Relative Smooth Topological Spaces

Abstract

In 1992, Ramadan introduced the concept of a smooth topological space and relativeness between smooth topological space and fuzzy topological space in Chang's (1968) view points. In this paper we give a new definition of smooth topological space. This definition can be considered as a generalization of the smooth topological space which was given by Ramadan. Some general properties such as relative smooth continuity and relative smooth compactness are studied.

1. Introduction

Let 𝑋 be a nonempty set and let ℒ,ℒ^′ be two lattice which will be copies of [0,1] or {0,1}. The family of all fuzzy sets on 𝑋 will be denoted by ℒ^𝑋 Zadeh [1].

In the consideration of the nature an observer can be modeled by an operator which evaluates each proposition by a number in the closed interval [0,1]; see Anvari and Molaei [2] and Molaei [3]. We assume that 𝜇 as a function from 𝑋 to ℒ is an observer of 𝑋 on lattice ℒ and denote [𝜇]={𝜆∈ℒ^𝑋∶𝜆⊆𝜇}, where 𝜆⊆𝜇 implies that 𝜆(𝑥)≤𝜇(𝑥) for all 𝑥∈𝑋.

Definition 1.1. Let 𝜇∈ℒ^𝑋. A relative smooth topological space or 𝜇-smooth topological space or 𝜇-STS for short is a triple (𝑋,𝜇,𝒯_𝜇), where 𝒯_𝜇∶[𝜇]→ℒ^′ is a mapping satisfying the following properties:(i)𝒯_𝜇(𝜇)=𝒯_𝜇(𝜒_𝜙)=1, where 𝜒 is the characteristic function;(ii)if 𝜆₁, 𝜆₂∈[𝜇], then 𝒯_𝜇(𝜆₁∩𝜆₂)≥𝒯_𝜇(𝜆₁)⋀𝒯_𝜇(𝜆₂), where ⋀ is the minimum operator in ℒ^′;(iii)𝒯_𝜇(∪{𝜆_𝑖∶𝑖∈𝐼})≥⋀{𝒯_𝜇(𝜆_𝑖)∶𝑖∈𝐼}. We call 𝒯_𝜇 a smooth topology from view point of 𝜇 or a 𝜇-smooth topology or a fuzzy family of 𝜇-open sets on 𝑋.

Remark 1.2. If 𝜇=𝜒_𝑋 then the 𝜒_𝑋-STS (𝑋,𝜒_𝑋,𝒯_𝜒𝑋) coincides with the smooth topological space (𝑋,𝜏) defined by Ramadan [4], and if we take ℒ=[0,1],ℒ^′={0,1}, and 𝜇=𝜒_𝑋 then the 𝜒_𝑋-STS coincides with the known definition of fuzzy topological space (𝑋,𝜏) defined by Chang [5]. If ℒ=ℒ^′={0,1}, and 𝜇=𝜒_𝑋 then 𝒯_𝜒𝑋 is a classical topology.

Definition 1.3. Let 𝜇∈ℒ^𝑋. A 𝜇-smooth cotopological space is a triple (𝑋,𝜇,ℱ_𝜇), where ℱ_𝜇∶[𝜇]→ℒ^′ is a mapping satisfying the following properties:(i)ℱ_𝜇(𝜇)=ℱ_𝜇(𝜒_𝜙)=1;(ii)if 𝜂₁, 𝜂₂∈[𝜇], then ℱ_𝜇(𝜂₁∪𝜂₂)≥ℱ_𝜇(𝜂₁)⋀ℱ_𝜇(𝜂₂);(iii)ℱ_𝜇(∩{𝜂_𝑖∶𝑖∈𝐼})≥⋀{ℱ_𝜇(𝜂_𝑖)∶𝑖∈𝐼}. We call ℱ_𝜇 a 𝜇-smooth co-topology or a fuzzy family of 𝜇-closed sets on 𝑋.

Theorem 1.4. Let (𝑋,𝜇,𝒯_𝜇) be a 𝜇-STS and ℱ_𝜇∶[𝜇]→ℒ^′ be a mapping defined by ℱ_𝜇(𝜂)=𝒯_𝜇(𝜂^′), where 𝜂^′=𝜇−𝜂. Then ℱ_𝜇 is a fuzzy family of 𝜇-closed sets.

Proof. (i)It is clear.(ii)It flows from (𝜂₁∪𝜂₂)^′=𝜇−(𝜂₁∪𝜂₂)=𝜇−sup{𝜂₁,𝜂₂}=inf{𝜇−𝜂₁,𝜇−𝜂₂}=𝜂^′₁∩𝜂^′₂. (1) So, ℱ_𝜇(𝜂₁∪𝜂₂)=𝒯_𝜇((𝜂₁∪𝜂₂)^′)=𝒯_𝜇(𝜂^′₁∩𝜂^′₂).(iii)It flows from (∩_𝑖∈𝐼𝜂_𝑖)^′=𝜇−∩_𝑖∈𝐼𝜂_𝑖=𝜇−inf{𝜂_𝑖∶𝑖∈𝐼}=sup{𝜇−𝜂_𝑖∶𝑖∈𝐼}=∪_𝑖∈𝐼𝜂^′_𝑖. (2) So, ℱ_𝜇(∩{𝜂_𝑖∶𝑖∈𝐼})=𝒯_𝜇(∪{𝜂^′_𝑖∶𝑖∈𝐼}).

Theorem 1.5. Let ℱ_𝜇 be a fuzzy family of 𝜇-closed sets and define 𝒯_ℱ𝜇∶[𝜇]→ℒ^′ by 𝒯_ℱ𝜇(𝜂)=ℱ_𝜇(𝜂^′). Then 𝒯_ℱ𝜇 is a 𝜇-STS on 𝑋.

Proof. The proof is similar to the previous theorem.

Corollary 1.6. Let 𝒯_𝜇 be a 𝜇-STS and ℱ_𝜇 a fuzzy family of 𝜇-closed sets. Then 𝒯_{ℱ𝒯𝜇=𝒯𝜇} and ℱ_{𝒯ℱ𝜇=ℱ𝜇}.

Proof. Suppose 𝜆,𝜂∈[𝜇] then we have 𝒯_ℱ𝒯𝜇(𝜆)=ℱ_𝒯𝜇(𝜆^′)=𝒯_𝜇(𝜆) and ℱ_𝒯ℱ𝜇(𝜂)=𝒯_ℱ𝜇(𝜂^′)=ℱ_𝜇(𝜂).

Example 1.7. Let 𝑋 be the set of all differentiable real-valued functions on (1,∞), with positive derivative of order one and let ℒ be the set of real-valued functions defined on (1,∞). Let 𝜇∶𝑋→ℒ be defined by 𝜇(𝑓)=𝑓^′+Exp, where Exp is the exponential function. For nonnegative integer 𝑛 define 𝜆^𝑛∶𝑋→ℒ by (𝜆^𝑛(𝑓))(𝑥)=𝑓^′(𝑥)+^𝑛_𝑖=1𝑥^𝑖−1.(𝑖−1)! (3) If we take ℒ^′=[0,1] and define 𝒯_𝜇∶[𝜇]→[0,1] by 𝒯_𝜇(𝜒_∅)=𝒯_𝜇(𝜇)=1;  𝒯_𝜇(𝜆^𝑛)=1−1/𝑛 for 𝑛=1,2,…. Then (𝑋,𝜇,𝒯_𝜇) is a 𝜇-STS. Since 𝜆^𝑛∩𝜆^𝑚=𝜆^𝑚,𝜆^𝑛∪𝜆^𝑚=𝜆^𝑛, where 𝑛>𝑚 and ∪𝜆^𝑛𝑖=𝜇 whenever 𝑛_𝑖 tends to +∞, so 𝒯_𝜇(𝜆^𝑛∩𝜆^𝑚)=𝒯_𝜇(𝜆^𝑚)≥𝒯_𝜇(𝜆^𝑛)𝒯_𝜇(𝜆^𝑚), (4) and for 𝐼⊆ℕ we find 𝒯_𝜇(∪{𝜆^𝑛∶𝑛∈𝐼})≥{𝒯_𝜇(𝜆^𝑛)∶𝑛∈𝐼}. (5)

Definition 1.8. Let 𝒯¹_𝜇 and 𝒯²_𝜇 be two 𝜇-smooth topological spaces on 𝑋. We say that 𝒯¹_𝜇 is finer than 𝒯²_𝜇 or 𝒯²_𝜇 is coarser than 𝒯¹_𝜇 and denoted by 𝒯¹_𝜇≻𝒯²_𝜇 if 𝒯¹_𝜇(𝜆)≥𝒯²_𝜇(𝜆) for every 𝜆∈[𝜇].

Theorem 1.9. Let {𝒯^𝑖_𝜇∶𝑖∈𝐼} be a family of 𝜇-STS on 𝑋. Then 𝒯_𝜇=⋀_𝑖∈𝐼𝒯^𝑖_𝜇 is also 𝜇-STS on 𝑋, where (_𝑖∈𝐼𝒯^𝑖_𝜇)(𝜆)=_𝑖∈𝐼𝒯^𝑖_𝜇(𝜆). (6)

Proof. (i)It is clear.(ii)For every 𝜆,𝜂∈[𝜇], 𝒯_𝜇(𝜆∩𝜂)=_𝑖∈𝐼𝒯^𝑖_𝜇(𝜆∩𝜂)≥_𝑖∈𝐼(𝒯^𝑖_𝜇(𝜆)𝒯^𝑖_𝜇=(𝜂))_𝑖∈𝐼(𝒯^𝑖_𝜇(𝜆))(_𝑖∈𝐼𝒯^𝑖_𝜇(𝜆))=𝒯_𝜇(𝜆)𝒯_𝜇(𝜂). (7)(iii)For Γ, 𝒯_𝜇(∪{𝜆_𝑗∶𝑗∈Γ})=_𝑖∈𝐼𝒯^𝑖_𝜇(∪{𝜆_𝑗≥∶𝑗∈Γ})_𝑖∈𝐼{𝒯^𝑖_𝜇(𝜆_𝑗=()∶𝑗∈Γ}_𝑖∈𝐼{𝒯^𝑖_𝜇(𝜆_𝑗=)∶𝑗∈Γ}){𝒯_𝜇(𝜆_𝑗)∶𝑗∈Γ}. (8)

Let 𝐴 be a subset of 𝑋 and 𝜆∈[𝜇]. The restriction of 𝜆 on 𝐴 is denoted by 𝜆|_𝐴.

Theorem 1.10. Let (𝑋,𝜇,𝒯_𝜇) be a 𝜇-STS and 𝐴⊂𝑋. Define a mapping 𝒯_𝜇|𝐴∶[𝜇]→ℒ^′ by 𝒯_𝜇|𝐴(𝜆)=∨{𝒯_𝜇(𝜂)∶𝜂∈[𝜇],𝜂|_𝐴=𝜆}. Then 𝒯_𝜇|𝐴 is a 𝜇-STS on 𝐴.

Proof. (i)It is clear that 𝒯_𝜇|𝐴(𝜒_∅)=𝒯_𝜇|𝐴(𝜇)=1. (ii)∀𝜆₁,𝜆₂∈ℒ^𝐴,𝜆₁,𝜆₂⊂𝜇. 𝒯_𝜇|𝐴(𝜆₁)𝒯_𝜇|𝐴(𝜆₂)=∨{𝒯_𝜇(𝜂₁)∶𝜂₁∈[𝜇],𝜂₁|_𝐴=𝜆₁}∨{𝒯_𝜇(𝜂₂)∶𝜂₂∈[𝜇],𝜂₂|_𝐴=𝜆₂}=∨{𝒯_𝜇(𝜂₁)𝒯_𝜇(𝜂₂)∶𝜂₁,𝜂₂∈[𝜇],𝜂₁∩𝜂₂|_𝐴=𝜆₁∩𝜆₂}≤∨{𝒯_𝜇(𝜂₁∩𝜂₂)∶𝜂₁,𝜂₂∈[𝜇],𝜂₁∩𝜂₂|_𝐴=𝜆₁∩𝜆₂}=𝒯_𝜇|𝐴(𝜆₁∩𝜆₂). (9)(iii)∀𝑖∈𝐼,𝒯_𝜇|𝐴{𝜆_𝑖∶𝑖∈𝐼}=∨{𝒯_𝜇(𝜂_𝑖)∶𝜂_𝑖∈[𝜇],𝜂_𝑖|_𝐴=𝜆_𝑖}. So 𝒯_𝜇|𝐴(∪{𝜆_𝑖∶𝑖∈𝐼})=∨{𝒯_𝜇(∪{𝜂_𝑖∶𝑖∈𝐼})∶𝜂_𝑖∈[𝜇],∪𝜂_𝑖|_𝐴=∪𝜆_𝑖}≤∨{{𝒯_𝜇(𝜂_𝑖)∶𝑖∈𝐼}∶𝜂_𝑖∈[𝜇],∪𝜂_𝑖|_𝐴=∪𝜆_𝑖}=𝒯_𝜇|𝐴{𝜆_𝑖∶𝑖∈𝐼}. (10)

Definition 1.11. The 𝜇-STS (𝐴,𝜇,𝒯_𝜇|𝐴) is called a subspace of (𝑋,𝜇,𝒯_𝜇) and 𝒯_𝜇|𝐴 is called the induced 𝜇-STS on 𝐴 from 𝒯_𝜇.

Theorem 1.12. Let (𝐴,𝜇,𝒯_𝜇|𝐴) be a 𝜇-smooth subspace of (𝑋,𝜇,𝒯_𝜇) and 𝜆∈ℒ^𝐴,𝜆∈[𝜇]. Then(a)ℱ_{𝒯𝐴𝜇|}(𝜆)=∨{ℱ_𝒯𝜇(𝜂)∶𝜂∈[𝜇],𝜂|_𝐴=𝜆},(b)if 𝐵⊂𝐴⊂𝑋, then 𝒯_𝜇|𝐵=(𝒯_𝜇|𝐴)_𝜇|𝐵.

Proof. (a)we have ℱ_{𝒯𝐴𝜇|}(𝜆)=𝒯_𝜇|𝐴(𝜆^′)=∨{𝒯_𝜇(𝜂)∶𝜂∈[𝜇],𝜂|_𝐴=𝜆^′}=∨{𝒯_𝜇(𝜂)∶𝜂^′∈[𝜇],𝜂^′|_𝐴=𝜆}=∨{ℱ_𝒯𝜇(𝜂^′)∶𝜂^′∈[𝜇],𝜂^′|_𝐴=𝜆}=∨{ℱ_𝒯𝜇(𝜉)∶𝜉∈[𝜇],𝜉|_𝐴=𝜆}. (11)(b)we have 𝒯_𝜇|𝐵(𝜆)=∨{𝒯_𝜇(𝜂)∶𝜂∈[𝜇],𝜂|_𝐵=𝜆}=∨{∨{𝒯_𝜇(𝜂)∶𝜂∈[𝜇],𝜂|_𝐴=𝜉}∶𝜉∈[𝜇]|_𝐴,𝜉|_𝐵=𝜆}=∨{𝒯_𝜇|𝐴(𝜉)∶𝜉∈[𝜇]|_𝐴,𝜉|_𝐵=𝜆}=(𝒯_𝜇|𝐴)_𝜇|𝐵. (12)

2. Relative Smooth Continuous Maps

The concept of continuity has been studied by Chang, Ramadan [4, 5] but here we shall study this concept from a different point of view.

Definition 2.1. Let ℎ∶ℒ→ℒ^′ be a linear isomorphism of vector lattices (or an order preserving one-to-one mapping when ℒ and ℒ^′ are copies of [0,1]) and (𝑋,𝜇,𝒯_𝜇),(𝑌,𝛾,𝒯_𝛾)  𝜇-STS and 𝛾-STS, respectively. A function 𝑓∶𝑋→𝑌 is called (ℎ,𝜇,𝛾)-smooth fuzzy continuous if 𝒯_𝜇(𝑓⁻¹(𝜂))≥𝒯_𝛾(𝜂) for all 𝜂∈𝒯_𝛾, where 𝑓⁻¹(𝜂)(𝑥)=(ℎ∘𝜂)(𝑓(𝑥))∩𝜇(𝑥) for all 𝑥∈𝑋. 𝑓⁻¹(𝛾)∩𝜇 is called the inverse image of 𝛾 relative to 𝜇.

Remark 2.2. When ℒ=ℒ^′=[0,1] and 𝜇=𝜒_𝑋 then the (𝑋,𝜒_𝑋,𝜏_𝜒𝑋), 𝜒_𝑋-RST coincides with the fuzzy topological space (𝑋,𝜏) defined by Chang [5–7].

Theorem 2.3. Let ℒ=ℒ^′=[0,1] and 𝑓∶𝑋→𝑌 be (𝐼,𝜒_𝑋,𝜒_𝑌)-fuzzy continuous, where 𝐼∶ℒ→ℒ^′ is the identity function. Then 𝑓 is continuous in Chang's view.

Proof. In Remark 2.2 we considered (𝑋,𝜒_𝑋,𝒯_𝜒𝑋) and (𝑌,𝜒_𝑌,𝒯_𝜒𝑌) as fuzzy topological spaces. Now let 𝛾 be an open set of smooth topology 𝒯_𝜒𝑌. Then 𝒯_𝜒𝑋(𝑓⁻¹(𝛾))=𝒯_𝜒𝑋((𝐼⁻¹∘𝛾∘𝑓)∩𝜒_𝑋)=𝒯_𝜒𝑋(𝛾∘𝑓)≥𝒯_𝜒𝑋(𝛾). (13) So 𝑓 is a fuzzy continuous function.

Theorem 2.4. Let 𝜇∈ℒ^𝑋,𝛾∈ℒ^′𝑌 where ℒ and ℒ^′ are copies of [0,1] and 𝑓∶𝑋→𝑌 a (ℎ,𝜇,𝛾)-fuzzy continuous functions where 𝑓⁻¹(𝛾)=𝜇. Then for every 𝛾-closed fuzzy set 𝜂,𝑓⁻¹(𝜂) is a 𝜇-closed fuzzy set.

Proof. Let 𝜂 be 𝛾-closed set. Then 𝜂^′ is a 𝛾-open set and we have (𝑓⁻¹(𝜂))^′=𝜇−inf{ℎ⁻¹∘𝜂∘𝑓,𝜇}=sup{𝜇−ℎ⁻¹∘𝜂∘𝑓,0}=𝜇−ℎ⁻¹∘𝜂∘𝑓=𝜇−𝑓⁻¹(𝜂). (14) Hence 𝒯_𝜇(𝑓⁻¹(𝜂^′))=𝒯_𝜇(𝑓⁻¹(𝛾−𝜂))=𝒯_𝜇(ℎ⁻¹∘(𝛾−𝜂)∘𝑓∩𝜇)=𝒯_𝜇((ℎ⁻¹∘𝛾∘𝑓−ℎ⁻¹∘𝜂∘𝑓)∩𝜇)=𝒯_𝜇(𝑓⁻¹(𝛾)−𝑓⁻¹(𝜂))=𝒯_𝜇(𝜇−𝑓⁻¹(𝜂))=ℱ_𝒯𝜇(𝑓⁻¹(𝜂)). (15) So 𝑓⁻¹(𝜂) is a 𝜇-closed fuzzy set.

Theorem 2.5. Let (𝑋_𝑖,𝜇_𝑖,𝒯_𝜇𝑖) be relative smooth topological spaces for 𝑖=1,2,3. If 𝑓∶𝑋₁→𝑋₂ and 𝑔∶𝑋₂→𝑋₃ are relative smooth continuous maps and 𝜇₁=𝑓⁻¹(𝜇₂) then so is 𝑔∘𝑓.

Proof. Using the relative smooth continuity of 𝑔 and 𝑓 it follows that 𝒯_𝜇1((𝑔∘𝑓)⁻¹(𝜂))=𝒯_𝜇1(𝑓⁻¹(𝑔⁻¹(𝜂)))≥𝒯_𝜇2(𝑔⁻¹(𝜂))≥𝒯_𝜇3(𝜂). (16) Since for every 𝑥∈𝑋,(𝑔∘𝑓)⁻¹(𝜂)(𝑥)=𝜂(𝑔∘𝑓)(𝑥)∩𝜇₁(𝑥)=𝜂(𝑔∘𝑓)(𝑥)∩𝜇₁(𝑥)∩𝜇₂(𝑓(𝑥))=𝑔⁻¹(𝜂(𝑓(𝑥))∩𝜇₁(𝑥))=𝑓⁻¹(𝑔⁻¹(𝜂)(𝑥))=(𝑓⁻¹∘𝑔⁻¹)(𝜂)(𝑥). (17)

Theorem 2.6. Let (𝑋,𝜇,𝒯_𝜇) and (𝑌,𝜈,𝒯_𝜈) be two relative smooth topological spaces, 𝑓∶𝑋→𝑌 a relative smooth continuous map, 𝐴⊂𝑋, and 𝑓⁻¹(𝜈)=𝜇. Then the 𝑓|_𝐴∶(𝐴,𝜇|_𝐴,𝒯_𝜇|𝐴)→(𝑌,𝜈,𝒯_𝜈) is also relative smooth continuous.

Proof. For each 𝜂∈[𝜈],𝒯_𝜇|𝐴((𝑓|_𝐴)⁻¹(𝜂))=∨{𝒯_𝜇(𝜆)∶𝜆∈[𝜇],𝜆|_𝐴=(𝑓|_𝐴)⁻¹(𝜂)}≥𝒯_𝜇(𝑓⁻¹(𝜂))≥𝒯_𝜈(𝜂). (18)

3. The Representation of a Relative Smooth Topology

Now we study the representation of a relative smooth topology 𝒯_𝜇.

Let (𝑋,𝜇,𝒯_𝜇) be a 𝜇-STS, 𝛼∈ℒ^′. Then we define

𝒯^𝛼_𝜇={𝜆∈[𝜇]∶𝒯_𝜇(𝜆)≥𝛼}. (19)

Theorem 3.1. Let (𝑋,𝜇,𝒯_𝜇) be a 𝜇-STS. Then for every 𝛼>0,𝒯^𝛼_𝜇 is a relative topological space. Moreover 𝛼₁≤𝛼₂ implies 𝒯^𝛼1_𝜇≥𝒯^𝛼2_𝜇.

Proof. It is clear that 𝜒_∅,𝜇∈𝒯^𝛼_𝜇. When 𝜆,𝜂∈𝒯^𝛼_𝜇, we have 𝒯_𝜇(𝜆)≥𝛼,𝒯_𝜇(𝜂)≥𝛼, (20) and so 𝒯_𝜇(𝜆∩𝜂)≥𝒯_𝜇(𝜆)𝒯_𝜇(𝜂)≥𝛼. (21) This implies that 𝜆∩𝜂∈𝒯^𝛼_𝜇. When 𝜆_𝑗∈𝒯^𝛼_𝜇 for each 𝑗∈𝐼 we have 𝒯_𝜇(∪𝜆_𝑗)≥𝒯_𝜇(𝜆_𝑗)≥𝛼. (22) Hence ∪𝜆_𝑗∈𝒯^𝛼_𝜇. So 𝒯^𝛼_𝜇 is a relative topology.
The second part is trivial to verify, since for 𝜆∈𝒯^𝛼2_𝜇, 𝒯_𝜇≥𝛼₂≥𝛼₁,𝜆∈𝒯^𝛼1_𝜇, so 𝒯^𝛼1_𝜇≥𝒯^𝛼2_𝜇.

Theorem 3.2. Let 𝒯^𝛼_𝜇,𝛼∈(0,1] be a family of 𝜇-fuzzy topology on 𝑋 such that 𝛼₁≥𝛼₂ implies 𝒯^𝛼1_𝜇≤𝒯^𝛼2_𝜇. Let 𝜏 be the ℒ^′-fuzzy set built by 𝜏_𝜇(𝜆)=∨{𝛼∶𝜆∈𝒯^𝛼_𝜇}. Then 𝜏_𝜇 is a 𝜇-smooth topology.

Proof. (a)𝜏_𝜇(𝜒_∅)=𝜏_𝜇(𝜇)=1 by the definition.(b)For every 𝜆,𝜂∈[𝜇] and 𝛼>0 if 𝜆,𝜂∈𝒯^𝛼_𝜇 then 𝜆∩𝜂∈𝒯^𝛼_𝜇. Therefore {𝛼∶𝜆∩𝜂∈𝒯^𝛼_𝜇}⊃{𝛼∶𝜆∈𝒯^𝛼_𝜇,𝜂∈𝒯^𝛼_𝜇} (23) implies that 𝜏_𝜇(𝜆∩𝜂)≥𝜏_𝜇(𝜆)𝜏_𝜇(𝜂). (24)(c)If every 𝜆_𝑖∈𝒯^𝛼_𝜇 then ∪𝜆_𝑖∈𝒯^𝛼_𝜇. Since {𝛼∶∪𝜆_𝑖∈𝒯^𝛼_𝜇}⊃{𝛼∶∪𝜆_𝑖∈𝒯^𝛼_𝜇,∀𝑖∈𝐼}, (25) then 𝜏_𝜇(∪_𝑖∈𝐼𝜆_𝑖)≥_𝑖∈𝐼𝜏_𝜇(𝜆_𝑖). (26)

For 𝜏_𝜇 being a relative ℒ^′-fuzzy set, with ℒ^′=[0,1], we can state a representation theorem.

Theorem 3.3. Let 𝒯_𝜇 be a relative smooth topology and 𝒯^𝛼_𝜇 the 𝛼 cut of 𝒯_𝜇. From the families of relative fuzzy topologies 𝒯^𝛼_𝜇 one built 𝒯_1𝜇(𝜆)=∨{𝛼∶𝜆∈𝒯^𝛼_𝜇}. Then 𝒯_1𝜇=𝒯_𝜇.

Proof. The proof is trivial from the preceding results and the well-known fact that ∨{𝛼∶𝜆∈𝒯^𝛼_𝜇}=∨{𝛼∶𝒯_𝜇(𝜆)≥𝛼}=𝒯_𝜇(𝜆). (27)

Definition 3.4. Let 𝜏 be a Chang fuzzy topology on 𝑋. Then a 𝜇-smooth topology 𝒯_𝜇 on 𝑋 is said to be compatible with 𝜏 if 𝜏={𝜆∈ℒ^𝑋∶𝒯_𝜇(𝜆∩𝜇)>0}.

Example 3.5. Let 𝑋 be a nonempty set and 𝒯_𝜇∶[𝜇]→ℒ^′ be a mapping defined by 𝒯_𝜇(𝜇)=𝒯_𝜇(𝜒_∅)=1,𝒯_𝜇(𝜆)=0 for every 𝜆∈[𝜇]⧵{𝜒_∅,𝜇}.

It is clear that 𝒯_𝜇 is the only relative smooth topology on 𝑋 compatible with the indiscrete fuzzy topology of Chang.

Example 3.6. Let 𝑋 be a nonempty set and define a mapping 𝒯_𝜇∶[𝜇]→ℒ^′ by 𝒯_𝜇(𝜇)=𝒯_𝜇(𝜒_∅)=1,𝒯_𝜇(𝜆)=𝛼 for every 𝜆∈[𝜇]⧵{𝜒_∅,𝜇}.

It is clear that 𝒯_𝜇 is a 𝜇-smooth topology on 𝑋 compatible with the discrete fuzzy topology of Chang.

4. Relative Smooth Compactness

Definition 4.1. Let (𝑋,𝜇,𝒯_𝜇) be a 𝜇-STS. 𝜆∈[𝜇],𝒜,ℬ⊂[𝜇].𝒜 is called a relative cover of 𝜆, if ∨𝒜≥𝜆 particularly, 𝒜 is called a cover (𝑋,𝜇,𝒯_𝜇) if 𝒜 is a cover of 𝜇.𝒜 is called a 𝜇-open cover of 𝜆, if 𝒜 is a family of 𝜇-open and 𝒜 is a cover of 𝜆.

For a cover 𝒜 of 𝜆,ℬ is called a subcover of 𝜆, if ℬ⊂𝒜 and ℬ is still a cover of 𝜆.

Definition 4.2. Let (𝑋,𝜇,𝒯_𝜇) be a 𝜇-STS. For every 𝛼∈[0,1), a family 𝒜⊂[𝜇] is called an 𝛼-cover, if for every 𝜆∈𝒜,𝒯_𝜇(𝜆)≥𝛼; 𝒜 is called a 𝜇-open 𝛼-cover if 𝒜 is a family of 𝜇-open set and 𝒜 is a 𝛼-cover; 𝒜₀⊂[𝜇] is called a sub-𝛼-cover of 𝒜 if 𝒜₀⊂𝒜 and 𝒜₀ is an 𝛼-cover.

Definition 4.3. Let 𝛼∈[0,1). A 𝜇-STS (𝑋,𝜇,𝒯_𝜇) is called 𝛼-compact if every 𝜇-open 𝛼-cover has a finite sub-𝛼-cover.

Theorem 4.4. Let 𝑓∶(𝑋,𝜇,𝒯_𝜇)→(𝑌,𝜈,𝒯_𝜈) be an onto 𝜇-smooth continuous mapping and 𝑓⁻¹(𝜈)=𝜇. If (𝑋,𝜇,𝒯_𝜇) is 𝛼-compact then so is (𝑌,𝜈,𝒯_𝜈).

Proof. Let {𝜆_𝑖∶𝑖∈𝐼} be a 𝜈-open 𝛼-cover of 𝜈. Now consider the family {𝑓⁻¹(𝜆_𝑖)∶𝑖∈𝐼}, since 𝑓 is 𝜇-smooth continuous, we have 𝜆∈𝒯^𝛼_𝜈⇔𝒯_𝜈≥𝛼⇔𝒯_𝜇(𝑓⁻¹(𝜆))≥𝛼⇔𝑓⁻¹(𝜆)∈𝒯^𝛼_𝜇. (28) It follows that {𝑓⁻¹(𝜆_𝑖)∶𝑖∈𝐼} is a 𝜇-open 𝛼-cover of 𝜇. Since (𝑋,𝜇,𝒯_𝜇) is 𝛼-compact there exists a finite subset 𝐼₀ of 𝐼 such that {𝑓⁻¹(𝜆_𝑖)∶𝑖∈𝐼₀} is a 𝜇-open 𝛼-cover of (𝑋,𝜇,𝒯_𝜇). Since 𝑓 is onto, then {𝜆_𝑖∶𝑖∈𝐼₀} is a 𝜈-open 𝛼-cover of (𝑌,𝜈,𝒯_𝜈), which concludes the proof.