Research Article On Topological Structures of Fuzzy Parametrized Soft

Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 164176, 8 pages http://dx.doi.org/10.1155/2014/164176

Research Article On Topological Structures of Fuzzy Parametrized Soft Sets Serkan Atmaca and Edris Zorlutuna Department of Mathematics, Cumhuriyet University, 58140 Sivas, Turkey Correspondence should be addressed to Serkan Atmaca; [email protected] Received 10 February 2014; Accepted 28 March 2014; Published 28 April 2014 Academic Editor: Naim Cagman Copyright © 2014 S. Atmaca and ˙Idris Zorlutuna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the topological structure of fuzzy parametrized soft sets and fuzzy parametrized soft mappings. We define the notion of quasi-coincidence for fuzzy parametrized soft sets and investigated its basic properties. We study the closure, interior, base, continuity, and compactness and properties of these concepts in fuzzy parametrized soft topological spaces.

1. Introduction In 1965, after Zadeh [1] generalized the usual notion of a set with the introduction of fuzzy set, the fuzzy set was carried out in the areas of general theories and applied to many real life problems in uncertain, ambiguous environment. In this manner, in 1968, Chang [2] gave the definition of fuzzy topology and introduced the many topological notions in fuzzy setting. In 1999, Molodtsov [3] introduced the concept of soft set theory which is a completely new approach for modelling uncertainty and pointed out several directions for the applications of soft sets, such as game theory, perron integrations, and smoothness of functions. To improve this concept, many researchers applied this concept on topological spaces (e.g., [4–9]), group theory, ring theory (e.g., [10–14]), and also decision making problems (e.g., [15–18]). Recently, researchers have combined fuzzy set and soft set to generalize the spaces and to solve more complicated problems. By this way, many interesting applications of soft set theory have been expanded. First combination of fuzzy set and soft set is fuzzy soft set and it was given by Maji et al. [19]. Then fuzzy soft set theory has been applied in several directions, such as topology (e.g., [20–23]), various algebraic structures (e.g., [24, 25]), and especially decision making (e.g., [26–29]). Another combination of fuzzy set ̆ and soft set was given by C ¸ agman et al. [30] who called it fuzzy parametrized soft set (for short FP-soft set). In that ̆ paper, C ¸ agman et al. defined operations on FP-soft sets

̆ and improved several results. After that, C ¸ agman and Deli [31, 32] applied FP-soft sets to define some decision making methods and applied these methods to problems that contain uncertainties and fuzzy object. In the present paper, we consider the topological structure of FP-soft sets. Firstly, we give some basic ideas of FP-soft sets and also studied results. We define FP-soft quasi-coincidence, as a generalization of quasi-coincidence in fuzzy manner [33], and use this notion to characterize concepts of FPsoft closure and FP-soft base in FP-soft topological spaces. We also introduce the notion of mapping on FP-soft classes and investigate the properties of FP-soft images and FP-soft inverse images of FP-soft sets. We define FP-soft topology in Chang’s sense. We study the FP-soft closure and FP-soft interior operators and properties of these concepts. Lastly we define FP-soft continuous mappings and we show that image of a FP-soft compact space is also FP-soft compact.

2. Preliminaries Throughout this paper 𝑋 denotes initial universe, 𝐸 denotes the set of all possible parameters which are attributes, characteristic, or properties of the objects in 𝑋, and the set of all subsets of 𝑋 will be denoted by 𝑃(𝑋). Definition 1 (see [1]). A fuzzy set 𝐴 in 𝑋 is a function defined as follows: 𝜇 (𝑥) : 𝑥 ∈ 𝑋} , (1) 𝐴={ 𝐴 𝑥 where 𝜇𝐴 : 𝑋 → [0, 1].

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Here 𝜇𝐴 is called the membership function of 𝐴, and the value 𝜇𝐴 (𝑥) is called the grade of membership of 𝑥 ∈ 𝑋. This value represents the degree of 𝑥 belonging to the fuzzy set 𝐴. A fuzzy point in 𝑋, whose value is 𝛼 (0 < 𝛼 ≤ 1) at the support 𝑥 ∈ 𝑋, is denoted by 𝑥𝛼 . A fuzzy point 𝑥𝛼 ∈ 𝐴, where 𝐴 is fuzzy set in 𝑋 if 𝛼 ≤ 𝜇𝐴 (𝑥). Definition 2 (see [3]). A pair (𝐹, 𝐸) is called a soft set over 𝑋 if 𝐹 is a mapping defined by 𝐹 : 𝐸 → 𝑃(𝑋). In other words, a soft set is a parametrized family of subsets of the set 𝑋. Each set 𝐹(𝑒), 𝑒 ∈ 𝐸, from this family may be considered as the set of 𝑒-elements of the soft set (𝐹, 𝐸). Definition 3 (see [30]). Let 𝐴 be a fuzzy set over 𝐸. A FP-soft set 𝐹𝐴 on the universe 𝑋 is defined as follows: 𝐹𝐴 = {(

𝜇𝐴 (𝑒) , 𝑓𝐴 (𝑒)) : 𝑒 ∈ 𝐸, 𝑓𝐴 (𝑒) ∈ 𝑃 (𝑋) , 𝑒 (2)

𝜇𝐴 (𝑒) ∈ [0, 1]} , where the function 𝑓𝐴 : 𝐸 → 𝑃(𝑋) is called approximate function such that 𝑓𝐴(𝑒) = ⌀ if 𝜇𝐴 (𝑒) = 0, and the function 𝜇𝐴 : 𝐸 → [0, 1] is called membership function of the set 𝐴. From now on, the set of all FP-soft sets over 𝑋 will be denoted by FPS(𝑋, 𝐸). Definition 4 (see [30]). Let 𝐹𝐴 ∈ FPS(𝑋, 𝐸). (1) 𝐹𝐴 is called the empty FP-soft set if 𝜇𝐴 (𝑒) = 0 for every 𝑒 ∈ 𝐸, denoted by 𝐹⌀ . (2) 𝐹𝐴 is called 𝐴-universal FP-soft set if 𝜇𝐴 (𝑒) = 1 and 𝑓𝐴(𝑒) = 𝑋 for all 𝑒 ∈ 𝐴, denoted by 𝐹𝐴̃. If 𝐴 = 𝐸, then 𝐴-universal FP-soft set is called universal FP-soft set, denoted by 𝐹𝐸̃. Definition 5 (see [30]). Let 𝐹𝐴, 𝐹𝐵 ∈ FPS(𝑋, 𝐸). (1) 𝐹𝐴 is called a FP-soft subset of 𝐹𝐵 if 𝐴 ≤ 𝐵 and ̃ 𝐹𝐵 . 𝑓𝐴(𝑒) ⊆ 𝑓𝐵 (𝑒) for every 𝑒 ∈ 𝐸 and one writes 𝐹𝐴⊂ (2) 𝐹𝐴 and 𝐹𝐵 are said to be equal, denoted by 𝐹𝐴 = 𝐹𝐵 if ̃ 𝐹𝐵 and 𝐹𝐵 ⊂ ̃ 𝐹𝐴. 𝐹𝐴⊂ ̃ 𝐹𝐵 , is the FP(3) The union of 𝐹𝐴 and 𝐹𝐵 , denoted by 𝐹𝐴∪ soft set, defined by the membership and approximate functions 𝜇𝐴∪𝐵 (𝑒) = max{𝜇𝐴 (𝑒), 𝜇𝐵 (𝑒)} and 𝑓𝐴∪𝐵 (𝑒) = 𝑓𝐴(𝑒) ∪ 𝑔𝐵 (𝑒) for every 𝑒 ∈ 𝐸, respectively. ̃ 𝐺𝐵 , (4) The intersection of 𝐹𝐴 and 𝐹𝐵 , denoted by 𝐹𝐴 ∩ is the FP-soft set, defined by the membership and approximate functions 𝜇𝐴∩𝐵 (𝑒) = min{𝜇𝐴 (𝑒), 𝜇𝐵 (𝑒)} and 𝑓𝐴∩𝐵 (𝑒) = 𝑓𝐴(𝑒) ∩ 𝑔𝐵 (𝑒) for every 𝑒 ∈ 𝐸, respectively. Definition 6 (see [30]). Let 𝐹𝐴 ∈ FPS(𝑋, 𝐸). Then the complement of 𝐹𝐴, denoted by 𝐹𝐴𝑐 , is the FP-soft set, defined by the membership and approximate functions 𝜇𝐴𝑐 (𝑒) = 1 − 𝜇𝐴(𝑒) and 𝑓𝐴𝑐 (𝑒) = 𝑋 − 𝑓𝐴(𝑒) for every 𝑒 ∈ 𝐸, respectively. Clearly (𝐹𝐴𝑐 )𝑐 = 𝐹𝐴, 𝐹𝐸𝑐̃ = 𝐹⌀ , and 𝐹⌀𝑐 = 𝐹𝐸̃ .

Proposition 7 (see [30]). Let 𝐹𝐴, 𝐹𝐵 , and 𝐹𝐶 ∈ 𝐹𝑃𝑆(𝑋, 𝐸). Then ̃ 𝐹𝐵 )𝑐 = 𝐹𝐴𝑐 ∩ ̃ 𝐹𝐵𝑐 ; (1) (𝐹𝐴 ∪ ̃ 𝐹𝐵 )𝑐 = 𝐹𝐴𝑐 ∪ ̃ 𝐹𝐵𝑐 ; (2) (𝐹𝐴 ∩ ̃ 𝐹𝐴 = 𝐹𝐴, 𝐹𝐴 ∪ ̃ 𝐹𝐴 = 𝐹𝐴; (3) 𝐹𝐴 ∩ ̃ 𝐹⌀ = 𝐹⌀ , 𝐹𝐴 ∩ ̃ 𝐹𝐸̃ = 𝐹𝐴; (4) 𝐹𝐴 ∩

̃ 𝐹𝐵 = 𝐹𝐵 ∩ ̃ 𝐹𝐴, 𝐹𝐴 ∪ ̃ 𝐹𝐵 = 𝐹𝐵 ∪ ̃ 𝐹𝐴; (5) 𝐹𝐴 ∩ ̃ (𝐹𝐵 ∩ ̃ 𝐹𝐶) = (𝐹𝐴 ∩ ̃ 𝐹𝐵 )∩ ̃ (𝐹𝐵 ∪ ̃ 𝐹𝐶) ̃ 𝐹𝐶, 𝐹𝐴∪ (6) 𝐹𝐴∩ ̃ 𝐹𝐵 )∪ ̃ 𝐹𝐶; (𝐹𝐴 ∪ ̃ 𝐹⌀ = 𝐹𝐴, 𝐹𝐴∪ ̃ 𝐹𝐸̃ = 𝐹𝐸̃ . (7) 𝐹𝐴∪

=

3. Some Properties of FP-Soft Sets and FP-Soft Mappings Definition 8. Let 𝐽 be an arbitrary index set and 𝐹𝐴 𝑖 ∈ FPS(𝑋, 𝐸) for all 𝑖 ∈ 𝐽. ̃ 𝑖∈𝐽 𝐹𝐴 , is the FP(1) The union of 𝐹𝐴 𝑖 ’s, denoted by ∪ 𝑖 soft set, defined by the membership and approximate functions 𝜇∪𝑖∈𝐽 𝐴 𝑖 (𝑒) = sup𝑖∈𝐽 {𝜇𝐴 𝑖 (𝑒)} and 𝑓∪𝑖∈𝐽 𝐴 𝑖 (𝑒) = ∪𝑖∈𝐽 𝑓𝐴 𝑖 (𝑒) for every 𝑒 ∈ 𝐸, respectively. ̃ 𝑖∈𝐽 𝐹𝐴 , is the FP(2) The intersection of 𝐹𝐴 𝑖 ’s, denoted by ∩ 𝑖 soft set, defined by the membership and approximate functions 𝜇∩𝑖∈𝐽 𝐴 𝑖 (𝑒) = inf 𝑖∈𝐽 {𝜇𝐴 𝑖 (𝑒)} and 𝑓∩𝑖∈𝐽 𝐴 𝑖 (𝑒) = ∩𝑖∈𝐽 𝑓𝐴 𝑖 (𝑒) for every 𝑒 ∈ 𝐸, respectively. Proposition 9. Let 𝐽 be an arbitrary index set and 𝐹𝐴 𝑖 ∈ FPS(𝑋, 𝐸) for all 𝑖 ∈ 𝐽. Then ̃ 𝑖∈𝐽 𝐹𝐴𝑐 ; ̃ 𝑖∈𝐽 𝐹𝐴 )𝑐 = ∩ (1) (∪ 𝑖 𝑖 ̃ 𝑖∈𝐽 𝐹𝐴 )𝑐 = ∪ ̃ 𝑖∈𝐽 𝐹𝐴𝑐 . (2) (∩ 𝑖 𝑖 ̃ 𝑖∈𝐽 𝐹𝐴 )𝑐 and 𝐹𝐶 = ∩ ̃ 𝑖∈𝐽 𝐹𝐴𝑐 . Then for all Proof. (1) Put 𝐹𝐵 = (∪ 𝑖 𝑖 𝑒 ∈ 𝐸, 𝜇𝐵 (𝑒) = 1 − 𝜇∪𝑖∈𝐽 𝐴 𝑖 (𝑒) = 1 − sup {𝜇𝐴 𝑖 (𝑒)} 𝑖∈𝐽

= inf {1 − 𝜇𝐴 𝑖 (𝑒)} = inf {𝜇𝐴𝑐𝑖 (𝑒)} = 𝜇𝐶 (𝑒) , 𝑖∈𝐽

𝑖∈𝐽

𝑓𝐵 (𝑒) = 𝑋 − 𝑓∪𝑖∈𝐽 𝐴 𝑖 (𝑒) = 𝑋 − ∪ 𝑓𝐴 𝑖 (𝑒) 𝑖∈𝐽

(3)

= ∩ (𝑋 − 𝑓𝐴 𝑖 (𝑒)) = ∩ 𝑓𝐴𝑐𝑖 (𝑒) 𝑖∈𝐽

𝑖∈𝐽

= 𝑓∩𝑖∈𝐽 𝐴𝑐 (𝑒) = 𝑓𝐶 (𝑒) . 𝑖

This completes the proof. The other can be proved similarly. Definition 10. The FP-soft set 𝐹𝐴 ∈ FPS(𝑋, 𝐸) is called FP-soft point if 𝐴 is fuzzy singleton and 𝑓𝐴(𝑒) ∈ 𝑃(𝑋) for 𝑒 ∈ supp 𝐴. If 𝐴 = {𝑒}, 𝜇𝐴 (𝑒) = 𝛼 ∈ (0, 1], then one denotes this FP-soft point by 𝑒𝛼𝑓 .


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Definition 11. Let 𝑒𝛼𝑓 , 𝐹𝐴 ∈ FPS(𝑋, 𝐸). One says that 𝑒𝛼𝑓 ∈̃ 𝐹𝐴 read as 𝑒𝛼𝑓 belongs to the FP-soft set 𝐹𝐴 if 𝛼 ≤ 𝜇𝐴 (𝑒) and 𝑓(𝑒) ⊆ 𝑓𝐴(𝑒). Proposition 12. Every nonempty FP-soft set 𝐹𝐴 can be expressed as the union of all the FP-soft points which belong to 𝐹𝐴. Proof. This follows from the fact that any fuzzy set is the union of fuzzy points which belong to it [33]. Definition 13. Let 𝐹𝐴, 𝐹𝐵 ∈ FPS(𝑋, 𝐸). 𝐹𝐴 is said to be FP-soft quasi-coincident with 𝐹𝐵 , denoted by 𝐹𝐴𝑞𝐹𝐵 , if there exists 𝑒 ∈ 𝐸 such that 𝜇𝐴 (𝑒) + 𝜇𝐵 (𝑒) > 1 or 𝑓𝐴(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒). If 𝐹𝐴 is not FP-soft quasi-coincident with 𝐹𝐵 , then one writes 𝐹𝐴𝑞𝐹𝐵 . Definition 14. Let 𝑒𝛼𝑓 , 𝐹𝐴 ∈ FPS(𝑋, 𝐸). 𝑒𝛼𝑓 is said to be FP-soft quasi-coincident with 𝐹𝐴, denoted by 𝑒𝛼𝑓 𝑞𝐹𝐴, if 𝛼 + 𝜇𝐴 (𝑒) > 1 or 𝑓(𝑒) is not subset of 𝑓𝐴𝑐 (𝑒). If 𝑒𝛼𝑓 is not FP-soft quasicoincident with 𝐹𝐴, then one writes 𝑒𝛼𝑓 𝑞𝐹𝐴. Proposition 15. Let 𝐹𝐴, 𝐹𝐵 ∈ FPS(𝑋, 𝐸). Then the following are true: ̃ 𝐹𝐵 ⇔ 𝐹𝐴𝑞𝐹𝐵𝑐 ; (1) 𝐹𝐴⊆

Proposition 16. Let {𝐹𝐴 𝑖 : 𝑖 ∈ 𝐽} be a family of FP-soft sets in FPS(𝑋, 𝐸), where 𝐽 is an index set. Then 𝑒𝛼𝑓 is FP-soft quasĩ 𝑖∈𝐽 𝐹𝐴 if and only if there exists some 𝐹𝐴 ∈ coincident with ∪ 𝑖 𝑖 {𝐹𝐴 𝑖 : 𝑖 ∈ 𝐽} such that 𝑒𝛼𝑓 𝑞𝐹𝐴 𝑖 . Proof. Obvious. Definition 17. Let FPS(𝑋, 𝐸) and FPS(𝑌, 𝐾) be families of all FP-soft sets over 𝑋 and 𝑌, respectively. Let 𝑢 : 𝑋 → 𝑌 and 𝑝 : 𝐸 → 𝐾 be two functions. Then a FP-soft mapping 𝑓𝑢𝑝 : FPS(𝑋, 𝐸) → FPS(𝑌, 𝐾) is defined as follows. (1) For 𝐹𝐴 ∈ FPS(𝑋, 𝐸), the image of 𝐹𝐴 under the FPsoft mapping 𝑓𝑢𝑝 is the FP-soft set 𝐺𝑆 over 𝑌 defined by the approximate function, ∀𝑘 ∈ 𝐾, −1 { ∪−1 𝑢 (𝑓𝐴 (𝑒)) , if 𝑝 (𝑘) ≠ ⌀; 𝑒∈𝑝 (𝑘) 𝑔𝑆 (𝑘) = { otherwise, {⌀,

̃ 𝐹𝐵 ≠ 𝐹⌀ . (2) 𝐹𝐴𝑞𝐹𝐵 ⇒ 𝐹𝐴∩ (3) 𝐹𝐴𝑞𝐹𝐴𝑐 ; (5) 𝑒𝛼𝑓 ∈̃𝐹𝐴𝑐 ⇔ 𝑒𝛼𝑓 𝑞𝐹𝐴; ̃ 𝐹𝐵 ⇒ if 𝑒𝛼𝑓 𝑞𝐹𝐴, then 𝑒𝛼𝑓 𝑞𝐹𝐵 for all 𝑒𝛼𝑓 ∈ FPS(𝑋, 𝐸); (6) 𝐹𝐴⊆

(2) For 𝐺𝑆 ∈ FPS(𝑌, 𝐾), then the preimage of 𝐺𝑆 under the FP-soft mapping 𝑓𝑢𝑝 is the FP-soft set 𝐹𝐴 over 𝑋 defined by the approximate function, ∀𝑒 ∈ 𝐸, 𝑓𝐴 (𝑒) = 𝑢−1 (𝑔𝑆 (𝑝 (𝑒))) ,

Proof. (1) Consider ̃ 𝐹𝐵 ⇐⇒ ∀𝑒 ∈ 𝐸, 𝐹𝐴⊆ ⇐⇒ ∀𝑒 ∈ 𝐸, 𝑓𝐴 (𝑒) ⊆ 𝑓𝐵 (𝑒)

(4)

⇐⇒ ∀𝑒 ∈ 𝐸, 𝜇𝐴 (𝑒) + 1 − 𝜇𝐵 (𝑒) ≤ 1,

(6)

where 𝑝−1 (𝑆) = 𝐴 is fuzzy set in 𝐸.

𝑓𝐴 (𝑒) ⊆ 𝑓𝐵 (𝑒)

𝜇𝐴 (𝑒) − 𝜇𝐵 (𝑒) ≤ 0,

(5)

where 𝑝(𝐴) = 𝑆 is fuzzy set in 𝐾.

(4) 𝐹𝐴𝑞𝐹𝐵 ⇔ there exists an 𝑒𝛼𝑓 ∈̃𝐹𝐴 such that 𝑒𝛼𝑓 𝑞𝐹𝐵 ;

𝜇𝐴 (𝑒) ≤ 𝜇𝐵 (𝑒) ,

Conversely, suppose 𝑒𝛼𝑓 𝑞𝐹𝐵 for some 𝑒𝛼𝑓 ∈̃𝐹𝐴. Then 𝛼 + 𝜇𝐵 (𝑒) > 1 or 𝑓(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒). Therefore, we have 𝜇𝐴 (𝑒) + 𝜇𝐵 (𝑒) > 1 or 𝑓𝐴(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒) for 𝑒 ∈ 𝐸. This shows 𝐹𝐴𝑞𝐹𝐵 . (5) It is obvious from (1). (6) Let 𝑒𝛼𝑓 , 𝐹𝐴 ∈ FPS(𝑋, 𝐸) and 𝑒𝛼𝑓 𝑞𝐹𝐴. Then 𝛼 + 𝜇𝐴(𝑒) > 1 ̃ 𝐹𝐵 , 𝛼 + 𝜇𝐵 (𝑒) > 1 or or 𝑓(𝑒) is not subset of 𝑓𝐴𝑐 (𝑒). Since 𝐹𝐴⊆ 𝑓(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒). Hence we have 𝑒𝛼𝑓 𝑞𝐹𝐵 .

𝑓𝐴 (𝑒) ⊆ 𝑓𝐵 (𝑒)

⇐⇒ 𝐹𝐴𝑞𝐹𝐵𝑐 . (2) Let 𝐹𝐴𝑞𝐹𝐵 . Then there exists an 𝑒 ∈ 𝐸 such that 𝜇𝐴 (𝑒)+ 𝜇𝐵 (𝑒) > 1 or 𝑓𝐴(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒). If 𝜇𝐴 (𝑒) + 𝜇𝐵 (𝑒) > 1, 𝐴 ∧ 𝐵 ≠ 0𝐸 and the proof is easy. If 𝑓𝐴(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒), ̃ 𝐹𝐵 ≠ 𝐹⌀ . then 𝑓𝐴(𝑒) ∩ 𝑓𝐵 (𝑒) ≠ ⌀. Hence 𝐹𝐴∩ (3) Suppose 𝐹𝐴𝑞𝐹𝐴𝑐 . Then there exists 𝑒 ∈ 𝐸 such that 𝜇𝐴 (𝑒) + 𝜇𝐴𝑐 (𝑒) > 1 or 𝑓𝐴(𝑒) is not subset of (𝑓𝐴𝑐 (𝑒))𝑐 . But this is impossible. (4) If 𝐹𝐴𝑞𝐹𝐵 , then there exists an 𝑒 ∈ 𝐸 such that 𝜇𝐴 (𝑒) + 𝜇𝐵 (𝑒) > 1 or 𝑓𝐴(𝑒) is not subset of 𝑓𝐵𝑐 (𝑒). Put 𝛼 = 𝜇𝐴 (𝑒) and 𝑓(𝑒) = 𝑓𝐴(𝑒). Then we have 𝑒𝛼𝑓 ∈̃𝐹𝐴 and 𝑒𝛼𝑓 𝑞𝐹𝐵 .

If 𝑢 and 𝑝 are injective, then the FP-soft mapping 𝑓𝑢𝑝 is said to be injective. If 𝑢 and 𝑝 are surjective, then the FP-soft mapping 𝑓𝑢𝑝 is said to be surjective. The FP-soft mapping 𝑓𝑢𝑝 is called constant, if 𝑢 and 𝑝 are constant. Theorem 18. Let 𝑋 and 𝑌 be crips sets 𝐹𝐴, 𝐹𝐴 𝑖 ∈ FPS(𝑋, 𝐸), 𝐺𝑆 , 𝐺𝑆𝑖 ∈ FPS(𝑌, 𝐾) ∀𝑖 ∈ 𝐽, where 𝐽 is an index set. Let 𝑓𝑢𝑝 : FPS(𝑋, 𝐸) → FPS(𝑌, 𝐾) be a FP-soft mapping. Then, ̃ 𝐹𝐴 then 𝑓𝑢𝑝 (𝐹𝐴 )⊂ ̃ 𝑓𝑢𝑝 (𝐹𝐴 ); (1) if 𝐹𝐴 1 ⊂ 2 1 2 −1 −1 ̃ 𝐺𝑆 then 𝑓𝑢𝑝 ̃ 𝑓𝑢𝑝 (2) if 𝐺𝑆1 ⊂ (𝐺𝑆1 )⊂ (𝐺𝑆2 ); 2 −1 ̃ 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐹𝐴)); the equality holds if 𝑓𝑢𝑝 is injective; (3) 𝐹𝐴⊂ −1 ̃ 𝐺𝑆 ; the equality holds if 𝑓𝑢𝑝 is surjective; (4) 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐺𝑆 ))⊂

̃ 𝑖∈𝐽 𝐹𝐴 ) = ∪ ̃ 𝑖∈𝐽 𝑓𝑢𝑝 (𝐹𝐴 ); (5) 𝑓𝑢𝑝 (∪ 𝑖 𝑖 ̃ 𝑖∈𝐽 𝐹𝐴 )⊂ ̃∩ ̃ 𝑖∈𝐽 𝑓𝑢𝑝 (𝐹𝐴 ); the equality holds if 𝑓𝑢𝑝 is (6) 𝑓𝑢𝑝 (∩ 𝑖 𝑖 injective; −1 ̃ −1 ̃ 𝑖∈𝐽 𝑓𝑢𝑝 (∪𝑖∈𝐽 𝐺𝑆𝑖 ) = ∪ (𝐺𝑆𝑖 ); (7) 𝑓𝑢𝑝

4

The Scientific World Journal −1 ̃ −1 ̃ 𝑖∈𝐽 𝑓𝑢𝑝 (8) 𝑓𝑢𝑝 (∩𝑖∈𝐽 𝐺𝑆𝑖 ) = ∩ (𝐺𝑆𝑖 );

−1 −1 (𝐺𝑆 ) = 𝐹𝐴 and 𝑓𝑢𝑝 (𝐺𝑆𝑐 ) = 𝐹𝐵 . Then for all (9) Put 𝑓𝑢𝑝 𝑒 ∈ 𝐸,

−1 −1 (9) (𝑓𝑢𝑝 (𝐺𝑆 ))𝑐 = 𝑓𝑢𝑝 (𝐺𝑆𝑐 );

𝑓𝐵 (𝑒) = 𝑓𝑝−1 (𝑆𝑐 ) (𝑒) = 𝑓(𝑝−1 (𝑆))𝑐 (𝑒) = 𝑓𝐴𝑐 (𝑒) ,

̃ 𝑓𝑢𝑝 (𝐹𝐴𝑐 ); (10) (𝑓𝑢𝑝 (𝐹𝐴))𝑐 ⊂

where 𝑝−1 (𝑆) and 𝑝−1 (𝑆𝑐 ) are fuzzy sets over 𝐸. This shows that the approximate functions of 𝐹𝐵 and 𝐹𝐴𝑐 are equal. This completes the proof. −1 (𝐺𝐾̃ ). Then for all 𝑒 ∈ 𝐸, (11) Put 𝐹𝐴 = 𝑓𝑢𝑝

−1 (11) 𝑓𝑢𝑝 (𝐺𝐾̃ ) = 𝐹𝐸̃ ; −1 (12) 𝑓𝑢𝑝 (𝐺⌀ ) = 𝐹⌀ ;

̃ 𝐺𝐾̃ ; the equality holds if 𝑓𝑢𝑝 is surjective; (13) 𝑓𝑢𝑝 (𝐹𝐸̃ )⊂ (14) 𝑓𝑢𝑝 (𝐹⌀ ) = 𝐺⌀ .

𝑓𝐴 (𝑒) = 𝑢−1 (𝐺𝐾̃ (𝑝 (𝑒))) = 𝑢−1 (𝑌) = 𝑋 = 𝑓𝐸 (𝑒) .

Proof. We only prove (3), (5), (7), (9), (11), and (12). The others can be proved similarly. −1 (3) Put 𝐺𝑆 = 𝑓𝑢𝑝 (𝐹𝐴) and 𝐹𝐵 = 𝑓𝑢𝑝 (𝐺𝑆 ). Since 𝐴 ≤ −1 −1 𝑝 (𝑝(𝐴)) = 𝑝 (𝑆) = 𝐵, it is sufficient to show that 𝑓𝐴(𝑒) ⊆ 𝑓𝐵 (𝑒) for all 𝑒 ∈ 𝐸, −1

𝑓𝐵 (𝑒) = 𝑢 (𝑔𝑆 (𝑝 (𝑒))) = 𝑢−1 (∪𝑒∈𝑝−1 (𝑝(𝑒)) 𝑢 (𝑓𝐴 (𝑒))) = ∪𝑒∈𝑝−1 (𝑝(𝑒)) 𝑢−1 (𝑢 (𝑓𝐴 (𝑒)))

(7)

This completes the proof. ̃ 𝑖∈𝐽 (𝐹𝐴 )). Then (5) Put 𝐺𝑆𝑖 = 𝑓𝑢𝑝 (𝐹𝐴 𝑖 ) and 𝐺𝑆 = 𝑓𝑢𝑝 (∪ 𝑖 𝑆 = 𝑝(∨𝐴 𝑖 ) = ∨𝑝(𝐴 𝑖 ) = ∨𝑆𝑖 and for all 𝑘 ∈ 𝐾,

∪ 𝑢 (𝑓𝐴 𝑖 (𝑒)) { ∪ = {𝑒∈𝑝−1 (𝑘) 𝑖∈𝐽 {⌀ {∪ = {𝑖∈𝐽 {⌀

∪

𝑒∈𝑝−1 (𝑘)

𝑢 (𝑓𝐴 𝑖 (𝑒))

if 𝑝−1 (𝑘) ≠ ⌀;

4. FP-Soft Topological Spaces Definition 19. A FP-soft topological space is a pair (𝑋, 𝜏), where 𝑋 is a nonempty set and 𝜏 is a family of FP-soft sets over 𝑋 satisfying the following properties: ̃ 𝐹𝐵 ∈ 𝜏; (T2) if 𝐹𝐴, 𝐹𝐵 ∈ 𝜏, then 𝐹𝐴∩ ̃ 𝑖∈𝐽 𝐹𝐴 ∈ 𝜏. (T3) if 𝐹𝐴 𝑖 ∈ 𝜏, ∀𝑖 ∈ 𝐽, then ∪ 𝑖 𝜏 is called a topology of FP-soft sets on 𝑋. Every member of 𝜏 is called FP-soft open in (𝑋, 𝜏). 𝐹𝐵 is called FP-soft closed in (𝑋, 𝜏) if 𝐹𝐵𝑐 ∈ 𝜏.

Example 21. Assume that 𝑋 = {𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 } is a universal set and 𝐸 = {𝑒1 , 𝑒2 , 𝑒3 } is a set of parameters. If

if 𝑝−1 (𝑘) ≠ ⌀; otherwise

𝐹𝐴 1 = {((𝑒1 )0,2 , {𝑥1 , 𝑥3 }) , ((𝑒2 )0,3 , {𝑥1 , 𝑥4 }) , (8)

otherwise

{ ∪ 𝑢 (𝑓𝐴 𝑖 (𝑒)) = ∪ {𝑒∈𝑝−1 (𝑘) 𝑖∈𝐽 {⌀

This shows that 𝐹𝐴 = 𝐹𝐸̃ . (12) Since 𝑝−1 (𝐾) is fuzzy empty set, that is, 0𝐸 , the proof is clear.

Example 20. 𝜏indiscrete = {𝐹⌀ , 𝐹𝐸̃ } is a FP-soft topology on 𝑋. 𝜏discrete = FPS(𝑋, 𝐸) is a FP-soft topology on 𝑋.

otherwise

if 𝑝−1 (𝑘) ≠ ⌀;

(11)

(T1) 𝐹⌀ , 𝐹𝐸̃ ∈ 𝜏;

⊇ 𝑓𝐴 (𝑒) .

{ ∪ 𝑢 ( ∪ 𝑓𝐴 (𝑒)) 𝑔𝑆 (𝑘) = {𝑒∈𝑝−1 (𝑘) 𝑖∈𝐽 𝑖 {⌀

(10)

((𝑒3 )0,4 , {𝑥2 })} , 𝐹𝐴 2 = {((𝑒1 )0,2 , {𝑥1 , 𝑥2 , 𝑥3 }) , ((𝑒2 )0,5 , {𝑥1 , 𝑥4 }) ,

if 𝑝−1 (𝑘) ≠ ⌀;

((𝑒3 )0,4 , {𝑥1 , 𝑥2 })} ,

otherwise

𝐹𝐴 3 = {((𝑒1 )0,7 , {𝑥1 , 𝑥3 }) , ((𝑒2 )0,3 , 𝑋) , ((𝑒3 )0,9 , {𝑥2 , 𝑥3 })} ,

= ∪ 𝑔𝑆𝑖 (𝑘) .

𝐹𝐴 4 = {((𝑒1 )0,7 , {𝑥1 , 𝑥2 , 𝑥3 }) , ((𝑒2 )0,5 , 𝑋) ,

𝑖∈𝐽

This completes the proof. −1 −1 ̃ (7) Put 𝐹𝐴 𝑖 = 𝑓𝑢𝑝 (𝐺𝑆𝑖 ) and 𝐹𝐴 = 𝑓𝑢𝑝 (∪𝑖∈𝐽 𝐺𝑆𝑖 ). Then 𝐴 = −1 −1 𝑝 (∨𝑆𝑖 ) = ∨𝑝 (𝑆𝑖 ) = ∨𝐴 𝑖 and for all 𝑒 ∈ 𝐸, −1

𝑓𝐴 (𝑒) = 𝑢 ( ∪ 𝑔𝑆𝑖 (𝑝 (𝑒))) 𝑖∈𝐼

= ∪ 𝑢−1 (𝑔𝑆𝑖 (𝑝 (𝑒))) 𝑖∈𝐼

= ∪ 𝑓𝐴 𝑖 (𝑒) . 𝑖∈𝐽

This completes the proof.

(9)

((𝑒3 )0,9 , {𝑥2 , 𝑥3 })} , (12) then 𝜏 = {𝐹⌀ , 𝐹𝐴 1 , 𝐹𝐴 2 , 𝐹𝐴 3 , 𝐹𝐴 4 , 𝐹𝐸̃ } is a FP-soft topology on 𝑋. Theorem 22. Let (𝑋, 𝜏) be a FP-soft topological space and let 𝜏󸀠 denote family of all closed sets. Then, (1) 𝐹⌀ , 𝐹𝐸̃ ∈ 𝜏󸀠 ; ̃ 𝐹𝐵 ∈ 𝜏󸀠 ; (2) if 𝐹𝐴, 𝐹𝐵 ∈ 𝜏󸀠 , then 𝐹𝐴∪ ̃ 𝑖∈𝐽 𝐹𝐴 ∈ 𝜏󸀠 . (3) if 𝐹𝐴 ∈ 𝜏󸀠 , ∀𝑖 ∈ 𝐽, then ∩ 𝑖

𝑖

The Scientific World Journal Proof. Straightforward. Definition 23. Let (𝑋, 𝜏) be a FP-soft topological space and 𝐹𝐴 ∈ FPS(𝑋, 𝐸). The FP-soft closure of 𝐹𝐴 in (𝑋, 𝜏), denoted by 𝐹𝐴, is the intersection of all FP-soft closed supersets of 𝐹𝐴. Clearly, 𝐹𝐴 is the smallest FP-soft closed set over 𝑋 which contains 𝐹𝐴, and 𝐹𝐴 is closed. Theorem 24. Let (𝑋, 𝜏) be a FP-soft topological space and 𝐹𝐴, 𝐹𝐵 ∈ FPS(𝑋, 𝐸). Then, (1) 𝐹⌀ = 𝐹⌀ and 𝐹𝐸̃ = 𝐹𝐸̃ ; ̃ 𝐹𝐴; (2) 𝐹𝐴⊂ (3) 𝐹𝐴 = 𝐹𝐴; ̃ 𝐹𝐵 , then 𝐹𝐴⊂ ̃ 𝐹𝐵 ; (4) if 𝐹𝐴⊂ (5) 𝐹𝐴 is a FP-soft closed set if and only if 𝐹𝐴 = 𝐹𝐴; ̃ 𝐹𝐵 = 𝐹𝐴∪ ̃ 𝐹𝐵 . (6) 𝐹𝐴∪ Proof. (1), (2), (3), and (4) are obvious from the definition of FP-soft closure. (5) Let 𝐹𝐴 be a FP-soft closed set. Since 𝐹𝐴 is the smallest ̃ 𝐹𝐴. Therefore, FP-soft closed set which contains 𝐹𝐴, then 𝐹𝐴⊂ 𝐹𝐴 = 𝐹𝐴. ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 and 𝐹𝐵 ⊂ ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 , then, by (4), (6) Since 𝐹𝐴⊂ ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 , 𝐹𝐵 ⊂ ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 , and hence 𝐹𝐴∪ ̃ 𝐹𝐵 ⊂ ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 . 𝐹𝐴⊂ ̃ 𝐹𝐵 Conversely, since 𝐹𝐴 and 𝐹𝐵 are FP-soft closed sets, 𝐹𝐴∪ ̃ 𝐹𝐵 ⊂ ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 , by (4), then is a FP-soft closed set. Again since 𝐹𝐴∪ ̃ 𝐹𝐵 ⊂ ̃ 𝐹𝐴∪ ̃ 𝐹𝐵 . 𝐹𝐴∪ Definition 25. Let (𝑋, 𝜏) be a FP-soft topological space. A FPsoft set 𝐹𝐴 in FPS(𝑋, 𝐸) is called FP-Q-neighborhood (briefly, FP-Q-nbd) of a FP-soft set 𝐹𝐵 if there exists a FP-soft open set ̃ 𝐹𝐴. 𝐹𝐶 in 𝜏 such that 𝐹𝐵 𝑞𝐹𝐶 and 𝐹𝐶⊆ Theorem 26. Let 𝑒𝛼𝑓 , 𝐹𝐴 ∈ FPS(𝑋, 𝐸). Then 𝑒𝛼𝑓 ∈̃𝐹𝐴 if and only if each FP-Q-nbd of 𝑒𝛼𝑓 is FP-soft quasi-coincident with 𝐹𝐴. Proof. Let 𝑒𝛼𝑓 ∈̃𝐹𝐴 . Suppose that 𝐹𝐶 is a FP-Q-nbd of 𝑒𝛼𝑓 and 𝐹𝐶𝑞𝐹𝐴. Then there exists a FP-soft open set 𝐹𝐵 such that ̃ 𝐹𝐶. Since 𝐹𝐶𝑞𝐹𝐴 , by Proposition 15(1), 𝐹𝐴⊆ ̃ 𝐹𝐶𝑐 ⊆ ̃ 𝐹𝐵𝑐 . 𝑒𝛼𝑓 𝑞𝐹𝐵 ⊆ 𝑓 𝑓 𝑐 Again since 𝑒𝛼 𝑞𝐹𝐵 , 𝑒𝛼 does not belong to 𝐹𝐵 . This is a ̃ 𝐹𝐵𝑐 . contradiction with 𝐹𝐴 ⊆ Conversely, let each Q-nbd of 𝑒𝛼𝑓 be FP-soft quasicoincident with 𝐹𝐴 . Suppose that 𝑒𝛼𝑓 does not belong to 𝐹𝐴. Then there exists a FP-soft closed set 𝐹𝐵 which is containing 𝐹𝐴 such that 𝑒𝛼𝑓 does not belong to 𝐹𝐵 . By Proposition 15(5), we have 𝑒𝛼𝑓 𝑞𝐹𝐵𝑐 . Then 𝐹𝐵𝑐 is a FP-Q-nbd of 𝑒𝛼𝑓 and, by Proposition 15(1), 𝐹𝐴𝑞𝐹𝐵𝑐 . This is a contradiction with the hypothesis. Definition 27. Let (𝑋, 𝜏) be a FP-soft topological space and 𝐹𝐴 ∈ FPS(𝑋, 𝐸). The FP-soft interior of 𝐹𝐴 denoted by 𝐹𝐴∘ is the union of all FP-soft open subsets of 𝐹𝐴. Clearly, 𝐹𝐴∘ is the largest fuzzy soft open set contained in 𝐹𝐴, and 𝐹𝐴∘ is FP-soft open.

5 Theorem 28. Let (𝑋, 𝜏) be a FP-soft topological space and 𝐹𝐴, 𝐹𝐵 ∈ FPS(𝑋, 𝐸). Then, (1) (𝐹⌀ )∘ = 𝐹⌀ and (𝐹𝐸̃ )∘ = 𝐹𝐸̃ ; ̃ 𝐹𝐴; (2) 𝐹𝐴∘ ⊂ ∘

(3) (𝐹𝐴∘ ) = 𝐹𝐴∘ ; ̃ 𝐹𝐵 , then 𝐹𝐴∘ ⊂ ̃ 𝐹𝐵∘ ; (4) if 𝐹𝐴⊂

(5) 𝐹𝐴 is a FP-soft open set if and only if 𝐹𝐴 = 𝐹𝐴∘ ; ∘

̃ 𝐹𝐵 ) = 𝐹𝐴∘ ∩ ̃ 𝐹𝐵∘ . (6) (𝐹𝐴∩ Proof. Similar to that of Theorem 24. Theorem 29. Let (𝑋, 𝜏) be a FP-soft topological space and 𝐹𝐴 ∈ FPS(𝑋, 𝐸). Then, 𝑐

(1) (𝐹𝐴∘ ) = 𝐹𝐴𝑐 ; 𝑐

∘

(2) (𝐹𝐴) = (𝐹𝐴𝑐 ) . Proof. We only prove (1). The other is similar. Consider 𝑐 ̃ 𝐹𝐵 })𝑐 ̃ {𝐹𝐵 | 𝐹𝐵 ∈ 𝜏, 𝐹𝐴⊂ (𝐹𝐴∘ ) = (∪

̃ 𝐹𝐵 } ̃ {𝐹𝐵𝑐 | 𝐹𝐵 ∈ 𝜏, 𝐹𝐴⊂ =∩ ̃ 𝐹𝐴𝑐 } ̃ {𝐹𝐵𝑐 | 𝐹𝐵𝑐 ∈ 𝜏󸀠 , 𝐹𝐵𝑐 ⊂ =∩

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= 𝐹𝐴𝑐 .

Theorem 30. Let 𝑐 : FPS(𝑋, 𝐸) → FPS(𝑋, 𝐸) be an operator satisfying the following: (c1) 𝑐(𝐹⌀ ) = 𝐹⌀ ;

̃ 𝑐(𝐹𝐴 ), ∀𝐹𝐴 ∈ FPS(𝑋, 𝐸); (c2) 𝐹𝐴⊆ ̃ 𝐹𝐵 ) = 𝑐(𝐹𝐴)∪ ̃ 𝑐(𝐹𝐵 ), ∀𝐹𝐴 , 𝐹𝐵 ∈ FPS(𝑋, 𝐸); (c3) 𝑐(𝐹𝐴 ∪ (c4) 𝑐(𝑐(𝐹𝐴 )) = 𝑐(𝐹𝐴 ), ∀𝐹𝐴 ∈ FPS(𝑋, 𝐸). Then one can associate FP-soft topology in the following way: 𝜏 = {𝐹𝐴𝑐 ∈ FPS (𝑋, 𝐸) | 𝑐 (𝐹𝐴) = 𝐹𝐴} .

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Moreover with this FP-soft topology 𝜏, 𝐹𝐴 = 𝑐(𝐹𝐴) for every 𝐹𝐴 ∈ FPS(𝑋, 𝐸). ̃ 𝑐(𝐹𝐸̃ ), so 𝑐(𝐹𝐸̃ ) = Proof. (T1) By (c1), 𝐹⌀𝑐 = 𝐹𝐸̃ ∈ 𝜏. By (c2) 𝐹𝐸̃ ⊆ 𝐹𝐸̃ and 𝐹⌀ ∈ 𝜏. (T2) Let 𝐹𝐴, 𝐹𝐵 ∈ 𝜏. By the definition of 𝜏, 𝑐(𝐹𝐴𝑐 ) = 𝑐 ̃ 𝐹𝐵 )𝑐 ) = 𝑐(𝐹𝐴𝑐 ∪ ̃ 𝐹𝐵𝑐 ) = 𝐹𝐴 and 𝑐(𝐹𝐵𝑐 ) = 𝐹𝐵𝑐 . By (c3), 𝑐((𝐹𝐴 ∩ 𝑐 𝑐 ̃ 𝑐 𝑐̃ 𝑐 ̃ ̃ 𝑐(𝐹𝐴 )∪𝑐(𝐹𝐵 ) = 𝐹𝐴∪𝐹𝐵 = (𝐹𝐴∩𝐹𝐵 ) . So 𝐹𝐴∩𝐹𝐵 ∈ 𝜏. (T3) Let {𝐹𝐴 𝑖 | 𝑖 ∈ 𝐽} ⊂ 𝜏. Since 𝑐 is order preserving ̃ 𝑐(𝐹𝐴𝑐 ) = 𝐹𝐴𝑐 . ̃ 𝐹𝐴𝑐 ,∀𝑘 ∈ 𝐽, then 𝑐(∩ ̃ 𝑖∈𝐽 𝐹𝐴𝑐 ⊆ ̃ 𝑖∈𝐽 𝐹𝐴𝑐 )⊆ and ∩ 𝑖 𝑘 𝑖 𝑘 𝑘 𝑐 𝑐 ̃∩ ̃ 𝑖∈𝐽 𝐹𝐴 )⊆ ̃ 𝑖∈𝐽 𝐹𝐴 . Conversely, by (c2) we Then we have 𝑐(∩ 𝑖 𝑖 ̃ 𝑐(∩ ̃ 𝑖∈𝐽 𝐹𝐴𝑐 ⊆ ̃ 𝑖∈𝐽 𝐹𝐴 )𝑐 ) = 𝑐(∩ ̃ 𝑖∈𝐽 𝐹𝐴𝑐 ) = ̃ 𝑖∈𝐽 𝐹𝐴𝑐 ). Hence, 𝑐((∪ have ∩ 𝑖 𝑖 𝑖 𝑖 𝑐 𝑐 ̃ 𝑖∈𝐽 𝐹𝐴 = (∪ ̃ 𝑖∈𝐽 𝐹𝐴 ) and ∪ ̃ 𝑖∈𝐽 𝐹𝐴 ∈ 𝜏. ∩ 𝑖 𝑖 𝑖 Now we will show that with this FP-soft topology 𝜏, 𝐹𝐴 = 𝑐(𝐹𝐴 ) for every 𝐹𝐴 ∈ FPS(𝑋, 𝐸). Let 𝐹𝐴 ∈ FPS(𝑋, 𝐸). Since

6


(𝐹𝐴)𝑐 ∈ 𝜏, then 𝑐(𝐹𝐴) = 𝐹𝐴. Since 𝑐 is order preserving, ̃ 𝑐(𝐹𝐴 ) = 𝐹𝐴 . Conversely, by (c4) we have (𝑐(𝐹𝐴 ))𝑐 ∈ 𝜏. 𝑐(𝐹𝐴 )⊆ ̃ 𝑐(𝐹𝐴 ) and 𝐹𝐴 is the smallest FP-soft closed set Then since 𝐹𝐴 ⊆ ̃ 𝑐(𝐹𝐴). over 𝑋 which contains 𝐹𝐴, 𝐹𝐴⊆ The operator 𝑐 is called the FP-soft closure operator. Remark 31. By Theorem 24(1), (2), (3), and (6) and Theorem 30, we see that with a FP-soft closure operator we can associate a FP-soft topology and conversely with a given FP-soft topology we can associate a FP-soft closure operator. Theorem 32. Let 𝑖 : FPS(𝑋, 𝐸) → FPS(𝑋, 𝐸) be an operator satisfying the following: (i1) 𝑖(𝐹𝐸̃ ) = 𝐹𝐸̃ ;

̃ 𝐹𝐴, ∀𝐹𝐴 ∈ FPS(𝑋, 𝐸); (i2) 𝑖(𝐹𝐴)⊆ ̃ 𝐹𝐵 ) = 𝑖(𝐹𝐴 )∩ ̃ 𝑖(𝐹𝐵 ), ∀𝐹𝐴, 𝐹𝐵 ∈ FPS(𝑋, 𝐸); (i3) 𝑖(𝐹𝐴∩ (i4) 𝑖(𝑖(𝐹𝐴)) = 𝑖(𝐹𝐴 ), ∀𝐹𝐴 ∈ FPS(𝑋, 𝐸). Then one can associate a FP-soft topology in the following way: 𝜏 = {𝐹𝐴 ∈ 𝐹𝑃𝑆 (𝑋, 𝐸) | 𝑖 (𝐹𝐴) = 𝐹𝐴} .

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Moreover, with this fuzzy soft topology 𝜏, (𝐹𝐴 )∘ = 𝑖(𝐹𝐴) for every 𝐹𝐴 ∈ FPS(𝑋, 𝐸). Proof. Similar to that of Theorem 30. The operator 𝑖 is called the FP-soft interior operator. Remark 33. By Theorem 28(1), (2), (3), and (6) and Theorem 32, we see that with a FP-soft interior operator we can associate a FP-soft topology and conversely with a given FP-soft topology we can associate a FP-soft interior operator. Definition 34. Let (𝑋, 𝜏) be a FP-soft topological space. A subcollection B of 𝜏 is called a base for 𝜏 if every member of 𝜏 can be expressed as a union of members of B. Example 35. If we consider the FP-soft topology 𝜏 in Example 21, then one easily sees that the family B = {𝐹⌀ , 𝐹𝐴 1 , 𝐹𝐴 2 , 𝐹𝐴 3 , 𝐹𝐸̃ } is a basis for 𝜏. Proposition 36. Let (𝑋, 𝜏) be a FP-soft topological space and B is subfamily of 𝜏. B is a base for 𝜏 if and only if for each 𝑒𝛼𝑓 in FPS(𝑋, 𝐸) and for each FP-soft open Q-nbd 𝐹𝐴 of 𝑒𝛼𝑓 , there ̃ 𝐹𝐴. exists a 𝐹𝐵 ∈ B such that 𝑒𝛼𝑓 𝑞𝐹𝐵 ⊆ Proof. Let B be a base for 𝜏, 𝑒𝛼𝑓 ∈̃ FPS(𝑋, 𝐸) and let 𝐹𝐴 be a FP-soft open Q-nbd of 𝑒𝛼𝑓 . Then there exists a subfamily B󸀠 ̃ {𝐹𝐵 | 𝐹𝐵 ∈ B󸀠 }. Suppose that 𝑒𝛼𝑓 𝑞𝐹𝐵 of B such that 𝐹𝐴 = ∪ 󸀠 for all 𝐹𝐵 ∈ B . Then 𝛼 + 𝜇𝐵 (𝑒) ≤ 1 and 𝑓(𝑒) ⊆ 𝑓𝐵𝑐 (𝑒) for every 𝐹𝐵 ∈ B󸀠 . Therefore, we have 𝛼 + 𝜇𝐴 (𝑒) ≤ 1 and 𝑓(𝑒) is not subset of 𝑓𝐴(𝑒) since 𝜇𝐴 (𝑒) = sup{𝜇𝐵 (𝑒) | 𝐹𝐵 ∈ B󸀠 } and 𝑓𝐴(𝑒) = ∪𝑓𝐵 (𝑒). This is a contradiction. Conversely, if B is not a base for 𝜏, then there exists a 𝐹𝐴 ∈ ̃ 𝐹𝐴} ≠ 𝐹𝐴 . Since 𝐹𝐶 ≠ 𝐹𝐴, ̃ {𝐹𝐵 ∈ B : 𝐹𝐵 ⊆ 𝜏 such that 𝐹𝐶 = ∪

there exists 𝑒 ∈ 𝐸 such that 𝜇𝐶(𝑒) < 𝜇𝐴 (𝑒) or 𝑓𝐶(𝑒) ⊂ 𝑓𝐴(𝑒). Put 𝛼 = 1 − 𝜇𝐶(𝑒) and 𝑓(𝑒) = 𝑓𝐶𝑐 (𝑒). Then in both cases, we obtain 𝑒𝛼𝑓 𝑞𝐹𝐴 and 𝑒𝛼𝑓 𝑞𝐹𝐶. Therefore, we have 𝜇𝐵 (𝑒) + 𝛼 ≤ 𝜇𝐶(𝑒) + 𝛼 = 1 and 𝑓(𝑒) ∩ 𝑓𝐵 (𝑒) ⊆ 𝑓(𝑒) ∩ 𝑓𝐶(𝑒) = ⌀, that is, 𝑒𝛼𝑓 𝑞𝐹𝐵 for all 𝐹𝐵 ∈ B which is contained in 𝐹𝐴. This is a contradiction. Definition 37. Let (𝑋, 𝜏1 ) and (𝑌, 𝜏2 ) be two FP-soft topological spaces. A FP-soft mapping 𝑓𝑢𝑝 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) is called −1 FP-soft continuous if 𝑓𝑢𝑝 (𝐺𝑆 ) ∈ 𝜏1 , ∀𝐺𝑆 ∈ 𝜏2 . Example 38. Assume that 𝑋 = {𝑥1 , 𝑥2 , 𝑥3 }, 𝑌 = {𝑦1 , 𝑦2 , 𝑦3 } are two universal sets, 𝐸 = {𝑒1 , 𝑒2 }, 𝐾 = {𝑘1 , 𝑘2 } are two parameter sets, and 𝑓𝑢𝑝 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) is a FP-soft mapping, where 𝑢(𝑥1 ) = 𝑦2 , 𝑢(𝑥2 ) = 𝑦1 , 𝑢(𝑥3 ) = 𝑦3 , and 𝑝(𝑒1 ) = 𝑘2 and 𝑝(𝑒2 ) = 𝑘1 . If we take 𝐹𝐴 = {((𝑒1 )0,3 , {𝑥2 , 𝑥3 }), ((𝑒2 )0,2 , {𝑥1 , 𝑥2 })}, 𝐺𝑆 = {((𝑘1 )0,2 , {𝑦1 , 𝑦2 }), ((𝑘2 )0,3 , {𝑦1 , 𝑦3 })}, 𝜏1 = {𝐹⌀ , 𝐹𝐸̃ , 𝐹𝐴}, and 𝜏2 = {0̃𝐾 , 1̃𝐾 , 𝐺𝑆 }, then 𝑓𝑢𝑝 is a FP-soft continuous mapping. The constant mapping 𝑓𝑢𝑝 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) is not continuous in general. Example 39. Assume that 𝑋 = {𝑥1 , 𝑥2 , 𝑥3 }, 𝑌 = {𝑦1 , 𝑦2 } are two universal sets, 𝐸 = {𝑒1 , 𝑒2 }, 𝐾 = {𝑘1 , 𝑘2 } are two parameter sets, and 𝑓𝑢𝑝 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) is a constant FP-soft mapping, where 𝑢(𝑥1 ) = 𝑢(𝑥2 ) = 𝑢(𝑥3 ) = 𝑦2 and 𝑝(𝑒1 ) = 𝑝(𝑒2 ) = 𝑘1 . If we take 𝐺𝑆 = {((𝑘1 )0,2 , {𝑦1 , 𝑦2 }), ((𝑘2 )0,5 , {𝑦2 , 𝑦3 })}, 𝜏1 = {𝐹⌀ , 𝐹𝐸̃ }, and 𝜏2 = −1 {𝐹⌀ , 𝐹𝐾̃ , 𝐺𝑆 }, 𝑓𝑢𝑝 is not a FP-soft continuous since 𝑓𝑢𝑝 (𝐺𝑆 ) ∉ 𝜏1 . Let 𝛼 ∈ [0, 1]. We denote by 𝛼𝐸 the constant fuzzy set on 𝐸; that is, 𝜇𝛼𝐸 (𝑒) = 𝛼 for all 𝑒 ∈ 𝐸 and 𝛼 ∈ [0, 1]. Definition 40. Let 𝐹𝐴 ∈ FPS(𝑋, 𝐸). 𝐹𝐴 is called 𝛼-𝐴-universal FP-soft set if 𝜇𝐴 (𝑒) = 𝛼 and 𝑓𝐴(𝑒) = 𝑋 for all 𝑒 ∈ 𝐴, denoted by 𝐹𝛼̃𝐴 . Definition 41 (see [21]). A FP-soft topology is called enriched if it satisfies 𝐹𝛼̃𝐴 ∈ 𝜏 and 𝐹𝛼̃𝑐𝐴 ∈ 𝜏 for all 𝛼 ∈ (0, 1]. Theorem 42. Let (𝑋, 𝜏1 ) be an enriched FP-soft topological space, (𝑌, 𝜏2 ) a FP-soft topological space, and 𝑓𝑢𝑝 : FPS(𝑋, 𝐸) → FPS(𝑌, 𝐾) a constant FP-soft mapping. Then 𝑓𝑢𝑝 is FP-soft continuous. −1 (𝐺𝑆 ) = 𝐹𝐴. Then 𝐴 = 𝑝−1 (𝑆) = 𝛼𝐸 , Proof. Let 𝐺𝑆 ∈ 𝜏2 . Put 𝑓𝑢𝑝 where 𝛼 = sup𝑘∈𝐾 {𝜇𝑆 (𝑘)} and

𝑋, 𝑓𝐴 (𝑒) = 𝑢−1 (𝑔𝑆 (𝑝 (𝑒))) = { ⌀,

𝑔𝑆 (𝑝 (𝑒)) ≠ ⌀ otherwise

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for all 𝑒 ∈ 𝐸. Hence 𝐹𝐴 = 𝐹𝛼̃𝐸 ∈ 𝜏1 or 𝐹𝐴 = 𝐹𝛼̃𝑐𝐸 ∈ 𝜏1 and so 𝑓𝑢𝑝 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) is FP-soft continuous. Theorem 43. Let (𝑋, 𝜏1 ) and (𝑌, 𝜏2 ) be two FP-soft topological spaces and let 𝑓𝑢𝑝 : FPS(𝑋, 𝐸) → FPS(𝑌, 𝐾) be a FP-soft mapping. Then the following are equivalent:

The Scientific World Journal (1) 𝑓𝑢𝑝 is FP-soft continuous; −1 (2) 𝑓𝑢𝑝 (𝐺𝑆 ) is FP-soft closed for every FP-closed set 𝐺𝑆 over 𝑌; ̃ 𝑓𝑢𝑝 (𝐹𝐴), ∀𝐹𝐴 ∈ FPS(𝑋, 𝐸); (3) 𝑓𝑢𝑝 (𝐹𝐴)⊂ −1 (𝐺 )⊂ ̃ −1 (4) 𝑓𝑢𝑝 𝑆 𝑓𝑢𝑝 (𝐺𝑆 ), ∀𝐺𝑆 ∈ FPS(𝑌, 𝐾); −1 −1 ̃ (𝑓𝑢𝑝 (5) 𝑓𝑢𝑝 (𝐺𝑆∘ )⊂ (𝐺𝑆 ))∘ , ∀𝐺𝑆 ∈ FPS(𝑌, 𝐾).

Proof. (1)⇒(2) It is obvious from Theorem 18(9). ∈ FPS(𝑋, 𝐸). Since (2)⇒(3) Let 𝐹𝐴 −1 −1 ̃ 𝑓𝑢𝑝 ̃ 𝑓𝑢𝑝 𝐹𝐴⊂ (𝑓𝑢𝑝 (𝐹𝐴))𝐹𝐴⊂ (𝑓𝑢𝑝 (𝐹𝐴)) ∈ 𝜏1󸀠 , therefore −1 ̃ 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐹𝐴)). By Theorem 18(4), we get we have 𝐹𝐴⊂

−1 ̃ 𝑓𝑢𝑝 (𝑓𝑢𝑝 ̃ 𝑓𝑢𝑝 (𝐹𝐴). (𝑓𝑢𝑝 (𝐹𝐴))⊂ 𝑓𝑢𝑝 (𝐹𝐴 )⊂ −1 (3)⇒(4) Let 𝐺𝑆 ∈ FPS(𝑌, 𝐾). If we choose 𝑓𝑢𝑝 (𝐺𝑆 ) instead −1 −1 ̃ 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐺𝑆 ))⊂ ̃ 𝐺𝑆 . Hence by of 𝐹𝐴 in (3), then 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐺𝑆 ))⊂

−1 (𝐺 )⊂ −1 ̃ −1 ̃ −1 Theorem 18(3), 𝑓𝑢𝑝 𝑆 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐺𝑆 )))⊂𝑓𝑢𝑝 (𝐺𝑆 ). (4)⇔(5) These follow from Theorem 18(9) and Theorem 29. (5)⇒(1) Let 𝐺𝑆 ∈ 𝜏2 . Since 𝐺𝑆 is a FP-soft open set, then −1 −1 −1 −1 ̃ 𝑓𝑢𝑝 ̃ (𝑓𝑢𝑝 (𝐺𝑆 ) = 𝑓𝑢𝑝 (𝐺𝑆∘ )⊂ (𝐺𝑆 ))∘ ⊂ (𝐺𝑆 ). Consequently, 𝑓𝑢𝑝 −1 𝑓𝑢𝑝 (𝐺𝑆 ) is a FP-soft open and so 𝑓𝑢𝑝 is FP-soft continuous.

Theorem 44. Let 𝑓𝑢𝑝 : (𝑋, 𝜏1 ) → (𝑌, 𝜏2 ) be a FP-soft mapping and let B be a base for 𝜏2 . Then 𝑓𝑢𝑝 is FP-soft −1 continuous if and only if 𝑓𝑢𝑝 (𝐺𝑆 ) ∈ 𝜏1 , ∀𝐺𝑆 ∈ B. Proof. Straightforward. Definition 45. A family C of FP-soft sets is a cover of a FP-soft ̃∪ ̃ {𝐹𝐴 : 𝐹𝐴 ∈ C, 𝑖 ∈ 𝐽}. It is a FP-soft open cover set 𝐹𝐴 if 𝐹𝐴 ⊆ 𝑖 𝑖 if each member of C is a FP-soft open set. A subcover of C is a subfamily of C which is also a cover. Definition 46. A family C of FP-soft sets has the finite intersection property if the intersection of the members of each finite subfamily of C is not empty FP-soft set. Definition 47. A FP-soft topological space (𝑋, 𝜏) is FPcompact if each FP-soft open cover of 𝐹𝐸̃ has a finite subcover. Example 48. Let 𝑋 = {𝑥1 , 𝑥2 , . . .}, 𝐸 = {𝑒1 , 𝑒2 , . . .}, and 𝐹𝐴 𝑛 = {((𝑒𝑖 )1/𝑛 , 𝑋 − {𝑥1 , 𝑥2 , . . . , 𝑥𝑛 }) : 𝑖 = 1, 2, . . .}. Then 𝜏 = {𝐹𝐴 𝑛 : 𝑛 = 1, 2, . . .} ∪ {𝐹⌀ , 𝐹𝐸̃ } is a FP-soft topology on 𝑋, and (𝑋, 𝜏) is FP-compact. Theorem 49. A FP-soft topological space is FP-soft compact if and only if each family of FP-soft closed sets with the finite intersection property has a nonempty FP-soft intersection. Proof. If C is a family of FP-soft sets in a FP-soft topological space (𝑋, 𝜏), then C is a cover of 𝐹𝐸̃ if and only if one of the following conditions hold: ̃ {𝐹𝐴 : 𝐹𝐴 ∈ C, 𝑖 ∈ 𝐽} = 𝐹𝐸̃ ; (1) ∪ 𝑖 𝑖 ̃ {𝐹𝐴 : 𝐹𝐴 ∈ C, 𝑖 ∈ 𝐽})𝑐 = 𝐹𝑐̃ = 𝐹⌀ ; (2) (∪ 𝑖 𝑖 𝐸

7 ̃ {𝐹𝐴𝑐 : 𝐹𝐴 ∈ C, 𝑖 ∈ 𝐽} = 𝐹⌀ . (3) ∩ 𝑖 𝑖 Hence the FP-soft topological space is FP-soft compact if and only if each family of FP-soft open sets over 𝑋 such that no finite subfamily covers 𝐹𝐸̃ fails to be a cover, and this is true if and only if each family of FP-soft closed sets which has the finite intersection property has a nonempty FP-soft intersection. Theorem 50. Let (𝑋, 𝜏1 ) and (𝑌, 𝜏2 ) be FP-soft topological spaces and let 𝑓𝑢𝑝 : FPS(𝑋, 𝐸) → FPS(𝑌, 𝐾) be a FPsoft mapping. If (𝑋, 𝜏1 ) is FP-soft compact and 𝑓𝑢𝑝 is FP-soft continuous surjection, then (𝑌, 𝜏2 ) is FP-soft compact. Proof. Let C = {𝐺𝑆𝑖 : 𝑖 ∈ 𝐽} be a cover of 𝐺𝐾̃ by FP-soft open sets. Then since 𝑓𝑢𝑝 is FP-soft continuous, the family of all FP−1 soft sets of the form 𝑓𝑢𝑝 (𝐺𝑆 ), for 𝐺𝑆 ∈ C, is a FP-soft open cover of 𝐹𝐸̃ which has a finite subcover. However, since 𝑓𝑢𝑝 is −1 surjective, then 𝑓𝑢𝑝 (𝑓𝑢𝑝 (𝐺𝑆 )) = 𝐺𝑆 for any FP-soft set 𝐺𝑆 over 𝑌. Thus, the family of images of members of the subcover is a finite subfamily of C which covers 𝐺𝐾̃ . Consequently, (𝑌, 𝜏2 ) is FP-soft compact.

5. Conclusion Topology is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics, but also in many real life applications. In this paper, we introduce the topological structure of fuzzy parametrized soft sets and fuzzy parametrized soft mappings. We study some fundamental concepts in fuzzy parametrized soft topological spaces such as closures, interiors, bases, compactness, and continuity. Some basic properties of these concepts are also presented. This paper will form the basis for further applications of topology.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

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