On Fuzzy Sp-Open Sets

Abstract

A new class of generalized fuzzy open sets in fuzzy topological space, called fuzzy sp-open sets, are introduced, and their properties are studied and the relationship between this new concept and other weaker forms of fuzzy open sets we discussed. Moreover, we introduce the fuzzy sp-continuous (resp., fuzzy sp-open) mapping and other stronger forms of sp-continuous (resp., fuzzy sp-open) mapping and establish their various characteristic properties. Finally, we study the relationships between all these mappings and other weaker forms of fuzzy continuous mapping and introduce fuzzy sp-connected. Counter examples are given to show the noncoincidence of these sets and mappings.

1. Introduction

In 1996, Dontchev and Przemski, [1] have introduced the concept of sp-open sets in general topology. In this paper, we extend the notion of sp-open sets to fuzzy topology space and study some notions based on this new concept. We further study the relation between fuzzy sp-open sets and other types of fuzzy open sets. We also introduce the concepts of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, other stronger forms of fuzzy sp-continuous (resp., fuzzy sp-open) mapping, and discuss their relation with other weaker forms of fuzzy continuous mapping.

2. Preliminaries

Throughout this paper, by (𝑋,𝜏) or simply by 𝑋 we mean a fuzzy topological space (fts, shorty) and 𝑓∶𝑋→𝑌 means a mapping 𝑓 from a fuzzy topological space 𝑋 to a fuzzy topological space 𝑌. If 𝑢 is a fuzzy set and 𝑝 is a fuzzy singleton in 𝑋, then 𝑁(𝑝), Int𝜆, cl𝑢, 𝑢^𝑐 denote, respectively, the neighborhood system of 𝑝, the interior of 𝑢, the closure of 𝑢, and complement of 𝑢.

Now, we recall some of the basic definitions and results in fuzzy topology.

Definition 2.1 (see [2]). A fuzzy singleton 𝑝 in 𝑋 is a fuzzy set defined by: 𝑝(𝑥)=𝑡, for 𝑥=𝑥₀ and 𝑝(𝑥)=0 otherwise, where 0<𝑡≤1. The point 𝑝 is said to have support 𝑥₀ and value 𝑡.

Definition 2.2. A fuzzy set 𝑢 in a fts X is called fuzzy 𝛼-open [3] (resp., Fuzzy preopen [4], Fuzzy 𝛽-open [5]) set if 𝑢≤ Int cl Int 𝑢 (resp., 𝑢≤ Int cl 𝑢, 𝑢≤ cl Int cl 𝑢, 𝑢≤ cl Int 𝑢). The family of all fuzzy 𝛼-open (resp., fuzzy preopen, fuzzy 𝛽-open, fuzzy semiopen) sets of 𝑋 is denoted by F𝛼O(𝑋) (resp., FPO(𝑋), F𝛽O(𝑋), FSO(𝑋)).

Definition 2.3 (see [4]). Let 𝑢 be any fuzzy set. Then,(i)pcl𝑢=∧{𝑣∶𝑣≥𝑢,𝑣isafuzzypreclosedsetof𝑋}is called preclosure,(ii)pInt𝑣=∨{𝑣∶𝑣≤𝑢,𝑣isafuzzypreopensetof𝑋} is called pre-Interior.The definitions of scl, sInt, 𝛼-cl, and 𝛼-Int are similar.

Theorem 2.4. For any fuzzy set 𝑢 in a fts𝑋, the following statements are true:(i)scl𝑢=𝑢∨Intcl𝑢 and sInt𝑢=𝑢∧clInt𝑢 [6],(ii)pcl𝑢≥𝑢∨clInt𝑢andpInt𝑢≤𝑢∧Intcl𝑢 [7].

Theorem 2.5 (see [3, 4]). The arbitrary union of fuzzy preopen (resp., fuzzy semiopen) sets is a fuzzy preopen (resp., fuzzy semiopen) set.

Definition 2.6. A mapping 𝑓∶(𝑋,𝜏)→(𝑌,𝜎) is said to be(i)fuzzy 𝛼-continuous [3] if 𝑓⁻¹(𝑣) is fuzzy 𝛼-open set in 𝑋 for each fuzzy open set 𝑣 in 𝑌,(ii)fuzzy semi continuous [3] if 𝑓⁻¹(𝑣) is fuzzy semiopen set in 𝑋 for each fuzzy open set 𝑣 in 𝑌,(iii)fuzzy 𝛽-continuous [5] if 𝑓⁻¹(𝑣) is fuzzy 𝛽-open set in 𝑋 for each fuzzy open set 𝑣 in 𝑌.

3. Fuzzy SP-Open Set

Definition 3.1. A fuzzy subset 𝑢 of fuzzy space 𝑋 is called fuzzy sp-open set if 𝑢≤Intcl𝑢∨clInt𝑢. The class of all fuzzy sp-open sets in 𝑋 will be denoted be FSP−𝑂(𝑋).
It is obvious that FPO(𝑋)∨FSO(𝑋)≤FSP−𝑂(𝑋)≤F𝛽O(𝑋).

Proposition 3.2. Let 𝑢 be fuzzy sp -open set such that Int𝑢=0. Then, 𝑢 is fuzzy preopen.

Using Theorem 2.5, we can easily prove the next corollary.

Corollary 3.3. Any union of fuzzy sp -open sets is a fuzzy sp-open set.

Remark 3.4. The intersection of fuzzy sp-open sets need not be fuzzy sp-open set. This is illustrated by the following example.

Example 3.5. Let 𝑋={𝑎,𝑏,𝑐} and 𝑣₁,𝑣₂,𝑣₃, and 𝑣₄ be fuzzy sets of 𝑋 defined as 𝑣₁(𝑎)=0.0,𝑣₁(𝑏)=0.0,𝑣₁(𝑐)𝑣=0.4,₂(𝑎)=0.9,𝑣₂(𝑏)=0.6,𝑣₂(𝑐)𝑣=0.0,₃(𝑎)=0.0,𝑣₃(𝑏)=0.3,𝑣₃(𝑐)𝑣=0.4,₄(𝑎)=0.9,𝑣₄(𝑏)=0.7,𝑣₄(𝑐)=0.2. (1) Let 𝜏={𝑜_𝑥,𝑣₁,𝑣₂,𝑣₁∨𝑣₂,1_𝑥}. Clearly, 𝜏 is a fuzzy topology on 𝑋, and by easy computation, it follows that 𝑣₃ and 𝑣₄ are fuzzy sp-open sets. But (𝑣₃∧𝑣₄) is not a fuzzy fuzzy sp-open set.

Theorem 3.6. For any fuzzy subset 𝑢 of a fuzzy space X, the following properties are equivalent:(i)𝑢 is fuzzy sp-open,(ii)𝑢≥pInt𝑢∨sInt𝑢.

Proof. (𝑖)⇒(𝑖𝑖). Let 𝑢 be a fuzzy sp-open, that is, 𝑢≤Intcl𝑢∨clInt𝑢. Then, we have ()∨()()pInt𝑢∨sInt𝑢≤𝑢∧Intcl𝑢𝑢∧clInt𝑢≤𝑢∧Intcl𝑢∨clInt𝑢≤𝑢∧𝑢=𝑢. (2)

Definition 3.7. A fuzzy subset 𝑢 of fuzzy space 𝑋 is called fuzzy sp-closed set if Intcl𝑢∧clInt𝑢≤𝑢. The class of all fuzzy sp-closed sets in 𝑋 will be denoted be FSP-𝐶(𝑋).

Definition 3.8. Let 𝑢 any fuzzy set. Then,(i)𝑠𝑝-cl𝑢=∧{𝑣∶𝑣≥𝑢,𝑣is a fuzzy𝑠𝑝-closedsetof𝑋} is called fuzzy sp-closure,(ii)𝑠𝑝-Int𝑣=∨{𝑣∶𝑣≤𝑢,𝑣isafuzzy𝑠𝑝-opensetof𝑋} is called fuzzy sp-Interior.By using Definitions 3.1, 3.7, and 3.8, we can prove the following theorems.

Theorem 3.9. Let 𝑢 and 𝑣 be the fuzzy sets in fts𝑋. Then, the following statements hold(i)𝑠𝑝-cl(𝑢) is fuzzy sp-closed,(ii)𝑢⊆𝑆𝑃-𝐶(𝑋)⇔𝑢=𝑠𝑝-cl(𝑢),(iii)𝑢≤𝑣⇒𝑠𝑝-cl(𝑢)≤𝑠𝑝-cl(𝑣),(iv)𝑠𝑝-Int𝑢 is fuzzy sp-open,(v)𝑢⊆𝑆𝑃-𝑂(𝑋)⇔𝑢=𝑠𝑝-Int(𝑢),(vi)𝑢≤𝑣⇒𝑠𝑝-Int(𝑢)≤𝑠𝑝-Int(𝑣),(vii)Int𝑢≤𝑠𝑝-Int𝑢≤𝑢≤𝑠𝑝-cl(𝑢)≤cl𝑢.

Theorem 3.10. For a fuzzy subset 𝜆 of a fuzzy space 𝑋, the following statements are holding:(i)𝑠𝑝-cl𝑢≥𝑢∨(Intcl𝑢∧clInt𝑢),(ii)𝑠𝑝-Int𝑢≤𝑢∧(Intcl𝑢∨clInt𝑢),(iii)𝑠𝑝-cl𝑢≥scl𝑢∧pcl𝑢,(iv)𝑠𝑝-Int𝑢≤sInt𝑢∨pInt𝑢.

Theorem 3.11. For any fuzzy subset 𝑢 of a fuzzy space X, the following statements are equivalent:(i)𝑢 is fuzzy sp-closed, (ii)𝑢^𝑐 is fuzzy sp-open,(iii)𝑢≥Intcl𝑢∧clInt𝑢, and (iv)𝑢^𝑐≤Intcl𝑢∨clInt𝑢.

Theorem 3.12. A fuzzy set 𝑢 in a fuzzy topology space 𝑋 is fuzzy sp -open if and only if for every fuzzy point 𝑝∈𝑢, there exists a fuzzy sp -open set 𝑣_𝑝≤𝑢 such that 𝑝∈𝑣_𝑝.

Proof. If 𝑢 is a fuzzy sp-open set, then we may take 𝑣_𝑝=𝑢 for every 𝑝∈𝑢.
Conversely, we have 𝑢=∪_𝑝∈𝑢{𝑝}≤∪_𝑝∈𝑢𝑣_𝑝≤𝑢 and, hence, 𝑢=∪_𝑝∈𝑢𝑣_𝑝. This shows that 𝑢 is a fuzzy sp-open set.

From Definitions 2.2 and 3.1, the above “Implication Figure 1” illustrates the relation of different classes of fuzzy open sets.

Figure 1 

Remark 3.13. The converse of these relations need not be true, in general as shown by the following examples.

Example 3.14. Let 𝑋={𝑎,𝑏,𝑐} and 𝑣₁,𝑣₂,𝑣₃, and 𝑣₄ be fuzzy sets of 𝑋 defined as 𝑣₁(𝑎)=0.5,𝑣₁(𝑏)𝑣=0.3,₂(𝑎)=0.5,𝑣₂(𝑏)𝑣=0.6,₃(𝑎)=0.6,𝑣₃(𝑏)𝑣=0.3,₄(𝑎)=0.6,𝑣₄(𝑏)=0.7. (3) Let 𝜏={𝑜_𝑥,𝑣₁,1_𝑥}. Clearly, 𝜏 is a fuzzy topological space on 𝑋, and by easy computation, we can see:(i)𝑣₂ is fuzzy sp-open set which is neither fuzzy 𝛼-open set nor fuzzy preopen,(ii)𝑣₃ is fuzzy sp-open which is not semiopen,(iii)𝑣₄ is fuzzy sp-open set which is not fuzzy open.

Example 3.15. Let 𝑋={𝑎,𝑏,𝑐} and 𝑣₁,𝑣₂,𝑎𝑛𝑑𝑣₃ fuzzy sets of 𝑋 defined as 𝑣₁(𝑎)=0.0,𝑣₁(𝑏)=0.0,𝑣₁(𝑐)𝑣=0.4,₂(𝑎)=0.9,𝑣₂(𝑏)=0.6,𝑣₂(𝑐)𝑣=0.0,₃(𝑎)=0.1,𝑣₃(𝑏)=0.0,𝑣₃(𝑐)=0.3. (4) Let 𝜏={𝑜_𝑥,𝑣₁,1_𝑥}. Clearly, 𝜏 is a fuzzy topological space on 𝑋, and by easy computation, it follows that 𝑣₃ is fuzzy 𝛽-open set which is not fuzzy sp-open.

4. Fuzzy SP-Continuous Mapping

Definition 4.1. A mapping 𝑓∶𝑋→𝑌 is said to be(i)fuzzy sp-continuous if 𝑓⁻¹(𝑣) is fuzzy sp-open set in 𝑋 for each fuzzy open set 𝑣 in 𝑌,(ii)fuzzy sp^⋆-continuous if 𝑓⁻¹(𝑣) is fuzzy sp-open set in 𝑋 for each fuzzy sp-open set 𝑣 in 𝑌,(iii)fuzzy sp^⋆⋆-continuous if 𝑓⁻¹(𝑣) is fuzzy open set in 𝑋 for each fuzzy sp-open set 𝑣 in 𝑌.

Theorem 4.2. For a mapping 𝑓∶(𝑋,𝜏)→(𝑌,𝜎), the following statements are equivalent: (i)𝑓 is fuzzy sp-continuous;(ii)for every fuzzy singleton 𝑝 in 𝑋 and every open set 𝑣 in 𝑌 such that 𝑓(𝑝)⊆𝑣, there exists a fuzzy sp -open set 𝑢⊆𝑋 such that 𝑝⊆𝑢 and 𝑢≤𝑓⁻¹(𝑣);(iii)for every fuzzy singleton 𝑝 in 𝑋 and every open set 𝑣 in 𝑌 such that 𝑓(𝑝)⊆𝑣, there exists a fuzzy sp -open set 𝑢⊆𝑋 such that 𝑝⊆𝑢 and 𝑓(𝑢)≤𝑣;(iv)the inverse image of each fuzzy closed set in 𝑌 is fuzzy sp-closed;(v)Intcl(𝑓⁻¹(𝑣))∧clInt(𝑓⁻¹(𝑣))≤(𝑓⁻¹(cl𝑣)) for each 𝑣⊆𝑌;(vi)𝑓[Intcl(𝑢)∧clInt(𝑢)]≤cl𝑓(𝑢) for each 𝑢⊆𝑋.

Proof. (i)⇒(ii) Let fuzzy singleton 𝑝 be in 𝑋 and every open set 𝑣 in 𝑌 such that 𝑓(𝑝)⊆𝑣, there exists a fuzzy open set 𝑚 be in 𝑌 such that 𝑓(𝑝)≤𝑚≤𝑣. Since 𝑓 is sp-continuous, 𝑢=𝑓⁻¹(𝑚) is fuzzy sp-open and we have 𝑝≤𝑓⁻¹(𝑓(𝑝))≤𝑓⁻¹(𝑚)≤𝑓⁻¹(𝑣) or 𝑝≤𝑢=𝑓⁻¹(𝑚)≤𝑓⁻¹(𝑣).
(ii)⇒(iii) Let fuzzy singleton 𝑝 be in 𝑋 and every fuzzy open set 𝑣 be in 𝑌 such that 𝑓(𝑝)⊆𝑣, there exists a fuzzy sp-open 𝑢 such that 𝑝≤𝑢 and 𝑢≤𝑓⁻¹(𝑣). So, we have 𝑝≤𝑢 and 𝑓(𝑢)≤𝑓⁻¹((𝑓(𝑣))≤𝑣.
(iii)⇒(i) Let 𝑣 be a fuzzy open set in 𝑌 and let us take 𝑝≤𝑓⁻¹(𝑣). This shows that 𝑓(𝑝)≤𝑓(𝑓⁻¹(𝑣))≤𝑣. Since 𝑣 is a fuzzy open set, then there exists a fuzzy sp-open set 𝑢 such that 𝑝≤𝑢 and 𝑓(𝑢)≤𝑣. This shows that 𝑝≤𝑢≤𝑓⁻¹(𝑓(𝑢))≤𝑓⁻¹(𝑣). By Theorem 3.12, it follows that 𝑓⁻¹(𝑣) is fuzzy sp-open set in 𝑋 and hence 𝑓 is fuzzy sp-continuous.
(i)⇒(iv) Let 𝑣 be a fuzzy closed in 𝑌. This implies that 𝐼_𝑌−𝑣 is fuzzy open set. Hence, 𝑓⁻¹(𝐼_𝑌−𝑣) is fuzzy sp-open set in 𝑋, that is, (𝐼_𝑋−𝑓⁻¹(𝑣)) is fuzzy sp-open set in 𝑋. Thus, 𝑓⁻¹(𝑣) is a fuzzy sp-closed set in 𝑋.
(iv)⇒(v) Let 𝑣⊆𝑌, then 𝑓⁻¹(cl𝑣) is sp-closed in 𝑋, that is, Intcl(𝑓⁻¹(𝑣))∧clInt(𝑓⁻¹(𝑣))≤Intcl(𝑓⁻¹(cl𝑣))∧clInt(𝑓⁻¹(cl𝑣))≤𝑓⁻¹(cl𝑣).
(v)⇒(vi) Let 𝑢⊆𝑋, put 𝑣=𝑓(𝑢) in (𝑣), then Intcl(𝑓⁻¹(𝑓(𝑢)))∧clInt(𝑓⁻¹(𝑓(𝑢)))≤𝑓⁻¹(cl(𝑓(𝑢))) so that Intcl(𝑢))∧clInt(𝑢)≤𝑓⁻¹(cl(𝑓(𝑢))). This gives 𝑓[Intcl(𝑢)∧clInt(𝑢)]≤cl𝑓(𝑢).
(vi)⇒(i) Let 𝑣⊆𝑌, be fuzzy open set. put 𝑢=𝐼_𝑌−𝑣 and 𝑢=𝑓⁻¹(𝑣), then 𝑓[Intcl(𝑓⁻¹(𝑣))∧clInt(𝑓⁻¹(𝑣))]≤cl𝑓(𝑓⁻¹(𝑣))≤cl(𝑣)=𝑣, that is, 𝑓⁻¹(𝑣)sp-closed set in 𝑋, so 𝑓 is sp-continuous mapping.

Definition 4.3. A mapping 𝑓∶𝑋→𝑌 is said to be(i)fuzzy sp-open (Fuzzy sp-closed) if 𝑓(𝑢) is fuzzy sp-open (fuzzy sp-closed) set in 𝑌 for each fuzzy open (fuzzy closed) set 𝑢 in 𝑋,(ii)fuzzy sp^⋆-open (Fuzzy sp^⋆-closed) if 𝑓(𝑢) is fuzzy sp-open (fuzzy sp-closed) set in 𝑌 for each fuzzy sp-open (fuzzy sp-closed) set 𝑢 in 𝑋,(iii)fuzzy sp^⋆⋆-open (Fuzzy sp^⋆⋆-closed) if 𝑓(𝑢) is fuzzy open (fuzzy closed) set in 𝑌 for each fuzzy sp-open (fuzzy sp-closed) set 𝑢 in 𝑋.

Remark 4.4. If 𝑓∶𝑋→𝑌 is fuzzy sp-continuous mapping and 𝑔∶𝑌→𝑍 is fuzzy sp-continuous mapping, then 𝑔𝑜𝑓∶𝑋→𝑍 may not be a fuzzy sp-continuous mapping; this can be show by the following example.

Example 4.5. Let 𝑋={𝑎,𝑏,𝑐} and 𝑣₁,𝑣₂,𝑣₃,𝑣₄ and 𝑣₅ be fuzzy sets of 𝑋 defined as, 𝑣₁(𝑎)=0.4,𝑣₁(𝑏)=0.6,𝑣₁(𝑐)𝑣=0.5,₂(𝑎)=0.6,𝑣₂(𝑏)=0.4,𝑣₂(𝑐)𝑣=0.4,₃(𝑎)=0.4,𝑣₃(𝑏)=0.6,𝑣₃(𝑐)𝑣=0.6,₄(𝑎)=0.7,𝑣₄(𝑏)=0.7,𝑣₄(𝑐)𝑣=0.8,₅(𝑎)=0.6,𝑣₅(𝑏)=0.4,𝑣₅(𝑐)=0.6. (5) Consider, fts𝜏₁, 𝜏₂, and 𝜏₃ where 𝜏₁={𝑜_𝑥,𝑣₁,𝑣₂,𝑣₁∩𝑣₂,𝑣₁∪𝑣₂,1_𝑥}, 𝜏₂={𝑜_𝑥,𝑣₄,1_𝑥}, and 𝜏₃={𝑜_𝑥,𝑣₃,1_𝑥} and the mapping 𝑓∶(𝑋,𝜏₁)→(𝑌,𝜏₂) and 𝑔∶(𝑋,𝜏₂)→(𝑌,𝜏₃) defined as 𝑓(𝑎)=𝑏,𝑓(𝑏)=𝑎,𝑓(𝑐)=𝑐 and 𝑔(𝑎)=𝑎,𝑔(𝑏)=𝑏,𝑔(𝑐)=𝑐. It is clear that 𝑓 and 𝑔 are fuzzy sp-continuous mapping. But 𝑔𝑜𝑓∶(𝑋,𝜏₁)→(𝑌,𝜏₃) is not a fuzzy sp-continuous mapping. This because (𝑔𝑜𝑓)⁻¹(𝑣₃)=𝑣₅ and 𝑣₅ is not fuzzy sp-open set, and hence 𝑔𝑜𝑓 is not fuzzy sp-continuous mapping.

Theorem 4.6. If 𝑓∶𝑋→𝑌 is fuzzy sp -continuous mapping and 𝑔∶𝑌→𝑍 is fuzzy continuous mapping, then 𝑔𝑜𝑓∶𝑋→𝑍 is fuzzy sp-continuous mapping.

Proof. Let 𝑣 be a fuzzy set of 𝑍. Then, (𝑔𝑜𝑓)⁻¹(𝑣)=𝑓⁻¹(𝑔⁻¹(𝑣)). And because 𝑔 is fuzzy continuous this implies that 𝑔⁻¹(𝑣) is a fuzzy open set of 𝑌 and hence 𝑓⁻¹(𝑔⁻¹(𝑣)) is a fuzzy sp-open set in 𝑋. Therefore, 𝑔𝑜𝑓 is a fuzzy sp- continuous mapping.

From Definitions 4.1 and 4.3, we can have the above “Implication Figure 2” illustrates the relation between different classes of fuzzy sp-continuous (fuzzy semi sp-open) mappings.

Figure 2 

The above “Implication Figure 3” illustrates the relation between fuzzy sp-continuous and different classes of fuzzy continuous mapping.

Figure 3 

Remark 4.7. We can see the converse of these relations need not be true, in general as shown by the following examples.

Example 4.8. Let 𝑋={𝑎,𝑏,𝑐} and 𝑣₁,𝑣₂,𝑣₃, and 𝑣₄ fuzzy sets of 𝑋 defined as 𝑣₁(𝑎)=0.4,𝑣₁(𝑏)=0.6,𝑣₁(𝑐)𝑣=0.5,₂(𝑎)=0.4,𝑣₂(𝑏)=0.4,𝑣₂(𝑐)𝑣=0.4,₃(𝑎)=0.6,𝑣₃(𝑏)=0.4,𝑣₃(𝑐)𝑣=0.6,₄(𝑎)=0.7,𝑣₄(𝑏)=0.7,𝑣₄(𝑐)=0.8. (6) Consider fts𝜏₁, 𝜏₂, and 𝜏₃ where 𝜏₁={𝑜_𝑥,𝑣₁,𝑣₂,𝑣₁∩𝑣₂,𝑣₁∪𝑣₂,1_𝑥}, 𝜏₂={𝑜_𝑥,𝑣₄,1_𝑥}, and 𝜏₃={𝑜_𝑥,𝑣₃,1_𝑥} and the mapping 𝑓∶(𝑋,𝜏₁)→(𝑌,𝜏₂) and 𝑔∶(𝑋,𝜏₁)→(𝑌,𝜏₃) defined as 𝑓(𝑎)=𝑎,𝑓(𝑏)=𝑏,𝑓(𝑐)=𝑐 and 𝑔(𝑎)=𝑎,𝑔(𝑏)=𝑏,g(𝑐)=𝑐. It is clear that:(i)𝑓 is a fuzzy sp-continuous mapping which is neither fuzzy 𝛼-continuous mapping nor fuzzy semi continuous mapping,(ii)𝑔 is a fuzzy 𝛽-continuous mapping which is not fuzzy sp-continuous mapping,(iii)𝑓 is a fuzzy sp-continuous mapping which is neither fuzzy sp^⋆-continuous mapping nor fuzzy sp^⋆⋆-continuous mapping, and this is because 𝑣₃ is fuzzy sp-open in 𝜏₂ which is neither fuzzy sp-open nor fuzzy open in 𝜏₁,(iv)if ℎ∶(𝑋,𝜏₃)→(𝑌,𝜏₁) defined as: ℎ(𝑎)=𝑎,ℎ(𝑏)=𝑏,ℎ(𝑐)=𝑐, it is clear that ℎ is a fuzzy sp-open mapping which is neither fuzzy sp^⋆-open mapping nor fuzzy sp^⋆⋆-open mapping.

Definition 4.9. A fuzzy set 𝑢 in an fts𝑋 is said to be fuzzy connected if 𝑢 cannot be expressed as the union of two fuzzy separated sets.

Now, we can generalize the definition of fuzzy connected to define fuzzy sp-connected as follows.

Definition 4.10. A fuzzy set 𝑣 in a fts(𝑋,𝜏) is said to be fuzzy sp-connected if and only if 𝑣 cannot be expressed as the union of two fuzzy sp-separated sets.

Theorem 4.11. Let 𝑓∶𝑋→𝑌 be a fuzzy sp -continuous surjective mapping. If 𝑣 is a fuzzy sp -connected subset in 𝑋 then, 𝑓(𝑣) is fuzzy connected in 𝑌.

Proof. Suppose that 𝑓(𝑚) is not connected in 𝑌. Then, there exist fuzzy separated subsets 𝑢 and 𝑣 in 𝑌 such that 𝑓(𝑚)=𝑢∪𝑣.
Since 𝑓 is fuzzy sp-continuous surjective mapping, 𝑓⁻¹(𝑢) and 𝑓⁻¹(𝑣) are fuzzy sp-open set in 𝑋 and 𝑚=𝑓⁻¹(𝑓(𝑚))=𝑓⁻¹(𝑢∪𝑣)=𝑓⁻¹(𝑢)∪𝑓⁻¹(𝑣).
It is clear that 𝑓⁻¹(𝑢) and 𝑓⁻¹(𝑣) are fuzzy sp-separated in 𝑋. Therefore, 𝑚 is not fuzzy sp-connected in 𝑋, which is a contradiction!!
Hence, 𝑌 is fuzzy connected.