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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Semi-Hurewicz-Type properties in ditopological texture spaces

Hafiz Ullah / Moiz ud Din Khan
Published Online: 2018-11-08 | DOI: https://doi.org/10.1515/math-2018-0104

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 (An)n<∞ of elements of 𝓐 there is a sequence (bn)n<∞ such that for each n, we have bnAn, and {bn}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 Sfin(𝓐, 𝓑) is defined as:

For each sequence (An)n<∞ of elements of 𝓐 there is a sequence (Bn)n<∞ such that for each n, we have Bn is finite subset of An, and n<BnB.

The property introduced in [2] by K. Menger in 1924 is equivalent to Sfin (𝓞, 𝓞) and was proved by W. Hurewicz in [3] in 1925. The property Sfin (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 Sfin (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] Ufin(𝓐, 𝓑): For each sequence {An}n∈ℕ of elements of 𝓐, which do not contain a finite sub-cover, there exist finite (possibly empty) subsets BnAn, n ∈ ℕ and {Bn}nNB. It was shown in [11] that the Hurewicz property is of the Sfin-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 Lfuzzy 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 AB 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|sA} and Qs={Pt|tS,sPt} 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 Abof A is defined by Ab={s ∈ 𝓢 ∣ AQs}. The set Abdoes not necessarily belong to ℑ.

If (S,P(s)),(L,2) are textures, then the product texture of (𝓢, P(𝓢)) and (L,2) 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 rP(S)2 satisfying:

(R1) r ⊈ Q(s,t) and PśQs implies r ⊈ Q(ś,t),

(R2) r ⊈ Q(s,t) then there is ś ∈ 𝓢 such that PsQś and r ⊈ Q(ś,t), is called relation and for RP(S)2 such that:

(CR1) P¯(s,t)R R and PsQś implies P¯(s´,t)R,

(CR2) If P¯(s,t)R then there exists ś ∈ 𝓢 such that PśQs 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, Aj1,jJ and Bj ∈ ℑ2,∀jJ. Then:

(1) r(jJBj)=jJrBjandR(jJAj)=jJRAj,

(2) r(jJAj)=jJrAjandR(jJBj)=jJRBj.

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,PsQs´tL with fQ¯(s,t) and P¯(s´,t)F.

(DF2) For t,tL and sS, fQ(s, t) and P¯(s,t)FPtQt.

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,fQ¯(s,t)AQs},

F(A)={Pt:s,P¯(s,t)FPsA},

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

f(B)={Ps:t,fQ¯(s,t)PtB},

F(B)={Qs:t,P¯(s,t)FBQt}.

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:

  1. (f, F) is a direlation.

  2. The following inclusion holds:

    1. f (F(A))⊆ AF(f(A)); ∀ A ∈ ℑ1, and

    2. f(F(B))⊆ BF(f (B)); ∀ B ∈ ℑ1

  3. 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) For t,tL,PtQtsS,Q(s,t) and P¯(s,t)F. Similarly, (f, F) is called injective if it satisfies the condition (INJ) For s, ś ∈ S, and t ∈ 𝓛 with fQ(s,t) and P¯(s´,t)FPsQs´.

We now recall the notion of ditopology on texture spaces.

Definition 2.5

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

  1. S, ∅ ∈ τ.

  2. G1, G2τ implies G1G2τ and

  3. Gατ, αI implies αGατ,

and κ ⊆ ℑ satisfies:

  1. S, ∅ ∈ κ.

  2. F1, F2κ implies F1F2κ and

  3. Fακ, αI impliesFακ,

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¯)={Fκ:AF},

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

(A)={Gτ:GA}

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):

  1. A ∈ ℑ is semi-open if and only if there exists a set GO(S) such that GAG.

  2. B ∈ ℑ is semi-closed if and only if there exists a set FC(S) such that (F)°⊆ BF.

  3. 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.

  4. Arbitrary join of semi-open sets is semi-open.

  5. 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:BSC(S), andAB}

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

(A)={B:BSO(S), andBA}.

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, τS,κS) → (T, ℑ2,τT, κT) is:

  1. continuous [17]; if F(G) ∈ τS where GτT;

  2. cocontinuous [17]; if f(K) ∈ κS where KκT;

  3. bicontinuous [17]; if it is continuous and cocontinuous.

Definition 2.8

[17] Let (Si, ℑi, τi, κi), i=1, 2 be ditoplogical texture spaces. A difunction (f, F):(S1, ℑ1) → (S2, ℑ2) is said to be:

  1. semi-continuous (semi-irresolute) if for each open (resp. semi-open) set A ∈ ℑ2, the inverse image F(A) ∈ ℑ1 is a semi-open set.

  2. semi-cocontinuous (semi-co-irresolute) if for each closed (resp. semi-closed) set B ∈ ℑ2, the inverse image f (B) ∈ ℑ1 is a semi-closed set.

  3. semi-bicontinuous if it semi-continuous and semi-cocontinuous.

  4. 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 {Gα:α ∈ ∇} is said to be semi-open cover of A if GαSO (S) for all α ∈ ∇, and A ⊆ ∨α∈∇Gα. Dually, we may speak of a semi-closed cocover of A, namely a family {𝓕α : α ∈ ∇} 𝓕αSC (S) satisfying αFαA.

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.

  1. A is said to have the semi-Hurewicz property (or s-Hurewicz property) for the ditopological texture space if for each sequence (Un)nN of semi-open covers of A there is a sequence (𝓥n)n∈ℕ such that for each n ∈ ℕ, 𝓥n is a finite subset of 𝓤n and nNm>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 by Ufin(sO,sO).

  2. 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)nN of semi-closed cocovers of A there is a sequencen)n∈ℕ such that for each n ∈ ℕ, κn is a finite subset of 𝓕n and nNm>n(κm)A. We say that (S, ℑ, τ, κ) is co-s-Hurewicz if the setis co-s-Hurewicz. This property is denoted by Ucfin(sC,sC), where sis 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 {Gα:α ∈ ∇} is semi-open cover of A, there is a finite subset0 of ∇, with A ⊆ ∨α∈∇0 Gα.

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 {𝓕α : α ∈ ∇} is a semi-closed cocover of A, there is a finite subset0 of ∇, with α0FαA A. 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 (An : n ∈ ℕ) of s-compact (s-cocompact) subsets of S such that nNAn=S (resp. nNAn=).

Theorem 3.5

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

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

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

Proof

(1). Let (Un)nN be a sequence of semi-open covers of S. Since S is σ-s-compact therefore it can be represented in the form S=iNAi, where each 𝓐i is s-compact AiAi+1 for all i ∈ ℕ. For each i ∈ ℕ, choose a finite set ViUi such that AiVi=Vi. Then the sequence (𝓥n)n∈ℕ shows that S is s-Hurewicz.

(2). Let (𝓕n)n∈ℕ be a sequence of semi-closed cocovers of ∅. We have =iNAi, where each Ai is s-cocompact and AiAi+1, i ∈ ℕ. For each i ∈ ℕ, choose a finite subset κiFi such that ∩κiAi. Then iNm>iκmiNAi=, 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). Then SUfin(sO,sO) if and only if Ucfin(sC,sC).

Proof

Let SUfin(sO,sO) and let (Fn)nN be a sequence of semi-closed cocovers of ∅. Then (σ (𝓕n) = σ (F) : F ∈ 𝓕n) and (σ(Fn)nN) is a sequence of semi-open covers of S. Since SUfin(sO,sO), there is a sequence (𝓥n)n∈ℕ of finite sets such that for each n, Vnσ(Fn) and nNm>n(Vm)=S. We have (σ (𝓥n) : n ∈ ℕ) is a sequence of finite sets, and also

=σ(S)=σ(nNm>n(Vm))=nNm>n(σ(Vm))

Hence Ucfin(sC,sC).

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

S=σ()=σ(nNm>n(κm))=nNm>n(σ(κm))

Hence SUfin(sO,sO). □

Example 3.7

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

Let (R,,τR,κR) be the real line with the texture ={(,r]:rR}{(,r):rR}{R,}, topology τR={(,r):rR}{R,} and cotopology κR={(,r]:rR}{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 cover U={(,n):nN} does not contain a finite subcover, nor the texture space is s-cocompact because its (closed and hence) semi-closed cocover {(,n]:nN} does not contain a finite cocover. But (R,,τR,κR) is s-Hurewicz and co-s-Hurewicz. Let us prove that this space is s-Hurewicz. Let (Un:nN) be a sequence of semi-open covers of ℝ. We note that semi-open sets can be of the form (−∞, r) and (−∞, r]. Write R={(,n]:nN}. For each n, 𝓤n is a semi-open cover of ℝ, hence there is some rn ∈ ℝ such that (,rn](,n]Un. Then the collection {(,rn]:nN} shows that (R,,τR,κR) is s-Hurewicz.

Evidently we have the following diagram:

scompactsHurewiczcompactHurewicz

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 Kκ with KS, K is s-Hurewicz if and only if G is co-s-Hurewicz for some Gτ and G≠∅.

Proof

(⇒) Let Gτ with G≠∅. Let (Fn)nN be a sequence of semi-closed cocovers of G. Set K=σ(G) and we obtain Kκ with KS. K is s-Hurewicz, for (σ(Fn))nN the sequence of semi-open covers of K there is a sequence (𝓥n)n∈ℕ of finite sets such that for each n, Vnσ(Fn) and nNm>n(Vm) is semi-open cover of K. Thus (σ(Vn))nN is a sequence of finite sets such that for each n, σ(Vn)Fn and σ(nNm>n(Vm))=nNm>n(σ(Vm))G which gives G is co-s-Hurewicz.

(⇐) Kκ with KS. Let (Un)nN be a sequence of semi-open covers of K. Since K=σ(G) but G is co-s-Hurewicz, so for (σ(Un))nN the sequence of semi-closed cocovers of G there is a sequence (κn)nN such that for each n ∈ ℕ, κn is a finite subset of (σ(Un) and nNm>n(κm)G. Thus σ(κn) is a sequence such that for each n ∈ ℕ, κn is a finite subset of 𝓤n and σ(nNm>n(κm))=nNm>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 ℑA={AK:K ∈ ℑ} of A is called the induced texture on A [21], and (A, ℑA) is called a principal subtexture of (S, ℑ).

Definition 4.1

[20]. Let (S, ℑ, τ, κ) be a ditopological texture space and (A, ℑA) be a principle subtexture of (S, ℑ) for A ∈ ℑ. Then (A, ℑA, τA, κA) is a subspace of a ditopological texture space (S, ℑ, τ, κ), where τA={AG:Gτ } and κA={AK:Kκ }.

Note that if Aτ, then A is said to be open subspace and if Aκ, then A is closed subspace.

Lemma 4.2

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

  1. 𝓑 is τAopen if and only if 𝓑 = AG for some τ-open set G.

  2. 𝓑 is κAclosed if and only if 𝓑 = AF 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 ASR(S), then (A, ℑA, τA, κA) is also s-Hurewicz.

Proof

Let (Un)nN 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 WnVn, n ∈ ℕ, such that S=nNm>nWm. Set Hn=Wn{σ(A)}, n ∈ ℕ. Then for each n ∈ ℕ, 𝓗n is a finite subset of 𝓤n and AnNm>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 ASR (S), then (A, ℑA, τA, κA) is also co-s-Hurewicz.

Proof

Let (𝓕n)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 WnKn, for each n ∈ ℕ, such that nNm>nWmS. Set Gn=Wn{σ(A)}, n ∈ ℕ. Then for each n ∈ ℕ, 𝓖n is a finite subset of 𝓕n and nNm>n(Gm)A, i.e., A is co-s-Hurewicz. □

Example 4.5

If a complemented ditopological texture space S is s-Hurewicz, and ASR (S) then the subtexture SA is also s-Hurewicz.

Letbe the real line textured by ={(,r]:rR}{(,r):rR}{R,}, with complemented ditopology (τR,κR) such that τR={(,r):rR}{R,} and κR={(,r]:rR}{R,}. Where SR(R)={{(,r):rR}{(,r]:rR}{R,}}. This ditopological texture space is s-Hurewicz and co-s-Hurewicz by Example 3.7. Let Y={(,r]:rR}, then (𝕐, 𝒴) is an induced subtexture of (ℝ, ℜ) with texturing Y={YA|A}={(,s):sY}{(,s]:sY}{Y,}, topology τY={GY:Gτ}={(,s):sY}{Y,} and cotopology κY={KY:Kκ}={(,s]:sY}{Y,}. Thus (Y,Y,τY,κY) is a ditopological texture space.

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

Lemma 4.6

[17] Let (Sj, ℑj, τj, κj), 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 (Si, ℑi, τi, κi), 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 ASO (S2).

(ii) s-cocontinuous; if f (B) ∈ C(S1) for each BSC (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 ∈ ℑ1 is s-Hurewicz, then f(A) ∈ ℑ2 is Hurewicz.

Proof

Let (𝓥n)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

AF(f(A))F(Vn)=F(Vn)

so that each F(𝓥n is a τ1−semi-open cover of A. As A is s-Hurewicz, therefore, for each n, there exist finite subsets WnVn such that AnNm>n(F(Wm)). Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

f(A)f(nNm>n(F(Wm))nNm>n(fF(Wm))nNm>nWm

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 ∈ ℑ1 is s-Hurewicz, then f(A) ∈ ℑ2 is also s-Hurewicz.

Proof

Let (𝓥n)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

AF(f(A))F(Vn)=F(Vn)

so that each F(𝓥n is a τ1-semi-open cover of A. As A is s-Hurewicz, therefore, for each n, there exist finite sets WnVn such that AnNm>n(F(Wm)). Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

f(A)f(nNm>n(F(Wm)))

nNm>n(fF(Wm))nNm>nWm

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) ∈ ℑ2 is also co-Hurewicz.

Proof

Let (𝓕n)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

FnF(A)f(Fn)f(F(A))Af(Fn)A

This gives each f(𝓕n is a κ1−closed cocover of A. As A is co-Hurewicz, therefore, for each n there exist finite sets KnFn such that nNm>nf(Km)A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

nNm>n(f(Km))AF(nNm>nf(Km))F(A)This impliesnNm>n(Ff(Km))F(A)nNm>nKmF(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 ∈ ℑ1 is co-s-Hurewicz, then F(A) ∈ ℑ2 is co-Hurewicz.

Proof

Let (𝓕n)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

FnF(A)f(Fn)f(F(A))Af(Fn)A

This gives that f(𝓕n is a κ1-semi-closed cocover of A. As A is co-s-Hurewicz, therefore, for each n, there exist finite sets KnFn such that nNm>nf(Km)A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

nNm>n(f(Km))AF(nNm>n(f(Km)))F(A)This givesnNm>n(Ff(Km))F(A)nNm>nKmF(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 ∈ ℑ1 is co-s-Hurewicz, then F(A) ∈ ℑ2 is co-s-Hurewicz.

Proof

Let (𝓕n : 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

FnF(A)f(Fn)f(F(A))Af(Fn)A

This gives f(𝓕n is a κ1-semi-closed cocover of A. As A is co-s-Hurewicz, therefore, for each n, there exist finite sets KnFn such that nNm>nf(Km)A. Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

nNm>n(f(Km))AF(nNm>nf(Km))F(A)This impliesnNm>n(Ff(Km))F(A)nNm>nKmF(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 ∈ ℑ1 is Hurewicz, then f(A) ∈ ℑ2 is s-Hurewicz.

Proof

Let (𝓥n)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

AF(f(A))F(Vn)=F(Vn),

This gives that F(𝓥n is a τ1−open cover of A. As A is Hurewicz, therefore, for each n, there exist finite sets WnF(Vn) such that AnNm>n(F(Wm)). Again by Lemma 2.3 (2b) and Lemma 2.1 (2) we have

f(A)f(nNm>n(F(Wm)))nNm>n(fF(Wm))nNm>nWm

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 ∈ ℑ1 is co-Hurewicz, then F(A) ∈ ℑ2 is co-s-Hurewicz.

Proof

Let (𝓕n)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

FnF(A)f(Fn)f(F(A))Af(Fn)A

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

nNm>n(f(Km))AF(nNm>nf(Km))F(A)This impliesnNm>n(Ff(Km))F(A)nNm>nKmF(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:

  1. Stable; if for every Kκ with KS is compact.

  2. Co-stable; if for every Gτ 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:

  1. Semi-stable; if for all FSC (S) with FS is semi-compact.

  2. Semi-co-stable; if for every GSO (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 Fκ with FS is Hurewicz.

(ii) H-co-stable; if for every Gτ 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 FSC (S) with FS is s-Hurewicz.

(ii) sH-co-stable; if for every GSO (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 texturing β={(0,b]:bB},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 let B=(0,1],β={(0,b]:bB},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 Kκ2 be such that KS2. 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 S2K. 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 Gτ2 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 ∅2G. 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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About the article

Received: 2018-01-11

Accepted: 2018-09-14

Published Online: 2018-11-08


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1243–1254, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0104.

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© 2018 Ullah and Din Khan, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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