary 3.6 is not equality, in general:

Example 3.7

Let X = ℝ, A = {a, b} and

τ={FSS(X,A):F(t)ufor every tA}

For every n ∈ ℕ, let FnSS(X,A) defined by Fn(a)=(-1n,1n) and Fn(b) = ∅︀. Since a0˜M(a0,τ)n˜Fn=a0, then M(a0 ) = a0. But M(a0 ) not even soft preopen.

The following question is natural:

Question 3.8

For an STS (X, τ,A), is it true that minss(X, τ, A) ∩ τ ⊆ min(X, τ,A)?

The following example is a negative answer for Question 3.8:

Example 3.9

Let X = {1, 2, 3}, A = {a, b} and τ = {FSS(X,A) : F(t) ∈ {∅︀, X, {1}, {1, 2}} for every tA}.

If b2∊̃Fτ, then 2 ∈ F (b) ∈ {∅︀,X, {1}, {1, 2}}, and so either F (b) = X or F (b) = {1, 2}. Then Mb2 (b) = {1, 2} and M(b2 ) (a) = ∅︀, and hence M(b2 ) ∈ minss(X, τ,A) ∩ τ. On the other hand, since b1τ – {0A} with b1 ⊆̃M(b2 ) and b1M(b2 ), then M(b2 )min(X, τ,A).

The following lemma will be used in the following main result:

Lemma 3.10

Let (X, τ,A) be a soft locally indiscrete STS. Then

• (a) minss(X, τ,A) ⊆ τ.

• (b) If M(ax ) ⊆̃M(by ), then by∊̃M(ax ).

• (c) If M(ax ) ⊆̃M(by ), then M(ax ) = M(by ).

Proof

(a) Let axSP(X,A). Since (X, τ,A) is soft locally indiscrete, then M(ax ) = ∩̃ {Fτc : ax∊̃F} and so M(ax )τc. Then again by soft local indiscreetness of (X, τ,A), we have M(ax )τ.

(b) Suppose to the contrary that M(ax ) ⊆̃M(by ) and by∊̃1AM(ax ). By (a), 1AM(ax )τ and so M(by ) ⊆̃1AM(ax ). Thus, M(by )∩̃M(ax ) = 0A. On the other hand, since M(ax ) ⊆̃M(by ), then M(by )∩̃M(ax ) = M(ax ) ≠ 0A, a contradiction.

(c) Suppose that M(ax ) ⊆̃M(by ), then by (b) by∊̃M(ax ) and so by (a), M(by ) ⊆̃M(ax ). It follows that M(ax ) = M(by ).

Theorem 3.11

For an STS (X, τ,A), the following are equivalent:

(a) (X, τ,A) is soft locally indiscrete.

(b) minss(X, τ,A) = min(X, τ,A).

(c) min (X, τ,A) is a soft base of (X, τ,A).

(d) (X, τ,A) has a soft base which forms a soft partition of 1A.

Proof

(a) =⇒ (b) By Corollary 3.6 we need only to show that minss(X, τ,A) ⊆ min(X, τ,A). Let axSP(X,A) and let Gτ – {0A} with G⊆̃M(ax ). Choose byG, then M(by ) ⊆̃G⊆̃M(ax ). So Lemma 3.10(c), M(by ) = G = M(ax ). This ends the proof.

(b) =⇒(c) By (b), ∪̃{F : F ∈ min (X, τ,A)} = 1A. Thus by Theorem 4.11 of [1], min (X, τ,A) is a soft base of (X, τ,A).

(c) =⇒(d) By (c) and Theorem 4.11 of [1], min (X, τ,A) is a soft base of (X, τ,A) which forms a soft partition of 1A.

(d) =⇒ (a) Let ℬ be a soft base which forms a soft partition of 1A. Let Gτ – {0A}, then there exists ℬ1⊆ ℬ such that ∪̃{F : F ∈ ℬ1} = G and so 1AG = ∪̃{1AF : F ∈ ℬ –ℬ1} ∈ τ. This shows Gτc. Therefore, (X, τ,A) is soft locally indiscrete.

Definition 3.12

Let (X, τ,A) be an STS. A soft set FSS (X,A) is said to be a soft minimal soft preopen set of (X, τ,A) if FPO(X, τ,A) and for all GPO(X, τ,A) with G⊆̃F either G = 0A or G = F. The family of all soft preopen sets in (X, τ,A) will be denoted by min(PO(X, τ,A)).

The following lemma will be necessary for proving the next main result:

Lemma 3.13

Let (X, τ,A) be an STS. If Fτ and GPO(X, τ,A), then F ∩̃GPO(X, τ,A).

Proof

Let Fτ and GPO(X, τ,A). Since GPO(X, τ,A), then there exists Kτ such that G⊆̃K⊆̃Clτ (G) and so F ∩̃G⊆̃F ∩̃K⊆̃F ∩̃Clτ (G). Since Fτ, then F ∩̃ Clτ (G)⊆̃Clτ (F ∩̃G). Hence, F ∩̃GPO(X, τ,A).

The following theorem shows that soft minimal soft preopen sets are the soft points soft preopen sets:

Theorem 3.14

For any STS (X, τ,A), min(PO(X, τ,A)) = PO(X, τ,A) ∩ SP(X,A).

Proof

By the definition of PO(X, τ,A), min(PO(X, τ,A)) ⊆ PO(X, τ,A). To show that min(PO(X, τ,A)) ⊆ SP(X,A), let F ∈ min(PO(X, τ,A)). Choose ax∊̃F. We are going to show that axPO(X, τ,A). By Lemma 3.13,

(1A-Clτ(ax))˜FPO(X,τ,A).

Since F ∈ min(PO(X, τ,A)), (1AClτ (ax))∩̃F ⊆̃F, and (1AClτ (ax))∩̃FF, then (1AClτ (ax))∩̃F = 0A and so F ⊆̃Clτ (ax). Hence, Clτ (F)⊆̃Clτ (ax) and thus ax ⊆̃F ⊆̃ Intτ (Clτ (F)) ⊆̃Intτ (Clτ (ax)) ⊆̃Clτ (ax). Therefore, axPO(X, τ,A). Now since F ∈ min(PO(X, τ,A)), then F = ax. It follows that FSP(X,A). Conversely, it is clear that PO(X, τ,A) ∩ SP(X,A) ⊆ min(PO(X, τ,A)).

Theorem 3.15

If F ∈ min (X, τ,A), then {ax : ax∊̃F} ⊆̃ min(PO(X, τ,A)).

Proof

Let F ∈ min(X, τ,A) and let ax∊̃F, then by Theorem 3.2 we have axPO(X, τ,A). Thus, axPO(X, τ,A) ∩SP(X,A). Hence, by Theorem 3.14 we have

axmin (PO(X,τ,A)).

The following example shows that the implication in Theorem 3.15 is not reversible, in general:

Example 3.16

Let X = ℝ and A = {a, b}. Let ℑ be the topology on X having the family {[−c, c] : c ∈ ℝ and c > 2} as a base and let

τ={FSS(X,A):F(t)for every tA}.

Then, a0 ∈ min(PO(X, τ,A)). On the other hand it is not difficult to check that min(X, τ,A) = ∅︀. Consider GSS(X,A), where G(a) = [−2, 2] and G(b) = ∅︀. Then {ax : ax∊̃G} ⊆̃ min(PO(X, τ,A)) but G ∉ min(X, τ,A).

Theorem 3.17

Let (X, τ,A) be an STS and axSP(X,A). Then ax ∈ min(PO(X, τ,A)) if and only if by ∈ min(PO(X, τ, A)) for every by∊̃M(ax ).

Proof
Necessity

Suppose that ax ∈ min(PO(X, τ,A)) and let by∊̃M(ax ).

Claim

ax∊̃Clτ (by).

Proof of Claim

Let Gτ such that ax∊̃G. Then M(ax ) ⊆̃G and so by∊̃G. Therefore, by = by∩̃G and hence by∩̃G ≠ 0A. It follows that ax∊̃Clτ (by).

Now by the above claim we have Clτ (ax)⊆̃Clτ (by). Since axPO(X, τ,A), then ax ⊆̃Intτ (Clτ (ax)) and so

by˜M(ax,τ)˜Intτ(Clτ(ax))˜Clτ(ax)˜Clτ(by).

Hence, byPO(X, τ,A). Therefore, by Theorem 3.14, we have by ∈ min(PO(X, τ,A)).

Sufficiency

Suppose that by ∈ min(PO(X, τ,A)) for every by∊̃M(ax ). Since ax∊̃M(ax ), then axPO(X, τ,A). Therefore, by Theorem 3.14, we have

axmin (PO(X,τ,A)).

Corollary 3.18

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ, A)), then M(ax )PO(X, τ,A).

Theorem 3.19

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ, A)) and there exists Gτ – {0A} such that G⊆̃M(ax ), then M(ax ) = G.

Proof

Suppose that ax ∈ min(PO(X, τ,A)) and G⊆̃M(ax ) for some Gτ – {0A}. Choose by∊̃G, then ax∊̃Clτ (by). Since axPO(X, τ,A), then there exists Fτ such that ax ⊆̃F ⊆̃Clτ (ax)⊆̃Clτ (by). Since ax∊̃Fτ and ax∊̃Clτ (by), then F ∩̃by ≠ 0A and so by∊̃F ⊆̃Clτ (ax). Since by∊̃Gτ and by∊̃Clτ (ax), then G∩̃ax ≠ 0A and hence ax∊̃G. It follows that M(ax ) ⊆̃G and thus, M(ax ) = G.

Corollary 3.20

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ, A)), then either M(ax ) ∈ min(X, τ,A) or Intτ (M(ax )) = 0A.

Theorem 3.21

Let (X, τ,A) be an STS and axSS (X,A). Then the following are equivalent:

• (a) M(ax ) ∈ min(X, τ,A).

• (b) ax ∈ min(PO(X, τ,A)) and Intτ (M(ax )) ≠ 0A.

Proof

(a) ⇒ (b) Suppose that M(ax ) ∈ min(X, τ,A), then obviously that Intτ (M(ax )) ≠ 0A. Also, by Theorem 3.15 we have ax ∈ min(PO(X, τ,A)).

(b) ⇒ (a) Follows directly from Corollary 3.20.

The following is the main concept of this section:

Definition 4.1

An STS (X, τ,A) is called soft prehomogeneous if for any ax, bySP (X,A), there exists a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that fpu (ax) = by.

Theorem 4.2

For an STS (X, τ,A), the following are equivalent:

(a) (X, τ,A) is soft prehomogeneous.

(b) For any ax, bySP (X,A), there is a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that p (x) = y and u (a) = b.

(c) For any two pairs (x, a), (y, b) ∈ X × A, there is a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that p (x) = y and u (a) = b.

Proof

Straightforward.

Remark 4.3

Soft homogeneous STSs are soft prehomogeneous.

Proof

Follows because soft homeomorphisms are soft prehomeomorphism.

Theorem 4.4

Every soft locally indiscrete STSs is soft prehomogeneous.

Proof

Let (X, τ,A) be a soft locally indiscrete STS. Let cz, dwSP (X,A). Define p : XX and u : AA as follows:

p(s)={wif s=zzif s=wsif szand sw

and

u(t)={dif t=ccif t=deif tcand td.

By Theorem 3.3, PO(X, τ,A) = SS(X,A) and thus fpu is a soft prehomeomorphism with fpu (ax) = by. Therefore, (X, τ,A) is soft prehomogeneous.

The following example shows that the converse of Remark 4.3 is not true, in general:

Example 4.5

Let X = ℝ, A = ℚ. Let FSS(X,A) defined by F (a) = ℕ for every aA. Let τ = {0A, 1A, F, 1AF}. Consider the STS (X, τ,A). Then τ = τc, and so (X, τ,A) is soft locally indiscrete. So by Theorem 4.4, (X, τ,A) is soft prehomogeneous. On the other hand, since min(X, τ,A) = {F, 1AF} and F(1) is countable but (1AF) (1) is uncountable, then by Theorem 5.38(b) of [1], (X, τ,A) is not soft homogeneous.

The following is an example of a soft prehomogeneous STS that is neither soft homogeneous nor soft locally indiscrete:

Example 4.6

Let X = ℕ, A = {a, b} and τ = {FSS(X,A) : F(s) ∈ {{∅︀} ∪ {{n, n + 1, ...} : n ∈ ℕ}} for all sA}.

Then,

(1) (X, τ,A) is not soft locally indiscrete: Take FSS(X,A) defined by F (s) = ℕ – {1} for all sA. Then Fττc, and hence (X, τ,A) is not soft locally indiscrete.

(2) (X, τ,A) is not soft homogeneous: If (X, τ,A) is (X, τ,A), then by Theorem 5.14 of [1] the TS (X, τa) is homogeneous and so there is a homeomorphism g : (X, τa) → (X, τa) such that g (1) = 2. But {1} ∈ (τa)c while g ({1}) = {2} (τa)c. It follows that (X, τ,A) is not soft homogeneous.

(3) (X, τ,A) is soft prehomogeneous: Let cx, dySP (X,A). Define p : XX and u : AA as follows:

p(s)={yif s=xxif s=ysif sxand sy

and

u(t)={dif t=ccif t=deif tcand td.

It is not difficult to check that PO(X, τ,A) = {FSS(X, A) : F(s) ∈ {∅︀,ℕ}} ∪ {FSS(X,A) : F(a) and F (b) are infinite sets }, and thus fpu is a soft prehomeomorphism with fpu (cx) = dy. Hence, (X, τ,A) is soft prehomogeneous.

Theorem 4.7

Let (X, τ,A) be an STS with min(PO(X, τ, A)) ≠ ∅︀. Then (X, τ,A) is soft prehomogeneous if and only if (X, τ,A) is soft locally indiscrete.

Proof
Necessity

Suppose that (X, τ,A) is soft prehomogeneous. We will apply Theorem 3.3(c). Let bySP(X,A). Pick ax ∈ min(PO(X, τ,A)). By soft prehomogeneity of (X, τ,A), there exists a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that fpu (ax) = by and so byPO(X, τ, A).

Sufficiency

Follows from Theorem 4.4.

Corollary 4.8

Let (X, τ,A) be an STS with min (X, τ,A) ≠ ∅︀. Then (X, τ,A) is soft prehomogeneous if and only if (X, τ,A) is soft locally indiscrete.

Proof

Follows from Theorems 3.15 and 4.7.

Corollary 4.9

Let (X, τ,A) be an STS such that τ is finite. Then (X, τ,A) is soft prehomogeneous if and only if (X, τ,A) is soft locally indiscrete.

Proof

Since τ is finite, then it is easy to see that min(X, τ, A) ≠ ∅︀. Thus, by Corollary 4.8 we get the result.

In the following result, τpr denotes the soft topology on X relative to A having PO(X, τ,A) as a subbase.

Theorem 4.10

If (X, τ,A) is soft prehomogeneous, then (X, τpr,A) is soft homogeneous.

Proof

Suppose that (X, τ,A) is soft prehomogeneous and let ax, bySS(X,A). Since (X, τ,A) is soft prehomogeneous, there exists a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that fpu (ax) = by. Let G be a soft basic soft open set in (X, τpr,A), then G=i=1n˜Gi where GiPO(X, τ,A) for all i = 1, 2, ..., n. Now, for every i ∈ {1, 2, ..., n}, fpu-1(Gi)PO(X,τ,A) and so fpu-1(G)=i=1n˜fpu-1(Gi)τpr. Therefore, fpu : (X, τpr,A) → (X, τpr, A) is soft continuous. By a similar way we can see that fp1u1 is soft continuous. Therefore, fpu : (X, τpr,A) → (X, τpr,A) is a soft homeomorphism with fpu (ax) = by. Hence, (X, τpr,A) is soft homogeneous.

By the end of this section, we raise the following question regarding the converse of Theorem 4.10:

Question 4.11

Let (X, τ,A) be an STS such that (X, τpr,A) is soft homogeneous. Is it true that (X, τ,A) is soft prehomogeneous?

Theorem 5.1

Let (X, τ,A) be an STS. Then for any axSP(X,A), M(ax )(a) = M(x,τa).

Proof

Let axSP(X,A), then

M(ax,τ)(a)=(˜{F:Fτand ax˜F})(a)={F(a):Fτand ax˜F}=({F(a):Fτand xF(a)})(a)={Uτa:xU}=M(x,τa).

Lemma 5.2

Let (X, τ,A) be an STS and let ℬ be a soft base of (X, τ,A). Then for any axSP(X,A), M(ax ) = ∩̃ {B ∈ ℬ : ax∊̃B}.

Proof

Let axSP(X,A), then clearly that

M(ax,τ)˜˜{B:ax˜B}.

Conversely, let by∊̃∩̃ {B ∈ ℬ : ax∊̃B} and let Gτ such that ax∊̃G. Choose B ∈ ℬ such that ax∊̃B⊆̃G, then by∊̃B⊆̃G. It follows that by∊̃M(ax ).

Theorem 5.3

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let σ=αAa. Then for every axSP(X,A), M(ax) = aM(x,ℑa).

Proof

By Theorem 3.5 of [1], {bY : bA and Y ∈ ℑb} is a soft base of (X, σ,A). So, by Lemma 5.2,

M(ax,σ)=˜{bY:bA,Yband ax˜bY}=˜{aY:Yaand xY}=a{Y:Yaand xY}=aM(x,a).

Corollary 5.4

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X. Then

minss (X,τ,A)={aY:aAand Ymins(X,a)}.

Lemma 5.5

Let (X, τ,A) be an STS. Then for every axSP(X,A), (Clτ (ax)) (a) = Clτa ({x}).

Proof

By Proposition 7 of [4], Clτa ({x}) ⊆ (Clτ (ax)) (a). To show that (Clτ (ax)) (a) ⊆ Clτa ({x}), suppose to the contrary that there exists y ∈ (Clτ (ax)) (a) – Clτa ({x}). Since y ∉ Clτa ({x}), then there exists Vτa such that yV and x ∉ V. Choose Fτ such that F (a) = V. Then ax∊̃1AF. On the other hand, since we have ay∊̃Fτ and ay∊̃Clτ (ax), then ax∊̃F, a contradiction.

Theorem 5.6

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ,A)), then {x} ∈ min (PO(X, τa)).

Proof

Suppose that ax ∈ min(PO(X, τ,A)). Then axPO(X, τ,A) and so there exists Fτ such that ax ⊆̃F ⊆̃Clτ (ax). Thus we have F (a) ∈ τa and {x} ⊆ F(a) ⊆ (Clτa (ax)) (a). On the other hand, by Lemma 5.5, (Clτ (ax)) (a) = Clτa ({x}). Hence, {x} ∈ PO(X, τa). Therefore, {x} ∈ min (PO(X, τa)).

The following lemma will be used in the following result:

Lemma 5.7

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let σ=αAa. Then for any GSS(X,A) and any aA, Cla (G(a)) = (Clσ(G)) (a).

Proof

Let GSS(X,A) and aA. By Theorem 3.7 of [1], σa = ℑa. So by Proposition 7 of [4], Cla (G(a)) ⊆ (Clσ(G)) (a). To show that (Clσ(G)) (a) ⊆ Cla (G(a)), let x ∈ (Clσ(G)) (a) and let U ∈ ℑa such that xU. Since ax∊̃aU and ax∊̃Clσ(G), then G∩̃aU ≠ 0A. Thus, (G∩̃aU) (a) = G(a) ∩ U ≠ ∅︀. It follows that xCl(G(a)).

Theorem 5.8

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let σ=αAa. Then GPO(X, σ,A) if and only if G(a) ∈ PO(X,ℑa) for all aA.

Proof
Necessity

Suppose that GPO(X, σ,A) and let aA. Choose Fσ such that G⊆̃F ⊆̃Clσ(G). Then by Lemma 5.7, G(a) ⊆ F(a) ⊆ (Clσ(G)) (a) = Cla (G(a)). Since Fσ, then by Theorem 3.7 of [1], F(a) ∈ ℑa. It follows that G(a) ∈ PO(X,ℑa).

Sufficiency

Suppose that G(a) ∈ PO(X,ℑa) for all aA. For each aA, choose Ua ∈ ℑa such that G(a) ⊆ UaCla (G(a)). Let FSS(X,A) defined by F(a) = Ua for all aA. Then Fσ and by Lemma 5.7, G(a) ⊆ F(a) ⊆ Cla (G(a)) = (Clσ(G)) (a) for all aA which implies that G⊆̃F ⊆̃Clσ(G). Hence, GPO(X, σ,A).

Corollary 5.9

Let (X,ℑ) be a TS and A be a set of parameters. Then GPO(X, τ(ℑ),A) if and only if G(a) ∈ PO(X,ℑ) for all aA.

Proof

For every aA, put ℑa = ℑ. Then τ()=aAa. So by Theorem 5.8, we get the result.

Theorem 5.10

Let X be an initial universe and A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X. Then min (PO(X,aAa,A))={ax:aAand {x}min(PO(X,a))}.

Proof

Let Gmin (PO(X,aAa,A)), then by Theorem 3.14 there exists axPO(X,aAa,A)SP(X,A) such that G = ax. By Theorem 3.7 of [1] and Theorem 5.6, we have {x} ∈ min (PO(X,ℑa)). This ends the proof that min (PO(X,aAa,A)){ax:aAand {x}min(PO(X,a))}.

Conversely, let aA and {x} ∈ min (PO(X,ℑa)). Then

(ax)(b)={{x},if b=a,,if ba,

and so (ax) (b) ∈ PO(X,ℑa) for all aA. Thus by Theorem 5.8, axPO(X,aAa,A). Therefore, by Theorem 3.14, we have axmin (PO(X,aAa,A)).

Corollary 5.11

Let X be an initial universe and A be a set of parameters. Then min (PO(X, τ(ℑ),A)) = {ax : aA and {x} ∈ min(PO(X,ℑ))}.

Proof

For every aA, put ℑa = ℑ. Then τ()=aAa. So by Theorem 5.10, we get the result.

Lemma 5.12

If fpu:(X,aAa,A)(X,aAa,A) is soft preirresolute with u (b) = b, then p : (X,ℑb) → (X,ℑb) is preirresolute.

Proof

Let UPO(X,ℑb). Then by Theorem 5.8, bUPO(X,aAa,A) and so, fpu-1(bU)PO(X,aAa,A). So again by Theorem 5.8, (fpu-1(bU))(b)PO(X,b). But (fpu-1(bU))(b)=p-1(bU(u-1(b))=p-1(U). Therefore, p : (X,ℑb) → (X,ℑb) is preirresolute.

Theorem 5.13

Let X be an initial universe and A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X. If (X,aAa,A) is soft prehomogeneous, then for every bA, (X,ℑb) is prehomogeneous.

Proof

Suppose that (X,aAa,A) is soft prehomogeneous and let bA. Let x, yX, then bx, bySP(X,A) and so there exists a soft prehomeomorphism fpu:(X,aAa,A)(X,aAa,A) such that fpu(bx) = by. Then p : XX is a bijection with p(x) = y. Since u (b) = u1(b) = b, then by Lemma 5.12, the functions p, p1 : (X,ℑb) → (X,ℑb) are preirresolute. It follows that p : (X,ℑb) → (X,ℑb) is a prehomeomorphism with p(x) = y. Hence, (X,ℑb) is prehomogeneous.

Question 5.14

Let {(X,ℑa) : aA} be an indexed family of prehomogeneous TSs. Is it true that (X,aAa,A) is soft prehomogeneous.

The following result gives a partial answer for Question 5.14:

Theorem 5.15

Let (X,ℑ) be a TS and A be a set of parameters. Then (X, τ (ℑ),A) is soft prehomogeneous if and only if (X,ℑ) is prehomogeneous.

Proof
Necessity

Suppose that (X, τ (ℑ),A) is soft prehomogeneous. For every bA, put ℑb = ℑ. Then τ(a)=bAb. Thus, by Theorem 5.13, we get the result.

Sufficiency

Suppose that (X,ℑ) is prehomogeneous. Let ax, bySP(X,A), then there is a prehomeomorphism p : (X,ℑ) → (X,ℑ) such that p(x) = y. Choose a bijection u : AA such that u(a) = b. Then fpu : (X, τ (ℑ),A) → (X, τ (ℑ),A) is a bijection. To show that fpu is soft preirresolute, let GPO(X, τ(ℑ),A), then by Corollary 5.9, G(c) ∈ PO(X,ℑ) for all cA. Since p : (X,ℑ) → (X,ℑ) is preirresolute, then for every dA, (fpu-1(G))(d)=p-1(G(u-1(d)))PO(X,). Therefore, again by Corollary 5.9, fpu-1(G)PO(X,τ(),A). Hence, fpu : (X, τ (ℑ),A) → (X, τ (ℑ),A) is soft preirresolute. Similarly, we can show that fp1u1 : (X, τ (ℑ),A) → (X, τ (ℑ),A) is soft preirresolute. It follows that (X, τ (ℑ),A) is soft prehomogeneous.

The following three questions are natural:

Question 5.16

Let fpu : (X, τ,A) → (X, τ,A) be soft prehomeomorphism and aA. Is it true that p : (X, τa) → (X, τa) is a prehomeomorphism?

Question 5.17

Let (X, τ,A) be soft prehomogeneous and aA. Is it true that (X, τa) is prehomogeneous?

Question 5.18

Let (X, τa) be prehomogeneous for all aA. Is it true that (X, τ,A) is soft prehomogeneous?

We will leave Questions 5.16 and 5.17 as open questions. On the other hand, the following example gives a negative answer for Question 5.18:

Several characterizations of soft minimal soft open sets are introduced, and two new kinds of soft sets are introduced and investigated. Moreover, the notion of soft prehomogeneity is introduced and investigated. Several relationships are studied. Also, some characterizations of soft locally indiscrete are introduced. Six open questions are raised. We expect that our results will be important for the forthcoming in SBTSs to build a good background for some practical applications and to answer the most intricate problems containing uncertainty in medical, engineering, economics, environment, and in general. In future studies, the following topics could be considered: (1) defining soft prehomogeneity components; and (2) defining soft semihomogeneity.

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Samer Al Ghour received the Ph.D. in Mathematics from University of Jordan, Jordan in 1999. Currently, he is a professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests is include general topology, fuzzy topology, and soft set theory.

E-mail: algore@just.edu.jo

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279

Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.269

Soft Minimal Soft Sets and Soft Prehomogeneity in Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Correspondence to:Samer Al Ghour (algore@just.edu.jo)

Received: July 4, 2021; Revised: August 16, 2021; Accepted: August 28, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft sets and soft minimal soft preopen sets as two new classes of soft sets in soft topological spaces, and we define soft prehomogeneity as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points, and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous soft topological space that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete soft topological space in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.

Keywords: Soft minimal soft open sets, Soft preopen soft sets, Soft locally indiscrete, Prehomogeneity, Generated soft topology

1. Introduction

This paper follows the notions and terminologies as appeared in [1] and [2]. In this paper, TS and STS will denote topological space and soft topological space, respectively. Molodtsov [3] defined soft sets in 1999. The soft set theory offers a general mathematical tool for dealing with uncertain objects. Let U be a universal set and E be a set of parameters. A soft set over U relative to E is a function G : E → ℘(U). SS (U,E) will denote the family of all soft sets over U relative to E. In this paper, the null soft set and the absolute soft set will be denoted by 0E and 1E, respectively. The structure of STSs was defined in [4] as follows: An STS is a triplet (U, τ,E), where τSS (U,E), τ contains 0E and 1E, τ is closed under finite soft intersection, and τ is closed under arbitrary soft union. Let (U, τ,E) be an STS and FSS(X,A), then F is said to be a soft open set in (U, τ,E) if Fτ and F is said to be a soft closed set in (U, τ,E) if 1AF is a soft open set in (U, τ,E). The family of all soft closed sets in (U, τ,E) will be denoted by τc. Soft topological concepts and their applications are still a hot area of research [1,2,526]. The concepts of minimal open sets in TSs were defined and investigated in [27]. Then research via this concept is continued by various researchers [2831]. Author in [32] defined and investigated the concepts of minimal sets and minimal preopen sets, in which he also introduced the notion of prehomogeneity. Soft minimal soft open sets in STSs were defined in [33], then research via them is continued in [1] and others.

In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft open sets and soft minimal soft preopen sets as two new classes of soft sets in STSs, and we define soft prehomogeneity. as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous STS that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.

2. Preliminaries

Herein, we recall several related definitions and results.

Let (X,ℑ) be a TS and let SX. Then ℑc will denote the family of all closed sets in (X,ℑ), Cl(S) (resp. Int (S)) will denote the closure of S in (X,ℑ) (resp. the interior of S in (X,ℑ)), and M(x,ℑ) will denote the set ∩{V ∈ ℑ : xV }.

Definition 2.1

Let (X,ℑ) be a TS and let SX. Then S is said to be

(a) [34] a preopen set in (X,ℑ) if SInt (Cl(S)). Equivalently: S is a preopen set in (X,ℑ) if and only if there exists V ∈ ℑ such that SVCl(S).

(b) [27] a minimal open set in (X,ℑ) if for all V ∈ ℑ with VS either V = ∅︀ or V = S.

(c) [32] a minimal set in (X,ℑ) if there exists xX such that S = M(x,ℑ).

(d) [32] a minimal preopen set in (X,ℑ) if S is a preopen set in (X,ℑ) and for all preopen set V in (X,ℑ) with VS either V = ∅︀ or V = S.

The family of all preopen sets (resp. minimal open sets, minimal sets, minimal preopen sets) in (X,ℑ) will be denoted by PO(X,ℑ) (resp. min(X,ℑ), min s(X,ℑ), min(PO(X,ℑ))).

Definition 2.2 [34]

A function is said to be

(a) preirresolute if f1(V )∈PO(X,ℑ) for all .

(b) prehomeomorphism if f is a bijection and both f and f1 are preirresolute.

Definition 2.3 [27]

A TS (X,ℑ) is called prehomogeneous if for any x, yX, there exists a prehomeomorphism f : (X,ℑ) → (X,ℑ) such that f (x) = y.

Definition 2.4 [35]

Let M,NSS (X,A).

(a) M is a soft subset of N, denoted by M⊆̃N, if M(a) ⊆ N(a) for each aA.

(b) M and N are said to be soft equal, denoted by F = G, if M⊆̃N and N⊆̃M.

(c) Soft union of M and N is denoted by M∪̃N and defined to be the soft set M∪̃NSS (X,A) where (M∪̃N) (a) = M(a) ∪ N (a) for each aA.

(d) Soft intersection of M and N is denoted by M∩̃N and defined to be the soft set M∩̃NSS (X,A) where (M∩̃N) (a) = M(a) ∩ N (a) for each aA.

(e) The difference of M and N is denoted by MN and defined to be the soft set MNSS(X,A) where (MN)(a) = M(a) – N(a) for each aA.

Definition 2.5 [36]

Let Δ be an arbitrary indexed set and {Gα : α ∈ Δ} ⊆ SS (X,A).

(a) The soft union of these soft sets is the soft set denoted by $∪α∈Δ˜Gα$ and defined by $(∪α∈Δ˜Gα) (a)=∪α∈ΔGα(a)$ for each aA.

(b) The soft intersection of these soft sets is the soft set denoted by $∩α∈Δ˜Gα$ and defined by $(∩α∈Δ˜Gα) (a)=∩α∈ΔGα(a)$ for each aA.

Definition 2.6 [1]

Let X be a universal set and A is a set of parameters. Then GSS(X,A) defined by

$G(a)={Y,if a=e,∅,if a≠e$

will be denoted by eY.

Definition 2.7 [37]

Let X be a universal set and A be a set of parameters. Then GSS(X,A) defined by

$G(a)={{x},if a=e,∅,if a≠e$

will be denoted by ex and will be called a soft point.

Definition 2.8 [37]

Let GSS (X,A) and axSP (X,A). Then ax is said to belong to F (notation: ax∊̃G) if ax ⊆̃G or equivalently: ax∊̃G if and only if xG(a).

Theorem 2.9 [4]

Let (X, τ,A) be an STS. Then the collection {F(a) : Fτ} defines a topology on X for every aA. This topology will be denoted by τa.

Theorem 2.10 [38]

Let (X,ℑ) be a TS. Then the collection

${F∈SS(X,A):F(a)∈ℑ for all a∈A}$

defines a soft topology on X relative to A. This soft topology will be denoted by τ (ℑ).

Let (X, τ,A) be an STS and let FSS(X,A). Then Clτ (F) (resp. Intτ (F)) will denote the soft closure of F in (X, τ,A) (resp. the soft interior of F in (X, τ,A)).

Definition 2.11

Let (X, τ,A) be a TS and let FSS(X,A). Then F is said to be

(a) [39] a soft preopen set in (X, τ,A) if FIntτ (Clτ (F)). Equivalently: F is a soft preopen set in (X, τ,A) if and only if there exists Gτ such that FGClτ (F).

(b) [33] a soft minimal soft open set in (X, τ,A) if for all G ∈ ℑ with G⊆̃F either G = 0A or G = F.

The family of all soft preopen sets (resp. soft minimal soft open sets) in (X, τ,A) will be denoted by PO(X, τ,A) (resp. min(X, τ,A)).

Definition 2.12 [40]

An STS (X, τ,A) is said to be soft locally indiscrete if τ = τc.

Definition 2.13

A soft mapping fpu : (X, τ,A) → (Y, σ,B) is said to be

(a) soft preirresolute if $fpu-1(F)∈PO (X,τ,A)$ for all FPO(Y, σ,B).

(b) soft prehomeomorphism if fpu is bijective, and fpu : (X, τ,A) → (Y, σ,B) and fp1u−1: (Y, σ,B) → (X, τ,A) are soft preirresolute.

3. Soft Minimal Soft Open Sets, Soft Minimal Soft Sets and Soft Minimal Soft Preopen Sets

We start this section by the following two characterizations of soft minimal soft open sets:

Theorem 3.1

Let (X, τ,A) and let FSS(X,A)–{0A}. Then the following are equivalent:

(a) F ∈ min (X, τ,A).

(b) For every GSS(X,A)–{0A} with G⊆̃F, F ⊆̃Clτ (G).

(c) For every GSS(X,A)–{0A} with G⊆̃F, Clτ (G) = Clτ (F).

Proof

(a) =⇒ (b) Let GSS(X,A) – {0A} with G⊆̃F. Let ax∊̃F and let Mτ such that ax∊̃M, then F ∩̃M ≠ 0A and so F ⊆̃M. Thus, 0AG = G∩̃M. Therefore, ax∊̃Clτ (G).

(b) =⇒ (c) Let GSS(X,A) – {0A} with G⊆̃F. Then Clτ (G)⊆̃Clτ (F). On the other hand, by (b) we have F ⊆̃Clτ (G), and so Clτ (F)⊆̃Clτ (Clτ (G)) = Clτ (G). Thus, Clτ (G) = Clτ (F).

(c) =⇒(a) Suppose to the contrary that there exists Gτ – {0A} such that G⊆̃F and GF. Choose ax∊̃FG. Then we have ax∊̃ (1AG) ∈ τc and so Clτ (ax)⊆̃Clτ (1AG) = 1AG. On the other hand, by (c), Clτ (ax) = Clτ (F) = Clτ (G), a contradiction.

Theorem 3.2

Let (X, τ,A) and let F ∈ min (X, τ,A). Then for every GSS(X,A) – {0A} with G⊆̃F, we have GPO(X, τ,A).

Proof

Let F ∈ min (X, τ,A) and GSS(X,A) – {0A} with G⊆̃F. Then by Theorem 3.1, G⊆̃F ⊆̃Clτ (G). Hence GPO(X, τ,A).

Theorem 3.3

For an STS (X, τ,A), the following are equivalent:

• (a) (X, τ,A) is soft locally indiscrete.

• (b) PO(X, τ,A) = SS(X,A).

• (c) SP(X,A) ⊆ PO(X, τ,A).

• (d) τcPO(X, τ,A).

Proof

(a) =⇒ (b) We need only to show that SS(X,A) ⊆ PO(X, τ,A). Let FSS(X,A), then Clτ (F) ∈ τc. So by (a), Clτ (F) ∈ τ. Since F ⊆̃Clτ (F)⊆̃Clτ (F), then FPO(X, τ,A).

(b) =⇒ (c) Since SP(X,A) ⊆ SS(X,A), then by (b) SP(X,A) ⊆ PO(X, τ,A).

(c)=⇒(d) LetGτc. Let ax∊̃G. By (c), axPO(X, τ,A) and so, ax ⊆̃Intτ (Clτ (ax)) ⊆̃Intτ (Clτ (G)). Hence, ax∊̃Intτ (Clτ (G)). Therefore, G⊆̃Intτ (Clτ (G)). Thus, GPO(X, τ,A).

(d) =⇒ (a) It is sufficient to show that τcτ. Let Gτc, then by (d), G⊆̃Intτ (Clτ (G)) = Intτ (G). But Intτ (G)⊆̃G is always true. It follows that Intτ (G) = G and hence Gτ.

Let (X, τ,A) be an STS. For each axSP(X,A), denote the soft set ∩̃ {F : Fτ and ax∊̃F} by M(ax ).

Definition 3.4

Let (X, τ,A) be an STS and GSS(X,A), then G is called a soft minimal soft set in (X, τ,A) if G = M(ax ) for some axSP(X,A). The family of all soft minimal soft sets in (X, τ,A) will be denoted by minss(X, τ,A).

Theorem 3.5

Let (X, τ,A) be an STS and G ∈ min(X, τ,A), then for every ax∊̃G, G = M(ax ).

Proof

Suppose that G ∈ min(X, τ,A) and let ax∊̃G. Then by definition of M(ax ) we have M(ax ) ⊆̃G. On the other hand, to see that G⊆̃M(ax ), suppose to the contrary that there is by∊̃GM(ax ). Then by∊̃G and there exists Fτ such that ax∊̃F and by∊̃1AF. Since ax∊̃G∩̃Fτ and G ∈ min(X, τ,A), then G⊆̃F and hence by∊̃F, a contradiction.

Corollary 3.6

For any STS (X, τ,A), min(X, τ,A) ⊆ minss(X, τ,A).

The following example will show that the inclusion in Corollary 3.6 is not equality, in general:

Example 3.7

Let X = ℝ, A = {a, b} and

$τ={F∈SS(X,A):F(t)∈ℑu for every t∈A}$

For every n ∈ ℕ, let FnSS(X,A) defined by $Fn(a)=(-1n,1n)$ and Fn(b) = ∅︀. Since $a0∈˜M(a0,τ)⊆∩n∈ℕ˜Fn=a0$, then M(a0 ) = a0. But M(a0 ) not even soft preopen.

The following question is natural:

Question 3.8

For an STS (X, τ,A), is it true that minss(X, τ, A) ∩ τ ⊆ min(X, τ,A)?

The following example is a negative answer for Question 3.8:

Example 3.9

Let X = {1, 2, 3}, A = {a, b} and τ = {FSS(X,A) : F(t) ∈ {∅︀, X, {1}, {1, 2}} for every tA}.

If b2∊̃Fτ, then 2 ∈ F (b) ∈ {∅︀,X, {1}, {1, 2}}, and so either F (b) = X or F (b) = {1, 2}. Then Mb2 (b) = {1, 2} and M(b2 ) (a) = ∅︀, and hence M(b2 ) ∈ minss(X, τ,A) ∩ τ. On the other hand, since b1τ – {0A} with b1 ⊆̃M(b2 ) and b1M(b2 ), then M(b2 )min(X, τ,A).

The following lemma will be used in the following main result:

Lemma 3.10

Let (X, τ,A) be a soft locally indiscrete STS. Then

• (a) minss(X, τ,A) ⊆ τ.

• (b) If M(ax ) ⊆̃M(by ), then by∊̃M(ax ).

• (c) If M(ax ) ⊆̃M(by ), then M(ax ) = M(by ).

Proof

(a) Let axSP(X,A). Since (X, τ,A) is soft locally indiscrete, then M(ax ) = ∩̃ {Fτc : ax∊̃F} and so M(ax )τc. Then again by soft local indiscreetness of (X, τ,A), we have M(ax )τ.

(b) Suppose to the contrary that M(ax ) ⊆̃M(by ) and by∊̃1AM(ax ). By (a), 1AM(ax )τ and so M(by ) ⊆̃1AM(ax ). Thus, M(by )∩̃M(ax ) = 0A. On the other hand, since M(ax ) ⊆̃M(by ), then M(by )∩̃M(ax ) = M(ax ) ≠ 0A, a contradiction.

(c) Suppose that M(ax ) ⊆̃M(by ), then by (b) by∊̃M(ax ) and so by (a), M(by ) ⊆̃M(ax ). It follows that M(ax ) = M(by ).

Theorem 3.11

For an STS (X, τ,A), the following are equivalent:

(a) (X, τ,A) is soft locally indiscrete.

(b) minss(X, τ,A) = min(X, τ,A).

(c) min (X, τ,A) is a soft base of (X, τ,A).

(d) (X, τ,A) has a soft base which forms a soft partition of 1A.

Proof

(a) =⇒ (b) By Corollary 3.6 we need only to show that minss(X, τ,A) ⊆ min(X, τ,A). Let axSP(X,A) and let Gτ – {0A} with G⊆̃M(ax ). Choose byG, then M(by ) ⊆̃G⊆̃M(ax ). So Lemma 3.10(c), M(by ) = G = M(ax ). This ends the proof.

(b) =⇒(c) By (b), ∪̃{F : F ∈ min (X, τ,A)} = 1A. Thus by Theorem 4.11 of [1], min (X, τ,A) is a soft base of (X, τ,A).

(c) =⇒(d) By (c) and Theorem 4.11 of [1], min (X, τ,A) is a soft base of (X, τ,A) which forms a soft partition of 1A.

(d) =⇒ (a) Let ℬ be a soft base which forms a soft partition of 1A. Let Gτ – {0A}, then there exists ℬ1⊆ ℬ such that ∪̃{F : F ∈ ℬ1} = G and so 1AG = ∪̃{1AF : F ∈ ℬ –ℬ1} ∈ τ. This shows Gτc. Therefore, (X, τ,A) is soft locally indiscrete.

Definition 3.12

Let (X, τ,A) be an STS. A soft set FSS (X,A) is said to be a soft minimal soft preopen set of (X, τ,A) if FPO(X, τ,A) and for all GPO(X, τ,A) with G⊆̃F either G = 0A or G = F. The family of all soft preopen sets in (X, τ,A) will be denoted by min(PO(X, τ,A)).

The following lemma will be necessary for proving the next main result:

Lemma 3.13

Let (X, τ,A) be an STS. If Fτ and GPO(X, τ,A), then F ∩̃GPO(X, τ,A).

Proof

Let Fτ and GPO(X, τ,A). Since GPO(X, τ,A), then there exists Kτ such that G⊆̃K⊆̃Clτ (G) and so F ∩̃G⊆̃F ∩̃K⊆̃F ∩̃Clτ (G). Since Fτ, then F ∩̃ Clτ (G)⊆̃Clτ (F ∩̃G). Hence, F ∩̃GPO(X, τ,A).

The following theorem shows that soft minimal soft preopen sets are the soft points soft preopen sets:

Theorem 3.14

For any STS (X, τ,A), min(PO(X, τ,A)) = PO(X, τ,A) ∩ SP(X,A).

Proof

By the definition of PO(X, τ,A), min(PO(X, τ,A)) ⊆ PO(X, τ,A). To show that min(PO(X, τ,A)) ⊆ SP(X,A), let F ∈ min(PO(X, τ,A)). Choose ax∊̃F. We are going to show that axPO(X, τ,A). By Lemma 3.13,

$(1A-Clτ(ax))∩˜F∈PO (X,τ,A).$

Since F ∈ min(PO(X, τ,A)), (1AClτ (ax))∩̃F ⊆̃F, and (1AClτ (ax))∩̃FF, then (1AClτ (ax))∩̃F = 0A and so F ⊆̃Clτ (ax). Hence, Clτ (F)⊆̃Clτ (ax) and thus ax ⊆̃F ⊆̃ Intτ (Clτ (F)) ⊆̃Intτ (Clτ (ax)) ⊆̃Clτ (ax). Therefore, axPO(X, τ,A). Now since F ∈ min(PO(X, τ,A)), then F = ax. It follows that FSP(X,A). Conversely, it is clear that PO(X, τ,A) ∩ SP(X,A) ⊆ min(PO(X, τ,A)).

Theorem 3.15

If F ∈ min (X, τ,A), then {ax : ax∊̃F} ⊆̃ min(PO(X, τ,A)).

Proof

Let F ∈ min(X, τ,A) and let ax∊̃F, then by Theorem 3.2 we have axPO(X, τ,A). Thus, axPO(X, τ,A) ∩SP(X,A). Hence, by Theorem 3.14 we have

$ax∈min (PO (X,τ,A)).$

The following example shows that the implication in Theorem 3.15 is not reversible, in general:

Example 3.16

Let X = ℝ and A = {a, b}. Let ℑ be the topology on X having the family {[−c, c] : c ∈ ℝ and c > 2} as a base and let

$τ={F∈SS(X,A):F(t)∈ℑ for every t∈A}.$

Then, a0 ∈ min(PO(X, τ,A)). On the other hand it is not difficult to check that min(X, τ,A) = ∅︀. Consider GSS(X,A), where G(a) = [−2, 2] and G(b) = ∅︀. Then {ax : ax∊̃G} ⊆̃ min(PO(X, τ,A)) but G ∉ min(X, τ,A).

Theorem 3.17

Let (X, τ,A) be an STS and axSP(X,A). Then ax ∈ min(PO(X, τ,A)) if and only if by ∈ min(PO(X, τ, A)) for every by∊̃M(ax ).

Proof
Necessity

Suppose that ax ∈ min(PO(X, τ,A)) and let by∊̃M(ax ).

Claim

ax∊̃Clτ (by).

Proof of Claim

Let Gτ such that ax∊̃G. Then M(ax ) ⊆̃G and so by∊̃G. Therefore, by = by∩̃G and hence by∩̃G ≠ 0A. It follows that ax∊̃Clτ (by).

Now by the above claim we have Clτ (ax)⊆̃Clτ (by). Since axPO(X, τ,A), then ax ⊆̃Intτ (Clτ (ax)) and so

$by⊆˜M(ax,τ)⊆˜Intτ(Clτ(ax))⊆˜Clτ(ax)⊆˜Clτ(by).$

Hence, byPO(X, τ,A). Therefore, by Theorem 3.14, we have by ∈ min(PO(X, τ,A)).

Sufficiency

Suppose that by ∈ min(PO(X, τ,A)) for every by∊̃M(ax ). Since ax∊̃M(ax ), then axPO(X, τ,A). Therefore, by Theorem 3.14, we have

$ax∈min (PO (X,τ,A)).$

Corollary 3.18

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ, A)), then M(ax )PO(X, τ,A).

Theorem 3.19

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ, A)) and there exists Gτ – {0A} such that G⊆̃M(ax ), then M(ax ) = G.

Proof

Suppose that ax ∈ min(PO(X, τ,A)) and G⊆̃M(ax ) for some Gτ – {0A}. Choose by∊̃G, then ax∊̃Clτ (by). Since axPO(X, τ,A), then there exists Fτ such that ax ⊆̃F ⊆̃Clτ (ax)⊆̃Clτ (by). Since ax∊̃Fτ and ax∊̃Clτ (by), then F ∩̃by ≠ 0A and so by∊̃F ⊆̃Clτ (ax). Since by∊̃Gτ and by∊̃Clτ (ax), then G∩̃ax ≠ 0A and hence ax∊̃G. It follows that M(ax ) ⊆̃G and thus, M(ax ) = G.

Corollary 3.20

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ, A)), then either M(ax ) ∈ min(X, τ,A) or Intτ (M(ax )) = 0A.

Theorem 3.21

Let (X, τ,A) be an STS and axSS (X,A). Then the following are equivalent:

• (a) M(ax ) ∈ min(X, τ,A).

• (b) ax ∈ min(PO(X, τ,A)) and Intτ (M(ax )) ≠ 0A.

Proof

(a) ⇒ (b) Suppose that M(ax ) ∈ min(X, τ,A), then obviously that Intτ (M(ax )) ≠ 0A. Also, by Theorem 3.15 we have ax ∈ min(PO(X, τ,A)).

(b) ⇒ (a) Follows directly from Corollary 3.20.

4. Soft Prehomogeneity

The following is the main concept of this section:

Definition 4.1

An STS (X, τ,A) is called soft prehomogeneous if for any ax, bySP (X,A), there exists a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that fpu (ax) = by.

Theorem 4.2

For an STS (X, τ,A), the following are equivalent:

(a) (X, τ,A) is soft prehomogeneous.

(b) For any ax, bySP (X,A), there is a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that p (x) = y and u (a) = b.

(c) For any two pairs (x, a), (y, b) ∈ X × A, there is a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that p (x) = y and u (a) = b.

Proof

Straightforward.

Remark 4.3

Soft homogeneous STSs are soft prehomogeneous.

Proof

Follows because soft homeomorphisms are soft prehomeomorphism.

Theorem 4.4

Every soft locally indiscrete STSs is soft prehomogeneous.

Proof

Let (X, τ,A) be a soft locally indiscrete STS. Let cz, dwSP (X,A). Define p : XX and u : AA as follows:

$p (s)={wif s=zzif s=wsif s≠z and s≠w$

and

$u (t)={dif t=ccif t=deif t≠c and t≠d.$

By Theorem 3.3, PO(X, τ,A) = SS(X,A) and thus fpu is a soft prehomeomorphism with fpu (ax) = by. Therefore, (X, τ,A) is soft prehomogeneous.

The following example shows that the converse of Remark 4.3 is not true, in general:

Example 4.5

Let X = ℝ, A = ℚ. Let FSS(X,A) defined by F (a) = ℕ for every aA. Let τ = {0A, 1A, F, 1AF}. Consider the STS (X, τ,A). Then τ = τc, and so (X, τ,A) is soft locally indiscrete. So by Theorem 4.4, (X, τ,A) is soft prehomogeneous. On the other hand, since min(X, τ,A) = {F, 1AF} and F(1) is countable but (1AF) (1) is uncountable, then by Theorem 5.38(b) of [1], (X, τ,A) is not soft homogeneous.

The following is an example of a soft prehomogeneous STS that is neither soft homogeneous nor soft locally indiscrete:

Example 4.6

Let X = ℕ, A = {a, b} and τ = {FSS(X,A) : F(s) ∈ {{∅︀} ∪ {{n, n + 1, ...} : n ∈ ℕ}} for all sA}.

Then,

(1) (X, τ,A) is not soft locally indiscrete: Take FSS(X,A) defined by F (s) = ℕ – {1} for all sA. Then Fττc, and hence (X, τ,A) is not soft locally indiscrete.

(2) (X, τ,A) is not soft homogeneous: If (X, τ,A) is (X, τ,A), then by Theorem 5.14 of [1] the TS (X, τa) is homogeneous and so there is a homeomorphism g : (X, τa) → (X, τa) such that g (1) = 2. But {1} ∈ (τa)c while g ({1}) = {2} (τa)c. It follows that (X, τ,A) is not soft homogeneous.

(3) (X, τ,A) is soft prehomogeneous: Let cx, dySP (X,A). Define p : XX and u : AA as follows:

$p (s)={yif s=xxif s=ysif s≠x and s≠y$

and

$u (t)={dif t=ccif t=deif t≠c and t≠d.$

It is not difficult to check that PO(X, τ,A) = {FSS(X, A) : F(s) ∈ {∅︀,ℕ}} ∪ {FSS(X,A) : F(a) and F (b) are infinite sets }, and thus fpu is a soft prehomeomorphism with fpu (cx) = dy. Hence, (X, τ,A) is soft prehomogeneous.

Theorem 4.7

Let (X, τ,A) be an STS with min(PO(X, τ, A)) ≠ ∅︀. Then (X, τ,A) is soft prehomogeneous if and only if (X, τ,A) is soft locally indiscrete.

Proof
Necessity

Suppose that (X, τ,A) is soft prehomogeneous. We will apply Theorem 3.3(c). Let bySP(X,A). Pick ax ∈ min(PO(X, τ,A)). By soft prehomogeneity of (X, τ,A), there exists a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that fpu (ax) = by and so byPO(X, τ, A).

Sufficiency

Follows from Theorem 4.4.

Corollary 4.8

Let (X, τ,A) be an STS with min (X, τ,A) ≠ ∅︀. Then (X, τ,A) is soft prehomogeneous if and only if (X, τ,A) is soft locally indiscrete.

Proof

Follows from Theorems 3.15 and 4.7.

Corollary 4.9

Let (X, τ,A) be an STS such that τ is finite. Then (X, τ,A) is soft prehomogeneous if and only if (X, τ,A) is soft locally indiscrete.

Proof

Since τ is finite, then it is easy to see that min(X, τ, A) ≠ ∅︀. Thus, by Corollary 4.8 we get the result.

In the following result, τpr denotes the soft topology on X relative to A having PO(X, τ,A) as a subbase.

Theorem 4.10

If (X, τ,A) is soft prehomogeneous, then (X, τpr,A) is soft homogeneous.

Proof

Suppose that (X, τ,A) is soft prehomogeneous and let ax, bySS(X,A). Since (X, τ,A) is soft prehomogeneous, there exists a soft prehomeomorphism fpu : (X, τ,A) → (X, τ,A) such that fpu (ax) = by. Let G be a soft basic soft open set in (X, τpr,A), then $G=∩i=1n˜Gi$ where GiPO(X, τ,A) for all i = 1, 2, ..., n. Now, for every i ∈ {1, 2, ..., n}, $fpu-1(Gi)∈PO (X,τ,A)$ and so $fpu-1(G)=∩i=1n˜fpu-1(Gi)∈τpr$. Therefore, fpu : (X, τpr,A) → (X, τpr, A) is soft continuous. By a similar way we can see that fp1u1 is soft continuous. Therefore, fpu : (X, τpr,A) → (X, τpr,A) is a soft homeomorphism with fpu (ax) = by. Hence, (X, τpr,A) is soft homogeneous.

By the end of this section, we raise the following question regarding the converse of Theorem 4.10:

Question 4.11

Let (X, τ,A) be an STS such that (X, τpr,A) is soft homogeneous. Is it true that (X, τ,A) is soft prehomogeneous?

Theorem 5.1

Let (X, τ,A) be an STS. Then for any axSP(X,A), M(ax )(a) = M(x,τa).

Proof

Let axSP(X,A), then

$M(ax,τ)(a)=(∩˜{F:F∈τ and ax∈˜F}) (a)=∩{F(a):F∈τ and ax∈˜F}=(∩{F(a):F∈τ and x∈F (a)}) (a)=∩{U∈τa:x∈U}=M(x,τa).$

Lemma 5.2

Let (X, τ,A) be an STS and let ℬ be a soft base of (X, τ,A). Then for any axSP(X,A), M(ax ) = ∩̃ {B ∈ ℬ : ax∊̃B}.

Proof

Let axSP(X,A), then clearly that

$M(ax,τ)⊆˜∩˜{B∈ℬ:ax∈˜B}.$

Conversely, let by∊̃∩̃ {B ∈ ℬ : ax∊̃B} and let Gτ such that ax∊̃G. Choose B ∈ ℬ such that ax∊̃B⊆̃G, then by∊̃B⊆̃G. It follows that by∊̃M(ax ).

Theorem 5.3

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let $σ=⊕α∈Aℑa$. Then for every axSP(X,A), M(ax) = aM(x,ℑa).

Proof

By Theorem 3.5 of [1], {bY : bA and Y ∈ ℑb} is a soft base of (X, σ,A). So, by Lemma 5.2,

$M(ax,σ)=∩˜{bY:b∈A,Y∈ℑb and ax∈˜bY}=∩˜{aY:Y∈ℑa and x∈Y}=a{Y:Y∈ℑa and x∈Y}=aM(x,ℑa).$

Corollary 5.4

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X. Then

$minss (X,τ,A)={aY:a∈A and Y∈ mins(X,ℑa)}.$

Lemma 5.5

Let (X, τ,A) be an STS. Then for every axSP(X,A), (Clτ (ax)) (a) = Clτa ({x}).

Proof

By Proposition 7 of [4], Clτa ({x}) ⊆ (Clτ (ax)) (a). To show that (Clτ (ax)) (a) ⊆ Clτa ({x}), suppose to the contrary that there exists y ∈ (Clτ (ax)) (a) – Clτa ({x}). Since y ∉ Clτa ({x}), then there exists Vτa such that yV and x ∉ V. Choose Fτ such that F (a) = V. Then ax∊̃1AF. On the other hand, since we have ay∊̃Fτ and ay∊̃Clτ (ax), then ax∊̃F, a contradiction.

Theorem 5.6

Let (X, τ,A) be an STS. If ax ∈ min(PO(X, τ,A)), then {x} ∈ min (PO(X, τa)).

Proof

Suppose that ax ∈ min(PO(X, τ,A)). Then axPO(X, τ,A) and so there exists Fτ such that ax ⊆̃F ⊆̃Clτ (ax). Thus we have F (a) ∈ τa and {x} ⊆ F(a) ⊆ (Clτa (ax)) (a). On the other hand, by Lemma 5.5, (Clτ (ax)) (a) = Clτa ({x}). Hence, {x} ∈ PO(X, τa). Therefore, {x} ∈ min (PO(X, τa)).

The following lemma will be used in the following result:

Lemma 5.7

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let $σ=⊕α∈Aℑa$. Then for any GSS(X,A) and any aA, Cla (G(a)) = (Clσ(G)) (a).

Proof

Let GSS(X,A) and aA. By Theorem 3.7 of [1], σa = ℑa. So by Proposition 7 of [4], Cla (G(a)) ⊆ (Clσ(G)) (a). To show that (Clσ(G)) (a) ⊆ Cla (G(a)), let x ∈ (Clσ(G)) (a) and let U ∈ ℑa such that xU. Since ax∊̃aU and ax∊̃Clσ(G), then G∩̃aU ≠ 0A. Thus, (G∩̃aU) (a) = G(a) ∩ U ≠ ∅︀. It follows that xCl(G(a)).

Theorem 5.8

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let $σ=⊕α∈Aℑa$. Then GPO(X, σ,A) if and only if G(a) ∈ PO(X,ℑa) for all aA.

Proof
Necessity

Suppose that GPO(X, σ,A) and let aA. Choose Fσ such that G⊆̃F ⊆̃Clσ(G). Then by Lemma 5.7, G(a) ⊆ F(a) ⊆ (Clσ(G)) (a) = Cla (G(a)). Since Fσ, then by Theorem 3.7 of [1], F(a) ∈ ℑa. It follows that G(a) ∈ PO(X,ℑa).

Sufficiency

Suppose that G(a) ∈ PO(X,ℑa) for all aA. For each aA, choose Ua ∈ ℑa such that G(a) ⊆ UaCla (G(a)). Let FSS(X,A) defined by F(a) = Ua for all aA. Then Fσ and by Lemma 5.7, G(a) ⊆ F(a) ⊆ Cla (G(a)) = (Clσ(G)) (a) for all aA which implies that G⊆̃F ⊆̃Clσ(G). Hence, GPO(X, σ,A).

Corollary 5.9

Let (X,ℑ) be a TS and A be a set of parameters. Then GPO(X, τ(ℑ),A) if and only if G(a) ∈ PO(X,ℑ) for all aA.

Proof

For every aA, put ℑa = ℑ. Then $τ (ℑ)=⊕a∈Aℑa$. So by Theorem 5.8, we get the result.

Theorem 5.10

Let X be an initial universe and A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X. Then $min (PO (X,⊕a∈Aℑa,A))={ax:a∈A and {x}∈min(PO(X,ℑa))}$.

Proof

Let $G∈min (PO (X,⊕a∈Aℑa,A))$, then by Theorem 3.14 there exists $ax∈PO (X,⊕a∈Aℑa,A)∩SP(X,A)$ such that G = ax. By Theorem 3.7 of [1] and Theorem 5.6, we have {x} ∈ min (PO(X,ℑa)). This ends the proof that $min (PO (X,⊕a∈Aℑa,A))⊆{ax:a∈A and {x}∈min(PO(X,ℑa))}$.

Conversely, let aA and {x} ∈ min (PO(X,ℑa)). Then

$(ax) (b)={{x},if b=a,∅,if b≠a,$

and so (ax) (b) ∈ PO(X,ℑa) for all aA. Thus by Theorem 5.8, $ax∈PO (X,⊕a∈Aℑa,A)$. Therefore, by Theorem 3.14, we have $ax∈min (PO (X,⊕a∈Aℑa,A))$.

Corollary 5.11

Let X be an initial universe and A be a set of parameters. Then min (PO(X, τ(ℑ),A)) = {ax : aA and {x} ∈ min(PO(X,ℑ))}.

Proof

For every aA, put ℑa = ℑ. Then $τ (ℑ)=⊕a∈Aℑa$. So by Theorem 5.10, we get the result.

Lemma 5.12

If $fpu:(X,⊕a∈Aℑa,A)→(X,⊕a∈Aℑa,A)$ is soft preirresolute with u (b) = b, then p : (X,ℑb) → (X,ℑb) is preirresolute.

Proof

Let UPO(X,ℑb). Then by Theorem 5.8, $bU∈PO (X,⊕a∈Aℑa,A)$ and so, $fpu-1(bU)∈PO(X,⊕a∈Aℑa,A)$. So again by Theorem 5.8, $(fpu-1(bU))(b)∈PO(X,ℑb)$. But $(fpu-1(bU)) (b)=p-1(bU (u-1 (b))=p-1 (U)$. Therefore, p : (X,ℑb) → (X,ℑb) is preirresolute.

Theorem 5.13

Let X be an initial universe and A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X. If $(X,⊕a∈Aℑa,A)$ is soft prehomogeneous, then for every bA, (X,ℑb) is prehomogeneous.

Proof

Suppose that $(X,⊕a∈Aℑa,A)$ is soft prehomogeneous and let bA. Let x, yX, then bx, bySP(X,A) and so there exists a soft prehomeomorphism $fpu:(X,⊕a∈Aℑa,A)→(X,⊕a∈Aℑa,A)$ such that fpu(bx) = by. Then p : XX is a bijection with p(x) = y. Since u (b) = u1(b) = b, then by Lemma 5.12, the functions p, p1 : (X,ℑb) → (X,ℑb) are preirresolute. It follows that p : (X,ℑb) → (X,ℑb) is a prehomeomorphism with p(x) = y. Hence, (X,ℑb) is prehomogeneous.

Question 5.14

Let {(X,ℑa) : aA} be an indexed family of prehomogeneous TSs. Is it true that $(X,⊕a∈Aℑa,A)$ is soft prehomogeneous.

The following result gives a partial answer for Question 5.14:

Theorem 5.15

Let (X,ℑ) be a TS and A be a set of parameters. Then (X, τ (ℑ),A) is soft prehomogeneous if and only if (X,ℑ) is prehomogeneous.

Proof
Necessity

Suppose that (X, τ (ℑ),A) is soft prehomogeneous. For every bA, put ℑb = ℑ. Then $τ(ℑa)=⊕b∈Aℑb$. Thus, by Theorem 5.13, we get the result.

Sufficiency

Suppose that (X,ℑ) is prehomogeneous. Let ax, bySP(X,A), then there is a prehomeomorphism p : (X,ℑ) → (X,ℑ) such that p(x) = y. Choose a bijection u : AA such that u(a) = b. Then fpu : (X, τ (ℑ),A) → (X, τ (ℑ),A) is a bijection. To show that fpu is soft preirresolute, let GPO(X, τ(ℑ),A), then by Corollary 5.9, G(c) ∈ PO(X,ℑ) for all cA. Since p : (X,ℑ) → (X,ℑ) is preirresolute, then for every dA, $(fpu-1(G)) (d)=p-1(G(u-1(d)))∈PO (X,ℑ)$. Therefore, again by Corollary 5.9, $fpu-1(G) ∈PO (X,τ(ℑ),A)$. Hence, fpu : (X, τ (ℑ),A) → (X, τ (ℑ),A) is soft preirresolute. Similarly, we can show that fp1u1 : (X, τ (ℑ),A) → (X, τ (ℑ),A) is soft preirresolute. It follows that (X, τ (ℑ),A) is soft prehomogeneous.

The following three questions are natural:

Question 5.16

Let fpu : (X, τ,A) → (X, τ,A) be soft prehomeomorphism and aA. Is it true that p : (X, τa) → (X, τa) is a prehomeomorphism?

Question 5.17

Let (X, τ,A) be soft prehomogeneous and aA. Is it true that (X, τa) is prehomogeneous?

Question 5.18

Let (X, τa) be prehomogeneous for all aA. Is it true that (X, τ,A) is soft prehomogeneous?

We will leave Questions 5.16 and 5.17 as open questions. On the other hand, the following example gives a negative answer for Question 5.18:

6. Conclusion

Several characterizations of soft minimal soft open sets are introduced, and two new kinds of soft sets are introduced and investigated. Moreover, the notion of soft prehomogeneity is introduced and investigated. Several relationships are studied. Also, some characterizations of soft locally indiscrete are introduced. Six open questions are raised. We expect that our results will be important for the forthcoming in SBTSs to build a good background for some practical applications and to answer the most intricate problems containing uncertainty in medical, engineering, economics, environment, and in general. In future studies, the following topics could be considered: (1) defining soft prehomogeneity components; and (2) defining soft semihomogeneity.

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