Semi-Hurewicz-Type properties in ditopological texture spaces

Hafiz Ullah; Moiz ud Din Khan

doi:10.1515/math-2018-0104

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Semi-Hurewicz-Type properties in ditopological texture spaces

Hafiz Ullah and Moiz ud Din Khan

Published/Copyright: November 8, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper we will define and discuss semi-Hurewicz type covering properties in ditopological texture spaces. We consider the behaviour of semi-Hurewicz and co-semi-Hurewicz selection properties under direlation and difunction between ditopological texture spaces.

Keywords: Semi-open set; Ditopological texture space; Semi-compact space; Semi-cocompact space; s-Hurewicz space; co-s-Hurewicz space

MSC 2010: 54A05; 54A10; 54C08; 54H12; 54D20

1 Introduction

The theory of selection principles can be traced back to the first half of 19th century. The general form of classical selection principles in topological spaces have been defined as follows:

Let 𝓢 be an infinite set and let 𝓐 and 𝓑 be collections of subsets of 𝓢. Then the symbol 𝓢₁(𝓐, 𝓑) defines the statement:

For each sequence (A_n)_n<∞ of elements of 𝓐 there is a sequence (b_n)_n<∞ such that for each n, we have b_n ∈ A_n, and {b_n}_n<∞ ∈ 𝓑.

If 𝓞 denotes the collection of open covers of a topological space (X, T) then the property 𝓢₁(𝓞, 𝓞) is called the Rothberger covering propertyand was introduced by Rothberger in [1].

Similarly, the selection hypothesis S_fin(𝓐, 𝓑) is defined as:

For each sequence (A_n)_n<∞ of elements of 𝓐 there is a sequence (B_n)_n<∞ such that for each n, we have B_n is finite subset of A_n, and ⋃n<∞Bn∈B.

The property introduced in [2] by K. Menger in 1924 is equivalent to S_fin (𝓞, 𝓞) and was proved by W. Hurewicz in [3] in 1925. The property S_fin (𝓞, 𝓞) is known as the Menger covering property.

In 2016, Sabah et al. in [4] proved that X has the s-Menger (resp. s-Rothberger covering property [5]) in topological space X, if X satisfies S_fin (s𝓞, s𝓞) (resp. S₁ (s𝓞, s𝓞)) where s𝓞 denotes the family of all semi-open covers of X.

For readers more interested in theory of selection principles and its relations with various branches of mathematics, we refer to see [6, 7, 8, 9, 10].

In this paper we study the properties of ditopological texture spaces related to the following classical Hurewicz property [3] U_fin(𝓐, 𝓑): For each sequence {A_n}_n∈ℕ of elements of 𝓐, which do not contain a finite sub-cover, there exist finite (possibly empty) subsets B_n ⊆ A_n, n ∈ ℕ and {⋃Bn}n∈N∈B. It was shown in [11] that the Hurewicz property is of the S_fin-type for appropriate classes of 𝓐 and 𝓑.

2 Preliminaries

L. M. Brown in 1992 at a conference on Fuzzy systems and artificial intelligence held in Trabzon introduced the notion of a texture space under the name fuzzy structure. Textures first arose in representation of connection of lattices of L−fuzzy sets and Hutton algebras under a point based settings. This representation provided a fruitful atmosphere to study complement free concepts in mathematics. We now recall the definition of the texture space as follows.

Texture space: [12] If 𝓢 is a set, a texturing ℑ ⊆ P(𝓢) is complete, point separating, completely distributive lattice containing S and ∅, and, for which finite join ⋁ coincides with union ⋃ and arbitrary meet ⋀ coincides with intersection ⋃. Then the pair (S, ℑ) is called the texture space.

A mapping σ : ℑ → ℑ satisfying σ²(A) = A, for each A ∈ ℑ and A ⊆ B implies σ(B) ⊆ σ(A), ∀ A, B ∈ ℑ is called a complementation on (𝓢, ℑ) and (𝓢, ℑ, σ) is then said to be a complemented texture [12]. The sets Ps=⋂{A∈ℑ|s∈A} and Qs=⋁{Pt|t∈S,s∉Pt} defines conveniently most of the properties of the texture space and are known as p-sets and q-sets respectively.

For A ∈ ℑ the core A^bof A is defined by A^b={s ∈ 𝓢 ∣ A ⊈ Q_s}. The set A^bdoes not necessarily belong to ℑ.

If (S,P(s)),(L,ℑ2) are textures, then the product texture of (𝓢, P(𝓢)) and (L,ℑ2) is P(𝓢) ⊗ℑ₂ for which P_(s,t) and Q_(s,t) denotes the p-sets and q-sets respectively. For s ∈ 𝓢, t ∈ 𝓛 we have p-sets and q-sets in the product space as following :

P¯(s,t)={s}×PtQ¯(s,t)=(S∖{s}×T)∪(S×Qt).

Direlation: [13] Let (S, ℑ₁), (𝓛, ℑ₂) be textures. Then for r∈P(S)⊗ℑ2 satisfying:

(R₁) r ⊈ Q_(s,t) and P_ś ⊈ Q_s implies r ⊈ Q_(ś,t),

(R₂) r ⊈ Q_(s,t) then there is ś ∈ 𝓢 such that P_s ⊈ Q_ś and r ⊈ Q_(ś,t), is called relation and for R∈P(S)⊗ℑ2 such that:

(CR₁) P¯(s,t)⊈RR and P_s ⊈ Q_ś implies P¯(s´,t)⊈R,

(CR₂) If P¯(s,t)⊈R then there exists ś ∈ 𝓢 such that P_ś ⊈ Q_s and P¯(s´,t)⊈R, is called a corelation from (𝓢, P(𝓢)) to (𝓛, ℑ₂). The pair (r, R) together is a direlation from (𝓢, ℑ₁) to (𝓛, ℑ₂).

Lemma 2.1

([13]) Let (r, R) be a direlation from (𝓢, ℑ₁) to (T, ℑ₂), J be an index set, Aj∈ℑ1,∀j∈J and B_j ∈ ℑ₂,∀j ∈ J. Then:

(1) r←(⋂j∈JBj)=⋂j∈Jr←BjandR→(⋂j∈JAj)=⋂j∈JR→Aj,

(2) r→(⋁j∈JAj)=⋁j∈Jr→AjandR←(⋁j∈JBj)=⋁j∈JR←Bj.

Difunction

Let (f, F) be a direlation from (𝓢, ℑ₁) to 𝓛, ℑ₂). Then (f, F): (𝓢, ℑ₁) → (𝓛, ℑ₂) is a difunction if it satisfies the following two conditions :

(DF1) For s,s´∈S,Ps⊈Qs´⟹t∈L with f⊈Q¯(s,t) and P¯(s´,t)⊈F.

(DF2) For t,t′∈L and s ∈ S, f ⊈ Q_{(s, t)} and P¯(s,t′)⊈F⟹Pt′⊈Qt.

Definition 2.2

[13] Let (f, F): (S, ℑ₁) → (𝓛, ℑ₂) be a difunction. For A ∈ ℑ₁, the image f^→(A) and coimage F^→(A) are defined as:

f⟶(A)=⋂{Qt:∀s,f⊈Q¯(s,t)⟹A⊆Qs},

F⟶(A)=⋁{Pt:∀s,P¯(s,t)⊈F⟹Ps⊆A},

and for B ∈ ℑ₂, the inverse image f^⇐(B) and inverse coimage F^⇐(B) are defined as:

f⟵(B)=⋁{Ps:∀t,f⊈Q¯(s,t)⟹Pt⊆B},

F⟵(B)=⋂{Qs:∀t,P¯(s,t)⊈F⟹B⊆Qt}.

For a difunction, the inverse image and the inverse coimage are equal, but the image and coimage are usually not.

Lemma 2.3

[13] For a direlation (f, F) from (S, ℑ₁) to (T, ℑ₂) the following are equivalent:

(f, F) is a direlation.
The following inclusion holds:
1. f^←(F^→(A))⊆ A ⊆ F^←(f^→(A)); ∀ A ∈ ℑ₁, and
2. f^←(F^←(B))⊆ B ⊆ F^→(f^←(B)); ∀ B ∈ ℑ₁
f^←(B)=F^←(B); ∀ B ∈ ℑ₂.

Definition 2.4

[13] Let (f, F): (S, ℑ₁) → (𝓛, ℑ₂) be a difunction. Then (f, F) is called surjective if it satisfies the condition:

(SUR) Fort,t′∈L,Pt⊈Qt′⟹∃s∈S,⊈Q(s,t′)andP¯(s,t)⊈F. Similarly, (f, F) is called injective if it satisfies the condition (INJ) For s, ś ∈ S, and t ∈ 𝓛 with f ⊈ Q_(s,t)andP¯(s´,t)⊈F⟹Ps⊈Qs´.

We now recall the notion of ditopology on texture spaces.

Definition 2.5

[14] A pair (τ, κ) of subsets of ℑ is said to be a ditopology on a texture space (S, ℑ), if τ ⊆ ℑ satisfies:

S, ∅ ∈ τ.
G₁, G₂ ∈ τ implies G₁ ∩ G₂ ∈ τ and
G_α ∈ τ, α ∈ I implies⋁αGα∈τ,

and κ ⊆ ℑ satisfies:

S, ∅ ∈ κ.
F₁, F₂ ∈ κ implies F₁∪ F₂ ∈ κ and
F_α ∈ κ, α ∈ I implies ⋂ F_α ∈ κ,

where the members of τ are called open sets and members of κ are closed sets. Also τ is called topology, κ is called cotopology and (τ, κ) is called ditopology. If (τ, κ) is a ditopology on (S, ℑ) then (S, ℑ, τ, κ) is called a ditopological texture space.

Note that in general we assume no relation between the open and closed sets in ditopology. In case of complemented texture space (S, ℑ, σ), τ and κ are connected by the relation κ=σ(τ), where σ is a complementation on (S, ℑ), that is an inclusion reversing involution σ :ℑ → ℑ, then we call (τ, κ) a complemented ditopology on (S, ℑ). A complemented ditopological texture space is denoted by (S, ℑ, σ, τ, κ). In this case we have σ (A) = (σ(A))° and σ(A°) = (σ (A)), where ()° denotes the interior and ( ) denotes the closure. Recall that for a ditopology (τ, κ) on (S, ℑ), for A ∈ ℑ the closure of A for the ditopology (τ, κ) is denoted by (A) and defined by

(A¯)=⋂{F∈κ:A⊆F},

and the interior of A is denoted by (A)° and defined by

(A)∘=⋁{G∈τ:G⊆A}

For terms not defined here, the reader is referred to see [6, 13, 15].

The idea of semi-open sets in topological spaces was first introduced by Norman Levine in 1963 in [16]. Ş Dost extended this concept of semi-open sets from topological spaces to ditopological texture spaces in 2012 in [17].

It is known from [17] that in a ditopological texture space (S, ℑ, τ, κ):

A ∈ ℑ is semi-open if and only if there exists a set G ∈ O(S) such that G ⊆ A ⊆ G.
B ∈ ℑ is semi-closed if and only if there exists a set F ∈ C(S) such that (F)°⊆ B ⊆ F.
O(S)⊆ SO (S) and C(S)⊆ SC (S). The collection of all semi-open (resp. semi-closed) sets in ℑ is denoted by SO (S, ℑ,τ, κ) or simply SO (S) (resp. SC (S, ℑ, τ, κ) or simply SC (S)). SR (S) is the collection of all the semi-regular sets in S. A set A is semi-regular if A is semi-open as well as semi-closed in S.
Arbitrary join of semi-open sets is semi-open.
Arbitrary intersection of semi-closed sets is semi-closed.

If A is semi-open in ditopoloical texture space (S, ℑ, τ, κ) then its complement may not be semi-closed. Every open set is semi-open, whereas a semi-open set may not be open. The intersection of two semi-open sets may not be semi-open, but intersection of an open set and a semi-open set is always semi-open.

In general there is no connection between the semi-open and semi-closed sets, but in case of complemented ditopological texture space (S, ℑ, σ, τ, κ), A ∈ ℑ is semi-open if and only if σ(A) is semi-closed. Where ()_∘ denotes the semi-Interior and ( ) denotes the semi-closure.

Definition 2.6

[18] Let (S, ℑ, τ, κ) be a ditopological texture space and A ∈ ℑ. We define:

(i) The semi-closure (A) of A under (τ, κ) by

(A_)=⋂{B:B∈SC(S), andA⊆B}

(ii) The semi-interior (A)_∘ of A under (τ, κ) by

(A)∘=⋁{B:B∈SO(S), andB⊆A}.

Lemma 2.7

[18] Let (S, ℑ, τ, κ) be a ditopological texture space. A set A ∈ ℑ is called:

(a) semi-open if and only if A ⊆ (A°)

(b) semi-closed if and only if (A)° ⊆ A.

A difunction (f, F): (S, ℑ₁, τ_S,κ_S) → (T, ℑ₂,τ_T, κ_T) is:

continuous [17]; if F^←(G) ∈ τ_S where G ∈ τ_T;
cocontinuous [17]; if f^←(K) ∈ κ_S where K ∈ κ_T;
bicontinuous [17]; if it is continuous and cocontinuous.

Definition 2.8

[17] Let (S_i, ℑ_i, τ_i, κ_i), i=1, 2 be ditoplogical texture spaces. A difunction (f, F):(S₁, ℑ₁) → (S₂, ℑ₂) is said to be:

semi-continuous (semi-irresolute) if for each open (resp. semi-open) set A ∈ ℑ_2,the inverse image F^←(A) ∈ ℑ₁is a semi-open set.
semi-cocontinuous (semi-co-irresolute) if for each closed (resp. semi-closed) set B ∈ ℑ_2,the inverse image f^←(B) ∈ ℑ₁is a semi-closed set.
semi-bicontinuous if it semi-continuous and semi-cocontinuous.
semi-bi-irresolute if it is semi-irresolute and semi-co-irresolute

Throughout this paper a space S is an infinite ditopological texture space (S, ℑ, τ, κ) on which no separation axioms are assumed unless otherwise stated.

3 Selection properties of ditopological texture spaces

In 2017 Kočinac and özçağ [19] introduced Hurewicz type properties in texture and ditopological texture spaces as a continuation of the studies of selection properties of texture structures initiated in 2015 in [14]. In this section, semi-Hurewicz selection properties of ditopological texture spaces are defined and studied.

Definition 3.1

Let (τ, κ) be a ditopology on the texture space (S, ℑ) and take A ∈ ℑ. The family {G_α:α ∈ ∇} is said to be semi-open cover of A if G_α ∈ SO (S) for all α ∈ ∇, and A ⊆ ∨_α∈∇G_α. Dually, we may speak of a semi-closed cocover of A, namely a family {𝓕_α : α ∈ ∇} 𝓕_α ∈ SC (S) satisfying∩α∈∇Fα⊆A.

Let s𝓞 denote the collection of all semi-open covers of a ditopological texture space (S, ℑ, τ, κ). Note that the class of semi-open covers contains the class of open covers of the ditopological texture space (S, ℑ, τ, κ).

Definition 3.2

Let (S, ℑ, τ, κ) be a ditopological texture space and A be a subset of S.

A is said to have the semi-Hurewicz property (or s-Hurewicz property) for the ditopological texture space if for each sequence(Un)n∈Nof semi-open covers of A there is a sequence (𝓥_n)_n∈ℕsuch that for each n ∈ ℕ, 𝓥_nis a finite subset of 𝓤_n and ⋁n∈N⋂m>n(⋃Vm)is a cover of A. We say that (S, ℑ, τ, κ) is s-Hurewicz if the set S is s-Hurewicz. This property is denoted byUfin(sO,sO).
A is said to have the co-semi-Hurewicz property (or co-s-Hurewicz property) for the ditopological texture space if for each sequence(Fn)n∈Nof semi-closed cocovers of A there is a sequence (κ_n)_n∈ℕsuch that for each n ∈ ℕ, κ_nis a finite subset of 𝓕_n and⋂n∈N⋁m>n(⋂κm)⊆A. We say that (S, ℑ, τ, κ) is co-s-Hurewicz if the set ∅ is co-s-Hurewicz. This property is denoted byUcfin(sC,sC), where s ℂ is the family of all semi-closed cocovers of ∅.

Every s-Hurewicz ditopological texture space (S, ℑ, τ, κ) is a Hurewicz ditopological texture space and co-s-Hurewicz ditopological texture space (S, ℑ, τ, κ) is co-Hurewicz ditopological texture space. Converses are not true in general.

Definition 3.3

[20]. Let (τ, κ) be a ditopology on the texture space (S, ℑ) and take A ∈ ℑ.

(1) A is said to be s-compact if whenever {G_α:α ∈ ∇} is semi-open cover of A, there is a finite subset ∇₀of ∇, with A ⊆ ∨_α∈∇₀G_α.

The ditopological texture space (S, ℑ, τ, κ) is s-compact if S is s-compact. Every s-compact space in the ditopologcal texture space is compact but not conversely.

(2) A is said to be s-cocompact if whenever {𝓕_α : α ∈ ∇} is a semi-closed cocover of A, there is a finite subset ∇₀of ∇, with∩α∈∇0Fα⊆AA. In particular, the ditopolgical texture space (S, ℑ, τ, κ) is s-cocompact if ∅ is s-cocompact. It is clear that s-cocompact ditopological texture space is cocompact.

In general s-compactness and s-cocompactness are independent.

Definition 3.4

A ditopological texture space (S, ℑ, τ, κ) is said to be σ-s-compact (resp. σ-s-cocompact) if there is a sequence (A_n : n ∈ ℕ) of s-compact (s-cocompact) subsets of S such that∨n∈NAn=S (resp. ∩n∈NAn=∅).

Theorem 3.5

Let (S, ℑ, τ, κ) be a ditopological texture space.

If (S, ℑ, τ, κ) is σ-s-compact, then (S, ℑ, τ, κ) has the s-Hurewicz property.
If (S, ℑ, τ, κ) is σ-s-cocompact, then (S, ℑ, τ, κ) has the co-s-Hurewicz property.

Proof

(1). Let (Un)n∈N be a sequence of semi-open covers of S. Since S is σ-s-compact therefore it can be represented in the form S=∨i∈NAi, where each 𝓐_i is s-compact A_i ⊆ A_i+1 for all i ∈ ℕ. For each i ∈ ℕ, choose a finite set Vi⊆Ui such that Ai⊆∨Vi=∪Vi. Then the sequence (𝓥_n)_n∈ℕ shows that S is s-Hurewicz.

(2). Let (𝓕_n)_n∈ℕ be a sequence of semi-closed cocovers of ∅. We have ∅=∩i∈NAi, where each A_i is s-cocompact and A_i ⊇ A_i+1, i ∈ ℕ. For each i ∈ ℕ, choose a finite subset κi⊆Fi such that ∩κ_i ⊆ A_i. Then ∩i∈N∨m>i∩κm⊆∩i∈NAi=∅, which means that S is co-s-Hurewicz. □

For complemented ditopological texture spaces we have:

Theorem 3.6

Let (S, ℑ, σ) be a texture with the complementation σ and let (τ, κ) be a complemented ditopology on (S, ℑ, σ). ThenS∈Ufin(sO,sO)if and only if∅∈Ucfin(sC,sC).

Proof

Let S∈Ufin(sO,sO) and let (Fn)n∈N be a sequence of semi-closed cocovers of ∅. Then (σ (𝓕_n) = σ (F) : F ∈ 𝓕_n) and (σ(Fn)n∈N) is a sequence of semi-open covers of S. Since S∈Ufin(sO,sO), there is a sequence (𝓥_n)_n∈ℕ of finite sets such that for each n, Vn⊆σ(Fn) and ∨n∈N∩m>n(∨Vm)=S. We have (σ (𝓥_n) : n ∈ ℕ) is a sequence of finite sets, and also

∅=σ(S)=σ(∨n∈N∩m>n(∨Vm))=∩n∈N∨m>n(∩σ(Vm))

Hence ∅∈Ucfin(sC,sC).

Conversely let ∅∈Ucfin(sC,sC) and (Un)n∈N be a sequence of semi-open covers of S. Then (σ(Un)={σ(U):U∈Un}) and (σ(Un)n∈N is a sequence of semi-closed cocovers of ∅. Since ∅∈Ucfin(sC,sC), there is a sequence (κ_n)_n∈ℕ of finite sets such that for each n, κn⊆σ(Un) and ∩n∈N∨m>n(∩κm)=∅. We have (σ (κ_n)_n∈ℕ) is a sequence of finite sets, such that

S=σ(∅)=σ(∩n∈N∨m>n(∩κm))=∨n∈N∩m>n(∨σ(κm))

Hence S∈Ufin(sO,sO). □

Example 3.7

There is a ditopological texture space which is s-Hurewicz, but not s-compact.

Let(R,ℜ,τR,κR)be the real line with the textureℜ={(−∞,r]:r∈R}∪{(−∞,r):r∈R}∪{R,∅}, topologyτR={(−∞,r):r∈R}∪{R,∅}and cotopologyκR={(−∞,r]:r∈R}∪{R,∅}. This ditopological texture space is not compact by (Example 2.3 [19]) and hence not s-compact because the (open and hence) semi-open coverU={(−∞,n):n∈N}does not contain a finite subcover, nor the texture space is s-cocompact because its (closed and hence) semi-closed cocover{(−∞,n]:n∈N}does not contain a finite cocover. But(R,ℜ,τR,κR)is s-Hurewicz and co-s-Hurewicz. Let us prove that this space is s-Hurewicz. Let(Un:n∈N)be a sequence of semi-open covers of ℝ. We note that semi-open sets can be of the form (−∞, r) and (−∞, r]. WriteR=∪{(−∞,n]:n∈N}. For each n, 𝓤_n is a semi-open cover of ℝ, hence there is some r_n ∈ ℝ such that(−∞,rn]⊆(−∞,n]∈Un. Then the collection{(−∞,rn]:n∈N} shows that (R,ℜ,τR,κR)is s-Hurewicz.

Evidently we have the following diagram:

s−compact⇒s−Hurewicz⇓⇓compact⇒Hurewicz

Theorem 3.8

Let (S, ℑ, σ) be a texture space with complementation σ and let (τ, κ) be a complemented ditopology on (S, ℑ, σ). Then for K ∈ κ with K≠S, K is s-Hurewicz if and only if G is co-s-Hurewicz for some G ∈ τ and G≠∅.

Proof

(⇒) Let G ∈ τ with G≠∅. Let (Fn)n∈N be a sequence of semi-closed cocovers of G. Set K=σ(G) and we obtain K ∈ κ with K≠S. K is s-Hurewicz, for (σ(Fn))n∈N the sequence of semi-open covers of K there is a sequence (𝓥_n)_n∈ℕ of finite sets such that for each n, Vn⊆σ(Fn) and ∨n∈N∩m>n(∪Vm) is semi-open cover of K. Thus (σ(Vn))n∈N is a sequence of finite sets such that for each n, σ(Vn)⊆Fn and σ(∨n∈N∩m>n(∪Vm))=∩n∈N∨m>n(∩σ(Vm))⊆G which gives G is co-s-Hurewicz.

(⇐) K ∈ κ with K≠S. Let (Un)n∈N be a sequence of semi-open covers of K. Since K=σ(G) but G is co-s-Hurewicz, so for (σ(Un))n∈N the sequence of semi-closed cocovers of G there is a sequence (κn)n∈N such that for each n ∈ ℕ, κ_n is a finite subset of (σ(Un) and ∩n∈N∨m>n(∩κm)⊆G. Thus σ(κ_n) is a sequence such that for each n ∈ ℕ, κ_n is a finite subset of 𝓤_n and σ(∩n∈N∨m>n(∩κm))=∨n∈N∩m>n(∪Vm)⊇K which gives K is s-Hurewicz. □

4 Operations in ditopological texture subspaces

Let (S, ℑ) be a texture space and A ∈ ℑ. The texturing ℑ_A={A ∩ K:K ∈ ℑ} of A is called the induced texture on A [21], and (A, ℑ_A) is called a principal subtexture of (S, ℑ).

Definition 4.1

[20]. Let (S, ℑ, τ, κ) be a ditopological texture space and (A, ℑ_A) be a principle subtexture of (S, ℑ) for A ∈ ℑ. Then (A, ℑ_A, τ_A, κ_A) is a subspace of a ditopological texture space (S, ℑ, τ, κ), where τ_A={A ∩ G:G ∈ τ } and κ_A={A ∩ K:K ∈ κ }.

Note that if A ∈ τ, then A is said to be open subspace and if A ∈ κ, then A is closed subspace.

Lemma 4.2

[20]. Let (A, ℑ_A, τ_A, κ_A) be a subspace of a ditopological texture space (S, ℑ, τ, κ) and 𝓑 ⊆ A then:

𝓑 is τ_A − open if and only if 𝓑 = A ∩ G for some τ-open set G.
𝓑 is κ_A − closed if and only if 𝓑 = A ∩ F for some κ-closed set F.

Theorem 4.3

Let (S, ℑ, σ, τ, κ) be a complemented ditopological texture space. If S is s-Hurewicz and A ∈ SR(S), then (A, ℑ_A, τ_A, κ_A) is also s-Hurewicz.

Proof

Let (Un)n∈N be a sequence of semi-open covers of A. Then for each n ∈ ℕ, Vn=Un∪{σ(A)} is semi-open cover of S. By hypothesis there are finite families Wn⊆Vn, n ∈ ℕ, such that S=∨n∈N∩m>n∪Wm. Set Hn=Wn∖{σ(A)}, n ∈ ℕ. Then for each n ∈ ℕ, 𝓗_n is a finite subset of 𝓤_n and A⊆∨n∈N∩m>n(∪Hm), i.e., 𝓐 is semi-Hurewicz. □

Theorem 4.4

Let (S, ℑ, σ, τ, κ) be a complemented ditopological texture space. If S is co-s-Hurewicz and A ∈ SR (S), then (A, ℑ_A, τ_A, κ_A) is also co-s-Hurewicz.

Proof

Let (𝓕_n)_n∈ℕ be a sequence of semi-closed cocovers of A. Then for each n ∈ ℕ, Kn=Fn∪{σ(A)} is a semi-closed cocover of S. But by hypothesis S is co-s-Hurewicz therefore, there are finite families Wn⊆Kn, for each n ∈ ℕ, such that ∩n∈N∨m>n∩Wm⊆S. Set Gn=Wn∖{σ(A)}, n ∈ ℕ. Then for each n ∈ ℕ, 𝓖_n is a finite subset of 𝓕_n and ∩n∈N∨m>n(∩Gm)⊆A, i.e., A is co-s-Hurewicz. □

Example 4.5

If a complemented ditopological texture space S is s-Hurewicz, and A ∈ SR (S) then the subtexture S_Ais also s-Hurewicz.

Let ℝ be the real line textured byℜ={(−∞,r]:r∈R}∪{(−∞,r):r∈R}∪{R,∅}, with complemented ditopology(τR,κR)such thatτR={(−∞,r):r∈R}∪{R,∅}andκR={(−∞,r]:r∈R}∪{R,∅}. WhereSR(R)={{(−∞,r):r∈R}∪{(−∞,r]:r∈R}∪{R,∅}}. This ditopological texture space is s-Hurewicz and co-s-Hurewicz by Example 3.7. LetY={(−∞,r]:r∈R}, then (𝕐, 𝒴) is an induced subtexture of (ℝ, ℜ) with texturing Y={Y∩A|A∈ℜ}={(−∞,s):s∈Y}∪{(−∞,s]:s∈Y}∪{Y,∅}, topologyτY={G∩Y:G∈τ}={(−∞,s):s∈Y}∪{Y,∅}and cotopologyκY={K∩Y:K∈κ}={(−∞,s]:s∈Y}∪{Y,∅}. Thus(Y,Y,τY,κY)is a ditopological texture space.

Now we have to show that 𝕐 is s-Hurewicz. Let(Un:n∈N)be a sequence of semi-open covers of 𝕐. Represent 𝕐 as∪{(−∞,n):n∈N}. For each n, 𝓤_nis a semi-open cover of 𝕐, hence there is some r_n ∈ ℝ such that(−∞,rn)⊆(−∞,n)∈Un. Then the collection{(−∞,rn):n∈N}shows that subtexture(Y,Y,τY,κY)is s-Hurewicz.

Lemma 4.6

[17] Let (S_j, ℑ_j, τ_j, κ_j), j = 1, 2, be ditopological texture space. (f, F) be a difunction between them if:

(1) The following statements are equivalent:

(a) (f, F) is semi-continuous.

(b) (F^→(A))° ⊆ F^→ (A_∘) ∀ A ∈ ℑ₁.

(c) f^←(B°) ⊆(f^←(B))_∘ ∀ B ∈ ℑ₂.

(2) The following statements are equivalent:

(a) (f, F) is semi-cocontinuous.

(b) f^→(A ⊆ (f^→(A)) ∀ A ∈ ℑ₁.

(c)(F^←(B)) ⊆ F^←(B) ∀ B ∈ ℑ₁.

Definition 4.7

Let (S_i, ℑ_i, τ_i, κ_i), i=1, 2 be ditoplogical texture spaces. A difunction (f, F):(S₁, ℑ₁) → (S₂, ℑ₂) is said to be:

(i) s-continuous; if F^←(A) ∈ O(S₁) for each A ∈ SO (S₂).

(ii) s-cocontinuous; if f^←(B) ∈ C(S₁) for each B ∈ SC (S₂).

(iii) s-bicontinuous; if it both s-continuous and s-cocontinuous.

Lemma 4.8

[17]. Let (f, F) : (S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be a difunction.

(1) if (f, F) is cocontinuous and coclosed then F^←(B) = (F^←(B))for all B ∈ ℑ₂.

(2) if (f, F) is continuous and open then F^←(B°) = (F^←(B))°; for all B ∈ ℑ₂.

Lemma 4.9

[17]. Let (f, F) : (S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be a difunction.

(1) if (f, F) is cocontinuous and coclosed then (f, F) is semi-irresolute.

(2) if (f, F) is continuous and open then (f, F) is semi-co-irresolute.

Theorem 4.10

Let (f, F):(S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be a semi-continuous difunction between the ditopological texture spaces. If A ∈ ℑ₁is s-Hurewicz, then f^→(A) ∈ ℑ₂is Hurewicz.

Proof

Let (𝓥_n)_n∈ℕ be a sequence of τ₂-open covers of f^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) along with semi-continuity of (f, F), for each n, we have

A⊆F←(f→(A))⊆F←(∨Vn)=∨F←(Vn)

so that each F^←(𝓥_n is a τ₁−semi-open cover of A. As A is s-Hurewicz, therefore, for each n, there exist finite subsets Wn⊆Vn such that A⊆∨n∈N∩m>n(∪F←(Wm)). Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

f→(A)⊆f→(∨n∈N∩m>n(∪F←(Wm))⟹∨n∈N∩m>n∪(f→F←(Wm))⊆∨n∈N∩m>n∪Wm

This proves that f^→(A) is a Hurewicz space. □

Theorem 4.11

Let (f, F):(S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be a semi-irresolute difunction between the ditopological texture spaces. If A ∈ ℑ₁is s-Hurewicz, then f^→(A) ∈ ℑ₂is also s-Hurewicz.

Proof

Let (𝓥_n)_n∈ℕ be a sequence of τ₂-semi-opoen covers of f^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) and semi-irresoluteness of (f, F), for each n, we have

A⊆F←(f→(A))⊆F←(∨Vn)=∨F←(Vn)

so that each F^←(𝓥_n is a τ₁-semi-open cover of A. As A is s-Hurewicz, therefore, for each n, there exist finite sets Wn⊆Vn such that A⊆∨n∈N∩m>n(∪F←(Wm)). Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

f→(A)⊆f→(∨n∈N∩m>n(∪F←(Wm)))

⟹∨n∈N∩m>n∪(f→F←(Wm))⊆∨n∈N∩m>n∪Wm

This proves that f^→(A) is s-Hurewicz. □

Theorem 4.12

Let (S₁, ℑ₁, τ₁, κ₁) and (S₂, ℑ₂, τ₂, κ₂) be ditopological texture spaces, and let (f, F) be a cocontinuous difunction between them. If A ∈ ℑ₁ is co-Hurewicz, then F^→(A) ∈ ℑ₂is also co-Hurewicz.

Proof

Let (𝓕_n)_n∈ℕ be a sequence of κ₂-closed cocovers of F^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) with cocontinuity of (f, F), for each n, we have

∩Fn⊆F→(A)⇒f←(∩Fn)⊆f←(F→(A))⊆A⇒∩f←(Fn)⊆A

This gives each f^←(𝓕_n is a κ₁−closed cocover of A. As A is co-Hurewicz, therefore, for each n there exist finite sets Kn⊆Fn such that ∩n∈N∨m>n∩f←(Km)⊆A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

∩n∈N∨m>n(∩f←(Km))⊆A⇒F→(∩n∈N∨m>n∩f←(Km))⊆F→(A)This implies∩n∈N∨m>n∩(F→f←(Km))⊆F→(A)⇒∩n∈N∨m>n∩Km⊆F→(A)

This proves that F^→(A) is co-Hurewicz. □

Theorem 4.13

Let (S₁, ℑ₁, τ₁, κ₁) and (S₂, ℑ₂, τ₂, κ₂) be ditopological texture spaces, and let (f, F) be a semi-cocontinuous difunction between them. If A ∈ ℑ₁is co-s-Hurewicz, then F^→(A) ∈ ℑ₂is co-Hurewicz.

Proof

Let (𝓕_n)_n∈ℕ be a sequence of κ₂-closed cocovers of F^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) with semi-cocontinuity of (f, F), for each n, we have

∩Fn⊆F→(A)⇒f←(∩Fn)⊆f←(F→(A))⊆A⇒∩f←(Fn)⊆A

This gives that f^←(𝓕_n is a κ₁-semi-closed cocover of A. As A is co-s-Hurewicz, therefore, for each n, there exist finite sets Kn⊆Fn such that ∩n∈N∨m>n∩f←(Km)⊆A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

∩n∈N∨m>n(∩f←(Km))⊆A⇒F→(∩n∈N∨m>n(∩f←(Km)))⊆F→(A)This gives∩n∈N∨m>n∩(F→f←(Km))⊆F→(A)⇒∩n∈N∨m>n∩Km⊆F→(A)

This proves that F^→(A) is co-Hurewicz. □

Theorem 4.14

Let (S₁, ℑ₁, τ₁, κ₁) and (S₂, ℑ₂, τ₂, κ₂) be ditopological texture spaces, and let (f, F) be a semi-co-irresolute difunction between them. If A ∈ ℑ₁is co-s-Hurewicz, then F^→(A) ∈ ℑ₂is co-s-Hurewicz.

Proof

Let (𝓕_n : n ∈ ℕ) be a sequence of κ₂−semi-closed cocovers of F^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) with semi-co-irresoluteness of (f, F), for each n, we have

∩Fn⊆F→(A)⇒f←(∩Fn)⊆f←(F→(A))⊆A⇒∩f←(Fn)⊆A

This gives f^←(𝓕_n is a κ₁-semi-closed cocover of A. As A is co-s-Hurewicz, therefore, for each n, there exist finite sets Kn⊆Fn such that ∩n∈N∨m>n∩f←(Km)⊆A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

∩n∈N∨m>n(∩f←(Km))⊆A⇒F→(∩n∈N∨m>n∩f←(Km))⊆F→(A)This implies∩n∈N∨m>n∩(F→f←(Km))⊆F→(A)⇒∩n∈N∨m>n∩Km⊆F→(A)

This proves that F^→(A) is co-s-Hurewicz. □

Theorem 4.15

Let (f, F):(S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be a s-continuous difunction between the ditopological texture spaces. If A ∈ ℑ₁is Hurewicz, then f^→(A) ∈ ℑ₂is s-Hurewicz.

Proof

Let (𝓥_n)_n∈ℕ be a sequence of τ₂ semi-open covers of f^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) with s-continuity of (f, F), for each n, we have

A⊆F←(f→(A))⊆F←(∨Vn)=∨F←(Vn),

This gives that F^←(𝓥_n is a τ₁−open cover of A. As A is Hurewicz, therefore, for each n, there exist finite sets Wn⊆F←(Vn) such that A⊆∨n∈N∩m>n(∪F←(Wm)). Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

f→(A)⊆f→(∨n∈N∩m>n(∪F←(Wm)))⟹∨n∈N∩m>n∪(f→F←(Wm))⊆∨n∈N∩m>n∪Wm

This proves that f^→(A) is s-Hurewicz. □

Theorem 4.16

Let (S₁, ℑ₁, τ₁, κ₁) and (S₂, ℑ₂, τ₂, κ₂) be ditopological texture spaces, and let (f, F) be an s-cocontinuous difunction between them. If A ∈ ℑ₁is co-Hurewicz, then F^→(A) ∈ ℑ₂is co-s-Hurewicz.

Proof

Let (𝓕_n)_n∈ℕ be a sequence of κ₂−semi-closed cocovers of F^→(A). Then by Lemma 2.3 (2a) and Lemma 2.1 (2) and s-cocontinuity of (f, F), for each n, we have

∩Fn⊆F→(A)⇒f←(∩Fn)⊆f←(F→(A))⊆A⇒∩f←(Fn)⊆A

This gives that f^←(𝓕_n is a κ₁−closed cocover of A. As A is co-Hurewicz, therefore, for each n, there exist finite sets Kn⊆Fn such that ∩n∈N∩f←(Kn)⊆A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

∩n∈N∨m>n(∩f←(Km))⊆A⇒F→(∩n∈N∨m>n∩f←(Km))⊆F→(A)This implies∩n∈N∨m>n∩(F→f←(Km))⊆F→(A)⇒∩n∈N∨m>n∩Km⊆F→(A)

This proves F^→(A) is co-s-Hurewicz. □

5 Semi-stability in ditopological texture spaces

Definition 5.1

[14]. A ditopology (τ, κ) on (S, ℑ) is said to be:

Stable; if for every K ∈ κ with K≠S is compact.
Co-stable; if for every G ∈ τ with G≠∅ is cocompact.

A ditopological texture space (S, ℑ, τ, κ) is called dicompact if it is compact, cocompact, stable and co-stable.

Definition 5.2

[22]. A ditopology (τ, κ) on (S, ℑ) is said to be:

Semi-stable; if for all F ∈ SC (S) with F≠S is semi-compact.
Semi-co-stable; if for every G ∈ SO (S) with G≠∅ is semi-cocompact.

A ditopological texture space (S, ℑ, τ, κ) is called s-dicompact if it is semi-compact, semi-cocompact, semi-stable and semi-co-stable.

Proposition 5.3

[22]. For a ditopological texture space (S, ℑ, τ, κ)

(1) semi-stable ⇒ stable

(2) semi-co-stable ⇒ co-stable.

Proof

It follows directly from the fact that O(S)⊆ SO (S) and C(S)⊆ SC (S). □

Definition 5.4

[19]. A ditopological texture space (S, ℑ, τ, κ) is said to be:

(i) H-stable; if for all F ∈ κ with F≠S is Hurewicz.

(ii) H-co-stable; if for every G ∈ τ with G≠∅ is co-Hurewicz.

A ditopological texture space (S, ℑ, τ, κ) is called di-Hurewicz if it is Hurewicz, co-Hurewicz, H-stable and H-co-stable.

Definition 5.5

A ditopological texture space (S, ℑ,τ, κ) is said to be:

(i) sH-stable; if for all F ∈ SC (S) with F≠S is s-Hurewicz.

(ii) sH-co-stable; if for every G ∈ SO (S) with G≠∅ is co-s-Hurewicz.

The following example shows that the ditopological texture spaces are sH-stable and sH-co-stable respectively.

Example 5.6

(1) Let 𝔹 = (0, 1] with texturingβ={(0,b]:b∈B},τB=β,κB={B,∅}. Where SO(𝔹) = τ and SC(𝔹) = κ. Then (τ, κ) is sH-stable because the only semi-closed set different from 𝔹 is ∅, and this set is trivially semi-compact and hence s-Hurewicz.

(2) Dually letB=(0,1],β={(0,b]:b∈B},τB={B,∅}κB=β, where SO(𝔹) = τ and SC(𝔹) = κ. Then (τ, κ) is sH-co-stable because the only semi-open set different from ∅ is 𝔹, and this set is semi-cocompact and hence co-s-Hurewicz.

A ditopological texture space (S, ℑ, τ, κ) is called s-di-Hurewicz if it is s-Hurewicz, s-co-Hurewicz, sH-stable and sH-co-stable.

Theorem 5.7

Let (f, F): (S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be bicontinuous surjective difunction between ditopological texture spaces. If (S₁, ℑ₁, τ₁, κ₁) is sH-stable then (S₂, ℑ₂, τ₂, κ₂) is also sH-stable.

Proof

Let K ∈ κ₂ be such that K≠S₂. Since (f, F) is cocontinuous we have that f^←(K) ∈ κ₁. Now we first show that f^←(K)≠S₁. On the contrary suppose that f^←(K)=S₁, since (f, F) is surjective so we have f^←(S₂)=S₁ which by [13] f^←(S₂)⊆ f^←(K). Since (f, F) is surjective, then by Corollary 2.33(1 ii) in [13] implies S₂ ⊆ K. This is contradiction so f^←(K)≠S₁.

Since (S₁, ℑ₁, τ₁, κ₁) is sH-stable so f^←(K) is s-Hurewicz set. Continuity of (f, F) and Theorem 12 implies f^→(f^←(K)) is s-Hurewicz in (S₂, ℑ₂, τ₂, κ₂). By [13], Corollary 2.33(1) the latter set is equal to K, and hence the proof. □

Theorem 5.8

Let (f, F): (S₁, ℑ₁, τ₁, κ₁) → (S₂, ℑ₂, τ₂, κ₂) be bicontinuous surjective difunction between ditopological texture spaces. If (S₁, ℑ₁, τ₁, κ₁) is sH-costable then (S₂, ℑ₂, τ₂, κ₂) is also sH- costable.

Proof

Let G ∈ τ₂ be such that G≠∅₂. Since (f, F) is continuous we have that F^←(G) ∈ τ₁. Now we first show that F^←(G)≠∅₁. Suppose that it is not true. f^←(K)=S₁, Since (f, F) is surjective so we have F^←(∅₂)=∅₁ which by [13] F^←(∅₂)⊆ F^←(G). Since (f, F) is surjective, then by Corollary 2.33(1 ii) in [13] implies ∅₂ ⊆ G. This contradiction shows F^←(G)≠∅₁.

Since (S₁, ℑ₁, τ₁, κ₁) is sH-stable so F^←(G) is co-s-Hurewicz set. Continuity of (f, F) and Theorem 12 implies F^→(F^←(G)) is co-s-Hurewicz in (S₂, ℑ₂, τ₂, κ₂). By [13], Corollary 2.33(1) the latter set is equal to G, and hence the proof. □

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Received: 2018-01-11

Accepted: 2018-09-14

Published Online: 2018-11-08

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0104

Keywords for this article

Semi-open set; Ditopological texture space; Semi-compact space; Semi-cocompact space; s-Hurewicz space; co-s-Hurewicz space

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BY-NC-ND 4.0