On several types of continuity and irresoluteness in

On several types of continuity and irresoluteness in 𝑳𝑳-topological Spaces

Samer H. Al Ghour

Dept. of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

*Corresponding author: algore@just.edu.jo

We use the concepts of 𝜔𝜔-open 𝐿𝐿-sets, 𝒩𝒩-open 𝐿𝐿-sets and 𝒟𝒟-open 𝐿𝐿-sets to define several new types of continuity and irresoluteness in 𝐿𝐿-topological spaces. Several results have been proved. In particular, decomposition theorems of the new irresoluteness concepts are introduced.

Keywords: 𝜔𝜔-open 𝐿𝐿-sets; 𝒩𝒩-open 𝐿𝐿-sets; 𝒟𝒟-open 𝐿𝐿-sets; fuzzy continuity; fuzzy irresoluteness.

Mathematics Subject Classification (2010): 54A05, 54A40.

1. Introduction and preliminaries

Fuzzy set theory still plays a vital role in research in almost all branches of mathematics and computer science (Rameez et al., 2017; Ahmad et al., 2016; Kia et al., 2016;

Pant et al., 2015). Introducing new types of continuity or irresoluteness is still a hot area of research in 𝐿𝐿-topological spaces (See Chettri & Chettri, 2017; Vadivel &

Swaminathan, 2017; Malathi & Uma, 2017; Swaminathan

& Balasubramaniyan, 2016; Deb Ray & Chettri, 2016;

Chettri et al., 2014; Kharal & Ahmad, 2013; Shukla, 2012;

Tripathy & Ray, 2014; Tripathy & Debnath, 2013;

Tripathy & Ray, 2013; Tripathy & Ray, 2012).

Throughout this paper, we follow the notions and terminologies as they appeared in (Al Ghour, 2017).

As a weaker form of openness, the class of 𝜔𝜔-open sets for a given topological space was introduced first by Hdeib in (Hdeib, 1982). This class of sets was used to introduce new topological notions in a number of previous works (See Al-Omari & Noorani, 2007; Al-Omari et al., 2009; Hdeib, 1989; Al Ghour, 2006; Al Ghour & Zareer, 2016; Darwesh, 2013; Al-Zoubi, 2003). In particular, 𝜔𝜔- continuity and 𝜔𝜔-irresoluteness were introduced in (Hdeib, 1989) and (Al-Zoubi, 2003), respectively. For the purpose of characterizing compactness, in Al-Omari & Noorani (2009) introduced 𝒩𝒩-open sets as a class of sets which lie between open sets and 𝜔𝜔-open sets. Via 𝒩𝒩-open sets, 𝒩𝒩- continuity and 𝒩𝒩-irresoluteness were introduced in (Al- Omari & Noorani, 2009) and (Al-Mshaqbeh, 2010), respectively.

For the purpose of introducing several types of compactness and Lindelöfness, and opening the door for future research in 𝐿𝐿-topological spaces, Al Ghour in Al Ghour (2017) fuzzified 𝜔𝜔-openness and 𝒩𝒩-openness by which he also introduced 𝒟𝒟-open 𝐿𝐿-sets as a weaker form of 𝒩𝒩-open 𝐿𝐿-sets as follows: Let (𝐿𝐿^!, 𝛿𝛿) be an 𝐿𝐿-ts and let 𝑊𝑊, 𝐴𝐴 ∈ 𝐿𝐿^!. 𝑊𝑊 is called an 𝜔𝜔-open (resp. 𝒩𝒩-open) 𝐿𝐿-set in (𝐿𝐿^!, 𝛿𝛿), if for every 𝑥𝑥!∈ 𝑀𝑀(𝐿𝐿^!) with 𝑥𝑥!≪ 𝑊𝑊, there exist 𝑈𝑈 ∈ 𝑄𝑄(𝑥𝑥!) and 𝐺𝐺 ∈ 𝑋𝑋!"! (resp. 𝐺𝐺 ∈ 𝑋𝑋!"#) such that 𝑥𝑥 ∈ 𝐺𝐺 and 𝑈𝑈 ∧ 𝒳𝒳!≤ 𝑊𝑊. 𝑊𝑊 is called a 𝒟𝒟-open 𝐿𝐿-set in 𝐿𝐿^!, 𝛿𝛿 if

for every 𝑥𝑥!∈ 𝑀𝑀(𝐿𝐿^!) with 𝑥𝑥!≪ 𝑊𝑊, there exists 𝑈𝑈 ∈ 𝑄𝑄 𝑥𝑥! such that 𝑈𝑈 ∧ 𝒳𝒳{!} ≤ 𝑊𝑊. Denote the class of 𝜔𝜔-open (resp. an 𝒩𝒩-open, a 𝒟𝒟-open) 𝐿𝐿-sets in (𝐿𝐿^!, 𝛿𝛿) by δ_! (resp. δ_𝒩𝒩, δ_𝒟𝒟). Author in (Al Ghour, 2017) proved that each of δ!, δ𝒩𝒩 and δ𝒟𝒟 forms an 𝐿𝐿-topology on 𝑋𝑋 and 𝛿𝛿 ⊆ δ𝒩𝒩⊆ δ! ⊆ δ𝒟𝒟 with each of the three inclusions being unreplaceable by equality in general. Also, via 𝜔𝜔- open 𝐿𝐿-sets, 𝒩𝒩-open 𝐿𝐿-sets and 𝒟𝒟-open 𝐿𝐿-sets, he introduced and investigated several new types of compactness and Lindelöfness.

This paper is devoted to using the classes of 𝜔𝜔- open 𝐿𝐿-sets, 𝒩𝒩-open 𝐿𝐿-sets and 𝒟𝒟-open 𝐿𝐿-sets to introduce and investigate 𝒩𝒩-continuity, 𝜔𝜔-continuity, 𝒟𝒟-continuity, 𝒩𝒩-irresoluteness, 𝜔𝜔-irresoluteness, and 𝒟𝒟-irresoluteness as new six 𝐿𝐿-topological properties.

In Section 2, we characterize all of the new six concepts in terms of continuity. Also, we introduce a decomposition theorem for each of 𝒩𝒩-irresoluteness and 𝜔𝜔-irresoluteness. Moreover, we give several relationships between the new six concepts and between them and the continuity concept, in particular, we show that 𝒟𝒟-continuity and 𝒟𝒟- irresoluteness concepts are equivalent, and give sufficient conditions related to when two or more of the new concepts together with continuity are coincident to each other. Although we give some sufficient conditions for the validity of the implication

"𝒩𝒩-irresoluteness ⇒ 𝜔𝜔-irresoluteness", we propose an open question about its validity in general. Moreover, we introduce composition and graph theorems related to the new concepts. In Section 3, we give examples to distinguish between our concepts as well as to highlight the significance of our results.

The following definition and proposition will be used in the sequel:

Definition 1.1. (Al Ghour, 2017) Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿- ts', 𝑓𝑓!→: 𝐿𝐿^!→ 𝐿𝐿^! be an 𝐿𝐿-mapping.

Abstract

We say 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is a continuous mapping or call 𝑓𝑓!→ continuous for short, if its 𝐿𝐿-reverse mapping 𝑓𝑓!←: 𝐿𝐿^!→ 𝐿𝐿^! maps every open 𝐿𝐿-subset in (𝐿𝐿^!, 𝜇𝜇) as an open 𝐿𝐿-subset in (𝐿𝐿^!, 𝛿𝛿), i.e. for every 𝑉𝑉 ∈ 𝜇𝜇, 𝑓𝑓_!^←(𝑉𝑉) ∈ 𝛿𝛿.

Proposition 1.2. (Liu & Luo, 1997) Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿-ts', 𝑓𝑓!→: 𝐿𝐿^!→ 𝐿𝐿^! be an 𝐿𝐿-mapping. Then the following conditions are equivalent:

(i) 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)is continuous.

(ii) For any subbase 𝜇𝜇₀ of 𝜇𝜇, 𝑓𝑓_!^←(𝑉𝑉) ∈ 𝛿𝛿 (∀ 𝑉𝑉 ∈ 𝜇𝜇₀).

2. Continuity and irresoluteness

Based on Definition 1.1 and by means of 𝜔𝜔-open, 𝒩𝒩-open, 𝒟𝒟-open L-sets in L-ts', we define six types of mappings:

Definition 2.1. Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿-ts', 𝑓𝑓!→: 𝐿𝐿^!→ 𝐿𝐿^! be an 𝐿𝐿-mapping.

(i) We say 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)is an 𝜔𝜔-continuous (resp.

𝒩𝒩-continuous, 𝒟𝒟-continuous) mapping or call 𝑓𝑓!→ ω- continuous (resp. 𝒩𝒩-continuous, 𝒟𝒟-continuous) for short, if its 𝐿𝐿-reverse mapping 𝑓𝑓_!^←: 𝐿𝐿^!→ 𝐿𝐿^! maps every open 𝐿𝐿- subset in (𝐿𝐿^!, 𝜇𝜇) as an 𝜔𝜔-open (resp. 𝒩𝒩-open, 𝒟𝒟-open) 𝐿𝐿- subset in (𝐿𝐿^!, 𝛿𝛿).

(ii) We say 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is an 𝜔𝜔-irresolute (resp. 𝒩𝒩-irresolute, 𝒟𝒟-irresolute) mapping or call 𝑓𝑓_!^→ ω- irresolute (resp. 𝒩𝒩-irresolute, 𝒟𝒟-irresolute) for short, if its 𝐿𝐿-reverse mapping 𝑓𝑓!←: 𝐿𝐿^!→ 𝐿𝐿^! maps every 𝜔𝜔-open (resp. 𝒩𝒩-open, 𝒟𝒟-open) 𝐿𝐿-subset in (𝐿𝐿^!, 𝜇𝜇) as an 𝜔𝜔-open (resp. 𝒩𝒩-open, 𝒟𝒟-open) 𝐿𝐿-subset in 𝐿𝐿^!, 𝛿𝛿 .

We start by showing that 𝒟𝒟-continuity and ω-

irresoluteness are coincident with each other:

Theorem 2.2. Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿-ts', 𝑓𝑓!→: 𝐿𝐿^!→ 𝐿𝐿^! be an 𝐿𝐿-mapping. Then the following conditions are equivalent:

(i) 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)is 𝒟𝒟-continuous.

(ii) 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿𝒟𝒟) → (𝐿𝐿^!, 𝜇𝜇)is continuous.

(iii) 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)is 𝒟𝒟-irresolute.

Proof. The proofs of (i) ⇔ (ii) and (iii) ⇒ (ii) are obvious, so we only show (ii) ⇒ (iii).

(ii) ⇒ (iii) : We apply Proposition 1.2. By Theorem 3.2 (vi) of (Al Ghour, 2017), 𝜇𝜇 ∪ 𝜏𝜏^!"#!! is a subbase of µ_𝒟𝒟. By (ii), for every 𝑉𝑉 ∈ 𝜇𝜇, , 𝑓𝑓!←(𝑉𝑉) ∈ δ𝒟𝒟, and for every 𝒳𝒳!∈ 𝜏𝜏^!"#!!, where 𝐵𝐵 ⊂ 𝑋𝑋, 𝑓𝑓!← 𝒳𝒳! = 𝒳𝒳!⁻¹(!)∈ 𝜏𝜏^!"#!!⊂ δ_𝒟𝒟.

Theorem 2.2 also characterizes 𝒟𝒟-continuity and 𝜔𝜔- irresoluteness in terms of continuity. As for continuity, Propositions 2.3, 2.4, and 2.5 will characterize 𝜔𝜔- continuity, 𝒩𝒩-continuity, 𝜔𝜔-irresoluteness, 𝒩𝒩- irresoluteness. In addition, Proposition 2.4 (resp.

Proposition 2.5) will give a nice decomposition theorem for 𝜔𝜔-irresoluteness (resp. 𝒩𝒩-irresoluteness).

Proposition 2.3. Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿-ts', 𝑓𝑓_!^→: 𝐿𝐿^!→ 𝐿𝐿^! be an 𝐿𝐿-mapping. Then

(i) 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)is 𝜔𝜔-continuous if and only if 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿!) → (𝐿𝐿^!, 𝜇𝜇)is continuous.

(ii) 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)is 𝒩𝒩-continuous if and only if 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿_𝒩𝒩) → (𝐿𝐿^!, 𝜇𝜇)is continuous.

Proposition 2.4. Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿-ts', 𝑓𝑓!→: 𝐿𝐿^!→ 𝐿𝐿^! be an 𝐿𝐿-mapping. Then the following conditions are equivalent:

(i) 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝜔𝜔-irresolute.

(ii) 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿_!) → (𝐿𝐿^!, µ_!)is continuous.

(iii) 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝜔𝜔-continuous, and 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿!) → (𝐿𝐿^!, 𝜏𝜏!"!")is continuous.

Proof. (i) ⇔ (ii) is clear.

(ii) ⇔ (iii) By Theorem 3.2 (vi) of (Al Ghour, 2017), 𝜇𝜇 ∪ 𝜏𝜏!"#" is a subbase of µ!. Applying Proposition 1.2, we get the result.

In view of the above theorem, we state the following result without proof.

Proposition 2.5. Let (𝐿𝐿^!, 𝛿𝛿), (𝐿𝐿^!, 𝜇𝜇) be 𝐿𝐿-ts', 𝑓𝑓!→: 𝐿𝐿^!→ 𝐿𝐿^! be an L-mapping. Then the following conditions are equivalent:

(i) 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝒩𝒩-irresolute.

(ii) 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿𝒩𝒩) → (𝐿𝐿^!, µ𝒩𝒩)is continuous. (iii) 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝒩𝒩-continuous, and 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿_𝒩𝒩) → (𝐿𝐿^!, 𝜏𝜏^!"#$)is continuous.

Regarding the relationships between our new six concepts and the old continuity concept, the following diagram summarizes the implications that follow directly from the definitions and Theorem 2.2.

𝒩𝒩-irresolute 𝜔𝜔-irresolute → 𝒟𝒟-irresolute

↓ ↓ ↕

𝒩𝒩-continuous → 𝜔𝜔-continuous → 𝒟𝒟-continuous

↑

continuous

Except the implication '𝒩𝒩-irresolute ⇒ 𝜔𝜔-irresolute', in section 3, we will give counter-examples which distinguish between the concepts in the above diagram.

Recall that a function 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 is called finite to one (resp. countable to one) if for each 𝑦𝑦 ∈ 𝑌𝑌, 𝑓𝑓⁻¹({𝑦𝑦}) is finite (resp. countable).

In each part of the next result, we give sufficient conditions for which two or more of the concepts in the above diagram are coincident to each other:

Theorem 2.6. Let 𝑓𝑓_!^→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)be an 𝐿𝐿-mapping.

Then

(i) if (𝐿𝐿^!, 𝛿𝛿) is 𝑇𝑇!, then the properties 𝑓𝑓!→is continuous and 𝑓𝑓_!^→ is 𝒩𝒩-continuous are equivalent.

(ii) if (𝐿𝐿^!, 𝛿𝛿) is 𝑇𝑇! and P-𝐿𝐿-ts, then the properties 𝑓𝑓!→ is continuous, 𝑓𝑓!→ is 𝒩𝒩-continuous and, 𝑓𝑓!→

is 𝜔𝜔-continuous are equivalent.

(iii) if (𝐿𝐿^!, 𝛿𝛿) is CS, then the properties 𝑓𝑓!→ is 𝜔𝜔- continuous and 𝑓𝑓!→ is 𝒟𝒟-continuous are equivalent.

(iv) if (𝐿𝐿^!, 𝛿𝛿) is FS, then the properties 𝑓𝑓_!^→ is 𝒟𝒟- continuous, 𝑓𝑓_!^→ is 𝜔𝜔-continuous and 𝑓𝑓_!^→ is 𝒟𝒟- continuous are equivalent.

(v) if 𝑓𝑓!→ is 𝒩𝒩-irresolute such that (𝐿𝐿^!, 𝛿𝛿) is 𝑇𝑇!, then 𝑓𝑓_!^→ is continuous.

(vi) if 𝑓𝑓_!^→ is 𝜔𝜔-irresolute such that (𝐿𝐿^!, 𝛿𝛿) is 𝑇𝑇_! and P-𝐿𝐿-ts, then 𝑓𝑓!→ is continuous.

(vii) if (𝐿𝐿^!, 𝛿𝛿) and (𝐿𝐿^!, 𝜇𝜇) are 𝑇𝑇!, then the properties 𝑓𝑓_!^→ is continuous and 𝑓𝑓_!^→ is 𝒩𝒩-irresolute are equivalent.

(viii) if (𝐿𝐿^!, 𝛿𝛿) and (𝐿𝐿^!, 𝜇𝜇) are 𝑇𝑇! and P-𝐿𝐿-ts, then the properties 𝑓𝑓!→ is continuous, 𝑓𝑓!→ is 𝒩𝒩- irresolute, and 𝜔𝜔-irresolute are equivalent.

(ix) if (𝐿𝐿^!, 𝜇𝜇) is 𝑇𝑇_!, then the properties 𝑓𝑓_!^→ is 𝒩𝒩- continuous and 𝑓𝑓!→ is 𝜔𝜔-irresolute are equivalent.

(x) if (𝐿𝐿^!, 𝜇𝜇) is 𝑇𝑇! and P-𝐿𝐿-ts, then the properties 𝑓𝑓_!^→ is 𝜔𝜔-continuous and 𝑓𝑓_!^→ is 𝜔𝜔-irresolute are equivalent.

(xi) if 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 is finite to one, and 𝑓𝑓!→ is 𝒩𝒩- continuous, then 𝑓𝑓_!^→ is 𝒩𝒩-irresolute.

(xii) if 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 is countable to one, and 𝑓𝑓!→ is 𝜔𝜔- continuous, then 𝑓𝑓!→ is 𝜔𝜔-irresolute.

Proof. The proofs of (i), (v), (vii), and (ix) follow directly by Theorem 3.4 of (Al Ghour, 2017).

The proofs of (ii), (vi), (viii), and (x) follow directly by Theorem 3.5 of (Al Ghour, 2017).

(iii) Follows by Theorem 3.7 (i) of (Al Ghour, 2017).

(iv) Follows by Theorem 3.7 (ii) of (Al Ghour, 2017).

(xi) By Theorem 4.4 of (Al Ghour, 2017), we only need to show 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿𝒩𝒩) → (𝐿𝐿^!, 𝜏𝜏^!"#$) is continuous. For every 𝐺𝐺 ∈ 𝑌𝑌_!"# 𝑓𝑓_!^← 𝒳𝒳_! = 𝒳𝒳_!^!!_! and 𝑋𝑋 − 𝑓𝑓⁻¹(𝐺𝐺) = 𝑓𝑓⁻¹(𝑌𝑌 − 𝐺𝐺) is finite, and thus 𝑓𝑓!← 𝒳𝒳! ∈ 𝜏𝜏^!"#$⊂ 𝛿𝛿𝒩𝒩. This ends the proof.

(xii) Is similar to that used in (xi).

Question 2.7. Is it true that every 𝒩𝒩-irresolute L-mapping is 𝜔𝜔-irresolute?

The following result gives three sufficient conditions for an 𝒩𝒩-irresolute 𝐿𝐿-mapping to be 𝜔𝜔-irresolute.

Theorem 2.8. Let 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇)be an 𝒩𝒩- irresolute 𝐿𝐿-mapping. Then,

(i) if 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 is countable to one, then 𝑓𝑓!→ is 𝜔𝜔-irresolute.

(ii) if (𝐿𝐿^!, 𝛿𝛿_!) is P-𝐿𝐿-ts, then 𝑓𝑓_!^→ is 𝜔𝜔-irresolute.

(iii) if (𝐿𝐿^!, 𝜇𝜇) is FS, then 𝑓𝑓!→ is 𝜔𝜔-irresolute.

Proof. (i) Since 𝑓𝑓!→ is 𝒩𝒩-irresolute, then by the above diagram 𝑓𝑓!→ is 𝜔𝜔-continuous. Therefore, by Theorem 2.7 (xii), it follows that 𝑓𝑓_!^→ is 𝜔𝜔-irresolute.

(ii) By Proposition 2.4, we only need to show that 𝑓𝑓!→∶ 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿!) → (𝐿𝐿^!, 𝜏𝜏!"#")is continuous. Let 𝐺𝐺 ∈ 𝑌𝑌!"!. It is not difficult to see that 𝒳𝒳!= ⋀{𝒳𝒳!!{!}: 𝑦𝑦 ∈ 𝑌𝑌 − 𝐺𝐺}. Since {𝒳𝒳_!!{!}: 𝑦𝑦 ∈ 𝑌𝑌 − 𝐺𝐺}⊂ µ_𝒩𝒩 and 𝑓𝑓_!^→ is 𝒩𝒩- irresolute, we have {𝑓𝑓!←(𝒳𝒳!!{!}): 𝑦𝑦 ∈ 𝑌𝑌 − 𝐺𝐺} ⊂ 𝛿𝛿𝒩𝒩. Since (𝐿𝐿^!, 𝛿𝛿!) is P-𝐿𝐿-ts and 𝑓𝑓!←(𝒳𝒳!) = ⋀{𝑓𝑓!←(𝒳𝒳!!{!}): 𝑦𝑦 ∈ 𝑌𝑌 − 𝐺𝐺}, then we have 𝑓𝑓_!^← 𝒳𝒳_! ∈ 𝛿𝛿_!.

(iii) Let 𝑉𝑉 ∈ µ!. Then by Theorem 3.7 (ii) of (Al Ghour, 2017), 𝑉𝑉 ∈ µ𝒩𝒩. Thus 𝑓𝑓!← 𝑉𝑉 ∈ 𝛿𝛿𝒩𝒩⊂ 𝛿𝛿!.

The composition of two 𝒩𝒩-continuous 𝐿𝐿-mappings do not need to be even 𝜔𝜔-continuous.

Example 2.9. Take 𝑋𝑋 = ℝ, 𝑌𝑌 = {0,1,2}, 𝑍𝑍 = {𝑎𝑎, 𝑏𝑏}, 𝐿𝐿 any F-lattice, 𝛿𝛿 = {0!, 1!}, 𝜇𝜇 = {0!1!, 𝒳𝒳!, 𝒳𝒳!,!}, and 𝛾𝛾 = {0!1!, 𝒳𝒳!}. Define the function 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 by

𝑓𝑓(𝑥𝑥) = 2 if 𝑥𝑥 = 0 and 𝑓𝑓(𝑥𝑥) = 1 otherwise, and define the function 𝑔𝑔: 𝑌𝑌 → 𝑍𝑍 by 𝑔𝑔(0) = 𝑔𝑔(2) = 𝑎𝑎 and 𝑔𝑔(1) = 𝑏𝑏.

Then 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) and 𝑔𝑔!→: (𝐿𝐿^!, 𝜇𝜇) → (𝐿𝐿^!, 𝛾𝛾) are 𝒩𝒩-continuous but (g ∘ f)!→ is not 𝒩𝒩-continuous since (g ∘ f)!← 𝒳𝒳! = 𝒳𝒳_!∘!^!! _! = 𝒳𝒳! ∉ 𝛿𝛿𝒩𝒩.

In what follows, we collect composition results related to our new continuity concepts. The results are easy to follow, and their proofs are left to the reader.

Proposition 2.10. Let 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) and 𝑔𝑔!→: (𝐿𝐿^!, 𝜇𝜇) → (𝐿𝐿^!, 𝛾𝛾) two 𝐿𝐿-mappings.

(i) If 𝑓𝑓_!^→ is 𝜔𝜔-continuous and 𝑔𝑔_!^→ is continuous, then (g ∘ f)!→ is 𝜔𝜔-continuous.

(ii) If 𝑓𝑓!→ is 𝒩𝒩-continuous and 𝑔𝑔!→ is continuous, then (g ∘ f)_!^→ is 𝒩𝒩-continuous.

(iii) If 𝑓𝑓!→ is 𝜔𝜔-irresolute and 𝑔𝑔!→ is 𝜔𝜔- continuous, then (g ∘ f)!→ is 𝜔𝜔-continuous.

(iv) If 𝑓𝑓!→ is 𝒩𝒩-irresolute and 𝑔𝑔!→ is 𝒩𝒩- continuous, then (g ∘ f)!→ is 𝒩𝒩-continuousIf.

(v) 𝑓𝑓_!^→ and 𝑔𝑔_!^→ are 𝜔𝜔-irresolute, then (g ∘ f)_!^→ is so.

(vi) If 𝑓𝑓!→ and 𝑔𝑔!→ are 𝒩𝒩-irresolute, then (g ∘ f)!→ is so.

(vii) If 𝑓𝑓_!^→ and 𝑔𝑔_!^→ are 𝒟𝒟-irresolute, then (g ∘ f)!→ is so.

The following lemma will be used in the proof of the next result. It generalizes a result in (Azad, 1981), where the result appears for the case 𝐿𝐿 = 𝐼𝐼. Its proof can be established using standard techniques, so it is omitted here for brevity.

Lemma 2.11. Let g: X → X×Y be the graph of the function 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌, and let 𝐿𝐿 be an F-lattice. If 𝐴𝐴 ∈ 𝐿𝐿^! and 𝐵𝐵 ∈ 𝐿𝐿^!, then g!← 𝐴𝐴×𝐵𝐵 = 𝐴𝐴 ∧ f!← 𝐵𝐵 .

In the following last result 𝛿𝛿×𝜇𝜇 will denote the product 𝐿𝐿-topology of 𝛿𝛿 and 𝜇𝜇 (for the definition of the product 𝐿𝐿- topology, one may refer to (Liu & Luo, 1997).

Theorem 2.12. Let 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) be an 𝐿𝐿- mapping and let 𝑔𝑔!→: (𝑋𝑋, 𝛿𝛿) → (𝑋𝑋×𝑌𝑌, 𝛿𝛿×𝜇𝜇) be its graph 𝐿𝐿-mapping.

(i) If 𝑔𝑔!→ is 𝜔𝜔-continuous, then so is 𝑓𝑓!→. (ii) If 𝑔𝑔!→ is 𝒩𝒩-continuous, then so is 𝑓𝑓!→. (iii) If 𝑔𝑔!→ is 𝒟𝒟-continuous, then so is 𝑓𝑓!→. (iv) If 𝑔𝑔!→ is 𝜔𝜔-irresolute, then so is 𝑓𝑓!→. (v) If 𝑔𝑔_!^→ is 𝒩𝒩-irresolute, then so is 𝑓𝑓_!^→.

Proof. The proofs of all the parts are similar, so we prove only (i).

(i) Let 𝑉𝑉 ∈ 𝜇𝜇. Then 1!×𝑉𝑉 ∈ 𝛿𝛿×𝜇𝜇. Since 𝑔𝑔!→ is 𝜔𝜔- continuous, then g_!^← 1_!×𝑉𝑉 ∈ 𝛿𝛿_!. By Lemma 2.12, we have g!← 1!×𝑉𝑉 = 1!∧ f!←(𝑉𝑉) = f!←(𝑉𝑉). Therefore, f!←(𝑉𝑉) ∈ 𝛿𝛿!. It follows that 𝑓𝑓!→ is 𝜔𝜔-continuous.

3. Examples

The following is an example of 𝒩𝒩-continuous function that is not continuous. For the symbols that are not defined in Example 3.1, we refer to (Liu & Luo, 1997).

Example 3.1. Let 𝐿𝐿 be any F-lattice which has a subset 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 satisfying 0 < 𝑎𝑎 < 𝑏𝑏 < 𝑐𝑐 < 1, 𝑎𝑎^!= 𝑐𝑐 and 𝑏𝑏^!= 𝑏𝑏 (in particular 𝐿𝐿 = 𝐼𝐼). Let 𝑋𝑋 = 𝑌𝑌 = 𝐼𝐼 𝐿𝐿 . Denote the 𝐿𝐿-

fuzzy unit interval on 𝑋𝑋 by 𝛿𝛿. Consider the mappings 𝑥𝑥: ℝ → 𝐿𝐿

𝑥𝑥 𝑡𝑡 = 1 𝑖𝑖𝑖𝑖 𝑡𝑡 ∈ −∞, 0 , 𝑎𝑎 𝑖𝑖𝑖𝑖 𝑡𝑡 ∈ 0,1 , 0 𝑖𝑖𝑖𝑖 𝑡𝑡 ∈ (1, ∞)

.

It is proved in (Al Ghour, 2017) that 𝒳𝒳_{! ! !{[!]}}∉ 𝛿𝛿. Let 𝜇𝜇 be the 𝐿𝐿-topology on 𝑌𝑌 generated by 𝛿𝛿 ∪ {𝒳𝒳! ! !{[!]}} as a subbase. Define 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 by 𝑓𝑓(𝑥𝑥) = 𝑥𝑥. Then 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝒩𝒩-continuous but not continuous.

The following is an example of an 𝜔𝜔-irresolute function that is not 𝒩𝒩-continuous. It shows also that 𝜔𝜔- irresoluteness does not imply 𝒩𝒩-irresoluteness, in general.

Example 3.2. Let 𝑋𝑋 = 𝑌𝑌 = ℝ, 𝐿𝐿 = 𝐼𝐼, 𝛿𝛿 = {0!} ∪ {𝐴𝐴: 𝐴𝐴(𝑥𝑥) > 0 for all 𝑥𝑥 ∈ 𝑋𝑋} and 𝜇𝜇 be the 𝐿𝐿-topology on 𝑌𝑌 generated by {𝐴𝐴: 0 < 𝐴𝐴(𝑦𝑦) ≤ 0.5 for all 𝑦𝑦 ∈ 𝑌𝑌} ∪ {0!, 1!, 𝒳𝒳{!!{!}} as a subbase. Define 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 by 𝑓𝑓(𝑥𝑥) = 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥. Then 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝜔𝜔-irresolute but not 𝒩𝒩-continuous.

𝒟𝒟-continuity does not imply 𝜔𝜔-continuity in general as it can be seen from the following example:

Example 3.3. Let 𝑋𝑋 = 𝑌𝑌 = (0,1], 𝐿𝐿 = 𝐼𝐼, 𝛿𝛿 be the 𝐿𝐿- topology on 𝑋𝑋 generated by {𝑎𝑎𝒳𝒳_!,! ∶ 𝑥𝑥 ∈ 𝑋𝑋, 𝑎𝑎 ∈ 𝐿𝐿} as a base and 𝜇𝜇 be the 𝐿𝐿-topology on 𝑌𝑌 generated by {𝑎𝑎𝒳𝒳!,! ∶ 𝑥𝑥 ∈ 𝑋𝑋, 𝑎𝑎 ∈ 𝐿𝐿} ∪ { 0.5 𝒳𝒳!.!} as a subbase.

Define 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 by 𝑓𝑓(𝑥𝑥) = 𝑥𝑥. Then 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝒟𝒟-continuous but not 𝜔𝜔-continuous.

The following is an example of a continuous function that is neither ω-irresolute nor 𝒩𝒩-irresolute. In particular, it shows that 𝜔𝜔-continuity does not imply 𝜔𝜔- irresoluteness, and also 𝒩𝒩-continuity does not imply 𝒩𝒩- irresoluteness, in general.

Example 3.4. Let 𝑋𝑋 = 𝑌𝑌 = ℝ, 𝐿𝐿 = 𝐼𝐼, 𝛿𝛿 = {0!} ∪ {𝐴𝐴: 𝐴𝐴(𝑥𝑥) > 0 for all 𝑥𝑥 ∈ 𝑋𝑋}, and 𝜇𝜇 = {𝐴𝐴: 0 < 𝐴𝐴(𝑦𝑦) ≤ 0.5 for all 𝑦𝑦 ∈ 𝑌𝑌} ∪ {0_!, 1_!}. Define 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 by 𝑓𝑓(𝑥𝑥) = 1 for 𝑥𝑥 ∈ ℚ and 𝑓𝑓(𝑥𝑥) = 2 for 𝑥𝑥 ∈ ℝ − ℚ. Then 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is continuous but neither 𝜔𝜔- irresolute nor 𝒩𝒩-irresolute. To see that 𝑓𝑓_!^→ is continuous, let 𝐵𝐵 ∈ 𝜇𝜇 − {0_!}. Then for all 𝑦𝑦 ∈ 𝑌𝑌, 0 < 𝐵𝐵(𝑦𝑦) ≤ 0.5. So for all 𝑥𝑥 ∈ 𝑋𝑋, we have (𝑓𝑓!← (𝐵𝐵))(𝑥𝑥) = 𝐵𝐵(𝑓𝑓(𝑥𝑥)) > 0.

Hence 𝑓𝑓!→ is continuous. Now, since 𝒳𝒳ℝ!{!}∈ 𝜇𝜇𝒩𝒩= 𝜇𝜇_𝒩𝒩∩ 𝜇𝜇_! but 𝑓𝑓_!^← 𝒳𝒳_{ℝ! !} = 𝒳𝒳_!^!!_{ℝ! !} = 𝒳𝒳_ℚ∉ 𝛿𝛿_!= 𝛿𝛿!∪ 𝛿𝛿𝒩𝒩, then 𝑓𝑓!→ is neither 𝜔𝜔-irresolute nor 𝒩𝒩- irresolute.

𝜔𝜔-irresoluteness and 𝒩𝒩-irresoluteness are not sufficient conditions to imply continuity as the following example clarifies:

Example 3.5. Let 𝑋𝑋 = 𝑌𝑌 = [0,1], 𝐿𝐿 = 𝐼𝐼

𝛿𝛿 = {0!, 1!} ∪ {𝐴𝐴: 𝑥𝑥⁴ < 𝐴𝐴(𝑥𝑥) ≤ 𝑥𝑥 for all 𝑥𝑥 ∈ 𝑋𝑋} and 𝜇𝜇 be the 𝐿𝐿-topology on 𝑌𝑌 generated by {0_!, 1_!} ∪ {𝐴𝐴: 𝑦𝑦² <

𝐴𝐴(𝑦𝑦) ≤ √𝑦𝑦 for all 𝑦𝑦 ∈ 𝑌𝑌} ∪ {𝒳𝒳!,!} as a subbase. Define 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 by 𝑓𝑓(𝑥𝑥) = 𝑥𝑥². Then 𝑓𝑓!→: (𝐿𝐿^!, 𝛿𝛿) → (𝐿𝐿^!, 𝜇𝜇) is 𝜔𝜔- irresolute and 𝒩𝒩-irresolute but not continuous

Example 3.5 is a particular example of an 𝒩𝒩-

continuous function that is not continuous.

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Submitted : 28/08/2017 Revised : 02/11/2017 Accepted : 28/11/2017

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ﺔﯾﻧدرﻷا ﺎﯾﺟوﻟوﻧﻛﺗﻟاو موﻠﻌﻟا ﺔﻌﻣﺎﺟ ، ،

،ﺪﺑرا ندرﻷا

algore@just.edu.jo

ﺺﺨﻠﻤﻟا

ﻢﯿھﺎﻔﻤﻟا مﺪﺨﺘﺴﻧ 𝜔𝜔 -open 𝐿𝐿-sets

و -open 𝐿𝐿-sets و 𝒩𝒩

-open 𝐿𝐿-sets ﻦﻣ دﺪﻋ ﻒﯾﺮﻌﺘﻟ 𝒟𝒟

تاءﺎﻀﻓ ﻲﻓ مﺰﻌﻟا ماﺪﻌﻧاو لﺎﺼﺗﻼﻟ ةﺪﯾﺪﺠﻟا عاﻮﻧﻷا ﺔﯿﺟﻮﻟﻮﺑﻮﻄﻟا L

.ﺞﺋﺎﺘﻨﻟا ﻦﻣ دﺪﻋ ﺔﻨھﺮﺑ ﻢﺗو .

.ةﺪﯾﺪﺠﻟا مﺰﻌﻟا ماﺪﻌﻧا ﻢﯿھﺎﻔﻤﻟ ﻚﯿﻜﻔﺘﻟا تﺎﯾﺮﻈﻧ ثاﺪﺤﺘﺳا ﻢﺗ صﻮﺼﺨﻟا ﮫﺟو ﻰﻠﻋ