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 0_E and 1_E, respectively. The structure of STSs was defined in [4] as follows: An STS is a triplet (U, τ,E), where τ ⊆ SS (U,E), τ contains 0_E and 1_E, τ is closed under finite soft intersection, and τ is closed under arbitrary soft union. Let (U, τ,E) be an STS and F ∈ SS(X,A), then F is said to be a soft open set in (U, τ,E) if F ∈ τ and F is said to be a soft closed set in (U, τ,E) if 1_A – F is a soft open set in (U, τ,E). The family of all soft closed sets in (U, τ,E) will be denoted by τ^c. Soft topological concepts and their applications are still a hot area of research [1,2,5–26]. The concepts of minimal open sets in TSs were defined and investigated in [27]. Then research via this concept is continued by various researchers [28–31]. 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 S ⊆ X. Then ℑ^c will denote the family of all closed sets in (X,ℑ), Cl_ℑ(S) (resp. Int_ℑ (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 : x ∈ V }.

Definition 2.1

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

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

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

(c) [32] a minimal set in (X,ℑ) if there exists x ∈ X 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 V ⊆ S 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 f⁻¹(V )∈PO(X,ℑ) for all .

(b) prehomeomorphism if f is a bijection and both f and f⁻¹ are preirresolute.

Definition 2.3 [27]

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

Definition 2.4 [35]

Let M,N ∈ SS (X,A).

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

(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∪̃N ∈ SS (X,A) where (M∪̃N) (a) = M(a) ∪ N (a) for each a ∈ A.

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

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

Definition 2.5 [36]

Let Δ be an arbitrary indexed set and {G_α : 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 a ∈ A.

(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 a ∈ A.

Definition 2.6 [1]

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

G(a)={Y,if a=e,∅,if a≠e

will be denoted by e_Y.

Definition 2.7 [37]

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

G(a)={{x},if a=e,∅,if a≠e

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

Definition 2.8 [37]

Let G ∈ SS (X,A) and a_x ∈ SP (X,A). Then a_x is said to belong to F (notation: a_x∊̃G) if a_x ⊆̃G or equivalently: a_x∊̃G if and only if x ∈ G(a).

Theorem 2.9 [4]

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

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 F ∈ SS(X,A). Then Cl_τ (F) (resp. Int_τ (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 F ∈ SS(X,A). Then F is said to be

(a) [39] a soft preopen set in (X, τ,A) if F ⊆ Int_τ (Cl_τ (F)). Equivalently: F is a soft preopen set in (X, τ,A) if and only if there exists G ∈ τ such that F ⊆ G ⊆ Cl_τ (F).

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

Definition 2.13

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

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

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

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

Theorem 3.1

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

(a) F ∈ min (X, τ,A).

(b) For every G ∈ SS(X,A)–{0_A} with G⊆̃F, F ⊆̃Cl_τ (G).

(c) For every G ∈ SS(X,A)–{0_A} with G⊆̃F, Cl_τ (G) = Cl_τ (F).

Theorem 3.2

Let (X, τ,A) and let F ∈ min (X, τ,A). Then for every G ∈ SS(X,A) – {0_A} with G⊆̃F, we have G ∈ PO(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) τ^c ⊆ PO(X, τ,A).

Definition 3.4

Let (X, τ,A) be an STS and G ∈ SS(X,A), then G is called a soft minimal soft set in (X, τ,A) if G = M_{(ax,τ )} for some a_x ∈ SP(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 a_x∊̃G, G = M_{(ax,τ )}.

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 F_n ∈ SS(X,A) defined by Fn(a)=(-1n,1n) and F_n(b) = ∅︀. Since a0∈˜M(a0,τ)⊆∩n∈ℕ˜Fn=a0, then M_{(a0,τ )} = a₀. But M_{(a0,τ )} 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 = {F ∈ SS(X,A) : F(t) ∈ {∅︀, X, {1}, {1, 2}} for every t ∈ A}.

If b₂∊̃F ∈ τ, then 2 ∈ F (b) ∈ {∅︀,X, {1}, {1, 2}}, and so either F (b) = X or F (b) = {1, 2}. Then M_b2 (b) = {1, 2} and M_{(b2,τ )} (a) = ∅︀, and hence M_{(b2,τ )} ∈ minss(X, τ,A) ∩ τ. On the other hand, since b₁ ∈ τ – {0_A} with b₁ ⊆̃M_{(b2,τ )} and b₁ ≠ M_{(b2,τ )}, then M_{(b2,τ )}∉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,τ )} ⊆̃M_{(by,τ )}, then b_y∊̃M_{(ax,τ )}.

• (c) If M_{(ax,τ )} ⊆̃M_{(by,τ )}, then M_{(ax,τ )} = M_{(by,τ )}.

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 1_A.

Definition 3.12

Let (X, τ,A) be an STS. A soft set F ∈ SS (X,A) is said to be a soft minimal soft preopen set of (X, τ,A) if F ∈ PO(X, τ,A) and for all G ∈ PO(X, τ,A) with G⊆̃F either G = 0_A 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 F ∈ τ and G ∈ PO(X, τ,A), then F ∩̃G ∈ PO(X, τ,A).

Theorem 3.14

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

Theorem 3.15

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

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 G ∈ SS(X,A), where G(a) = [−2, 2] and G(b) = ∅︀. Then {a_x : a_x∊̃G} ⊆̃ min(PO(X, τ,A)) but G ∉ min(X, τ,A).

Theorem 3.17

Let (X, τ,A) be an STS and a_x ∈ SP(X,A). Then a_x ∈ min(PO(X, τ,A)) if and only if b_y ∈ min(PO(X, τ, A)) for every b_y∊̃M_{(ax,τ )}.

Corollary 3.18

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

Theorem 3.19

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

Corollary 3.20

Let (X, τ,A) be an STS. If a_x ∈ min(PO(X, τ, A)), then either M_{(ax,τ )} ∈ min(X, τ,A) or Int_τ (M_{(ax,τ )}) = 0_A.

Theorem 3.21

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

• (a) M_{(ax,τ )} ∈ min(X, τ,A).

• (b) a_x ∈ min(PO(X, τ,A)) and Int_τ (M_{(ax,τ )}) ≠ 0_A.

The following is the main concept of this section:

Definition 4.1

An STS (X, τ,A) is called soft prehomogeneous if for any a_x, b_y ∈ SP (X,A), there exists a soft prehomeomorphism f_pu : (X, τ,A) → (X, τ,A) such that f_pu (a_x) = b_y.

Theorem 4.2

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

(a) (X, τ,A) is soft prehomogeneous.

(b) For any a_x, b_y ∈ SP (X,A), there is a soft prehomeomorphism f_pu : (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 f_pu : (X, τ,A) → (X, τ,A) such that p (x) = y and u (a) = b.

Remark 4.3

Soft homogeneous STSs are soft prehomogeneous.

Theorem 4.4

Every soft locally indiscrete STSs is soft prehomogeneous.

Example 4.5

Let X = ℝ, A = ℚ. Let F ∈ SS(X,A) defined by F (a) = ℕ for every a ∈ A. Let τ = {0_A, 1_A, F, 1_A – F}. Consider the STS (X, τ,A). Then τ = τ^c, 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, 1_A – F} and F(1) is countable but (1_A – F) (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 = {F ∈ SS(X,A) : F(s) ∈ {{∅︀} ∪ {{n, n + 1, ...} : n ∈ ℕ}} for all s ∈ A}.

Then,

(1) (X, τ,A) is not soft locally indiscrete: Take F ∈ SS(X,A) defined by F (s) = ℕ – {1} for all s ∈ A. Then F ∈ τ – τ^c, 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, τ_a) is homogeneous and so there is a homeomorphism g : (X, τ_a) → (X, τ_a) such that g (1) = 2. But {1} ∈ (τ_a)^c while g ({1}) = {2} ∉ (τ_a)^c. It follows that (X, τ,A) is not soft homogeneous.

(3) (X, τ,A) is soft prehomogeneous: Let c_x, d_y ∈ SP (X,A). Define p : X → X and u : A → A 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) = {F ∈ SS(X, A) : F(s) ∈ {∅︀,ℕ}} ∪ {F ∈ SS(X,A) : F(a) and F (b) are infinite sets }, and thus f_pu is a soft prehomeomorphism with f_pu (c_x) = d_y. 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.

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.

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.

Theorem 4.10

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

Question 4.11

Let (X, τ,A) be an STS such that (X, τ_pr,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 a_x ∈ SP(X,A), M_{(ax,τ )}(a) = M_(x,τ^a).

Lemma 5.2

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

Theorem 5.3

Let X be an initial universe and let A be a set of parameters. Let {ℑ_a : a ∈ A} be an indexed family of topologies on X and let σ=⊕α∈Aℑa. Then for every a_x ∈ SP(X,A), M_(ax,σ) = a_M(x,ℑa).

Corollary 5.4

Let X be an initial universe and let A be a set of parameters. Let {ℑ_a : a ∈ A} 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 a_x ∈ SP(X,A), (Cl_τ (a_x)) (a) = Cl_τa ({x}).

Theorem 5.6

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

Lemma 5.7

Let X be an initial universe and let A be a set of parameters. Let {ℑ_a : a ∈ A} be an indexed family of topologies on X and let σ=⊕α∈Aℑa. Then for any G ∈ SS(X,A) and any a ∈ A, Cl_ℑa (G(a)) = (Cl_σ(G)) (a).

Theorem 5.8

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

Corollary 5.9

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

Theorem 5.10

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

Corollary 5.11

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

Lemma 5.12

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

Theorem 5.13

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

Question 5.14

Let {(X,ℑ_a) : a ∈ A} 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.

Question 5.16

Let f_pu : (X, τ,A) → (X, τ,A) be soft prehomeomorphism and a ∈ A. Is it true that p : (X, τ_a) → (X, τ_a) is a prehomeomorphism?

Question 5.17

Let (X, τ,A) be soft prehomogeneous and a ∈ A. Is it true that (X, τ_a) is prehomogeneous?

Question 5.18

Let (X, τ_a) be prehomogeneous for all a ∈ A. 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.