EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 999-1017 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Some Properties of Fuzzy Supra Soft Topological Spaces A. M. Abd El-latif1,2 1 Faculty of Arts and Science, Northern Border University, Rafha, P. O. Box, 840, K. S. A. 2 Mathematics Department, Faculty of Education, Ain Shams University, Roxy, 11341, Cairo, Egypt Abstract. In this paper, we introduce the notion of fuzzy supra soft topological spaces, which is a generalization to the notion of fuzzy soft topological spaces and supra soft topological spaces. Also, we consider the notion of fuzzy supra soft continuity as a generalization to fuzzy soft continuity, supported by examples and counterexamples. These examples illustrating the notions used in the paper are included. So we can see that all these concepts are independent from each other or does implies the other. Finally, as a direct application to fuzzy supra soft topological spaces, we introduce the notion of fuzzy supra soft compact (resp. fuzzy supra soft lindelöf) spaces to such spaces as a generalization to fuzzy soft compactness. Furthermore, we establish some interesting properties of this notion. 2010 Mathematics Subject Classifications: 54A40, 06D72, 54D30, 03E72 Key Words and Phrases: Fuzzy soft set, Fuzzy supra soft topological space, Fuzzy supra soft continuous mapping, Fuzzy supra soft compactness 1. Introduction Many theories like theory of probability, theory of fuzzy sets, theory of intuitionistic fuzzy sets, theory of rough sets etc. can be considered as mathematical tools for dealing with uncertain data, obtained in various fields of engineering, physics, computer science, economics, social science, medical science, and of many other diverse fields. But all these theories have their own difficulties. The most appropriate theory for dealing with un- certainties is the theory of fuzzy sets, introduced by Zadeh [32] in 1965. This theory brought a paradigmatic change in mathematics. But, there exists difficulty, how to set the membership function in each particular case. The theory of intuitionistic fuzzy sets is a more generalized concept than the theory of fuzzy sets, but this theory has the same difficulties. All the above mentioned theories are successful to some extent in dealing with problems arising due to vagueness present in the real world. But there are also cases where these theories failed to give satisfactory DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3440 Email addresses: alaa 8560@yahoo.com, alaa.ali@nbu.edu.sa (A. M. Abd El-latif) http://www.ejpam.com 999 c© 2019 EJPAM All rights reserved. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1000 results, possibly due to inadequacy of the parametrization tool in them. As a necessary supplement to the existing mathematical tools for handling uncertainty, in 1999, Molodtsov [26] initiated the theory of soft sets as a new mathematical tool to deal with uncertainties while modeling the problems in engineering, physics, computer science, economics, social sciences, and medical sciences. In 1968, Chang [14] introduced fuzzy topological space and in 2011, subsequently Cagman et al. [13] and Shabir et al. [31] introduced soft topological spaces and they defined basic notions of soft topological spaces. In 2011, Tanay et al. [10] introduced the notion of fuzzy soft topological spaces, which is extended in [15, 21, 28, 29]. Recently, Some researchers [11, 12, 17, 24, 27, 30] studied on the fuzzy soft compact topological spaces. In 1984, Mashhour et al. [25] introduced supra topological space, subsequently El- sheikh et al. [16] introduced supra soft topological spaces and they defined basic notions of supra soft topological spaces, which is extended in [4, 5, 20]. In 1987, Abd El-Monsef et al. introduced fuzzy supra topological space and defined basic notions of fuzzy supra topological spaces. Our aim of this paper, is to introduce the notion of fuzzy supra soft topological spaces, which is a generalization to the notion of fuzzy soft topological spaces [21] and supra soft topological spaces [16]. Also, we consider the notion of fuzzy supra soft continuity as a generalization to fuzzy soft continuity [8], fuzzy semi-soft continuity [19], fuzzy pre- soft continuity [1], fuzzy α-soft continuity [2], fuzzy β-soft continuity [3] and fuzzy b-soft continuity [6], supported by examples and counterexamples. We also introduce and study the concepts of fuzzy supra open (resp. closed) soft functions a generalization to fuzzy open (resp. closed) soft functions [30]. Finally, we introduce the notion of fuzzy supra soft compact (resp. fuzzy supra soft lindelöf) spaces as a generalization to such introduced in [11, 12, 17, 24, 27, 30]. We also introduce some basic definitions of fuzzy supra soft compact spaces and theorems of the concept. 2. Preliminaries From the literature, we recall the following definitions and results for the development of fuzzy soft set theory and fuzzy soft topological spaces, which will be needed in this paper. Definition 1. [32] A fuzzy set A in a non-empty set X is characterized by a membership function µA : X −→ [0, 1] = I whose value µA(x) represents the ”degree of membership” of x in A for x ∈ X. Here, IX denotes the family of all fuzzy sets on X. Definition 2. [26] Let X be an initial universe and E be a set of parameters. Let P (X) denote the power set of X and A be a non-empty subset of E. A pair (F,A), denoted by FA, is called a soft set over X , where F is a mapping given by F : A → P (X). For a particular e ∈ A , F (e) may be considered the set of e-approximate elements of the soft set (F,A) and if e 6∈ A, then F (e) = φ i.e FA = {(e, F (e)) : e ∈ A ⊆ E, F : A → P (X)}. The family of all these soft sets over X denoted by SS(X)A. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1001 Definition 3. [22] Let A ⊆ E. A pair (f,A), denoted by fA, is called a fuzzy soft set over X , where f is a mapping given by f : A → IX defined by fA(e) = µefA where µefA = 0 if e 6∈ A and µefA 6= 0 if e ∈ A, where 0 is the membership function of null fuzzy set over X, which takes value 0 for all x ∈ X i.e 0(e) = 0 ∀ x ∈ X. The family of all these fuzzy soft sets over X denoted by FSS(X)A. Definition 4. [29] The complement of a fuzzy soft set (f,A), denoted by (f,A)c, is defined by (f,A)c = (f c, A), f cA : E → IX is a mapping given by µefcA = 1 − µefA ∀ e ∈ A, where 1(e) = 1 ∀ x ∈ X. Clearly (f cA)c = fA. Definition 5. [23] A fuzzy soft set fA over X is said to be a null fuzzy soft set, denoted by 0̃A, if for all e ∈ A, fA(e) = 0. Definition 6. [23] A fuzzy soft set fA over X is said to be an absolute fuzzy soft set, denoted by 1̃A, if for all e ∈ A, fA(e) = 1, where 1 is the membership function of absolute fuzzy set over X, which takes value 1 for all for all x ∈ X. Clearly, we have (1̃A)c = 0̃A and (0̃A)c = 1̃A. Definition 7. [29] Let fA, gB ∈ FSS(X)E. Then, fA is fuzzy soft subset of gB, denoted by fA v gB, if A ⊆ B and µefA ⊆ µ e gB ∀ e ∈ A, i.e. µefA(x) ≤ µegB (x) ∀ x ∈ X and ∀ e ∈ A. Definition 8. [29]. The union of two fuzzy soft sets fA and gB over the common universe X is also a fuzzy soft set hC , where C = A ∪B and for all e ∈ C, hC(e) = µehc = µefA ∨ µ e gB ∀e ∈ C. Here, we write hC = fA t gB. Definition 9. [29]. The intersection of two fuzzy soft sets fA and gB over the common universe X is also a fuzzy soft set hC , where C = A ∩B and for all e ∈ C, hC(e) = µehc = µefA ∧ µ e gB ∀e ∈ C. Here, we write hC = fA u gB. Definition 10. [10]. Let T be a collection of fuzzy soft sets over a universe X with a fixed set of parameters E, then T is called a fuzzy soft topology on X if (1) 1̃E , 0̃E ∈ T, where 0̃E(e) = 0 and 1̃E(e) = 1, ∀e ∈ E, (2) The union of any members of T, belongs to T, (3) The intersection of any two members of T, belongs to T. The triplet (X,T, E) is called a fuzzy soft topological space over X. Also, each member of T is called a fuzzy open soft in (X,T, E). We denote the set of all fuzzy open soft sets by FOS(X,T, E), or FOS(X). Definition 11. [10] Let (X,T, E) be a fuzzy soft topological space. A fuzzy soft set fA over X is said to be fuzzy closed soft set in X, if its relative complement f cA is fuzzy open soft set. We denote the set of all fuzzy closed soft sets by FCS(X,T, E), or FCS(X). A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1002 Definition 12. [28] Let (X,T, E) be a fuzzy soft topological space and fA ∈ FSS(X)E. The fuzzy soft closure of fA, denoted by Fcl(fA) is the intersection of all fuzzy closed soft super sets of fA. i.e., Fcl(fA) = u{hD : hD is fuzzy closed soft set and fA v hD}. The fuzzy soft interior of gB, denoted by Fint(gB) is the fuzzy soft union of all fuzzy open soft subsets of gB.i.e., Fint(gB) = t{hD : hD is fuzzy open soft set and hD v gB}. Definition 13. [21] The fuzzy soft set fA ∈ FSS(X)E is called fuzzy soft point if for the element e ∈ E, µefA(x) 6= 0 and µe ′ fA (x) = 0 for each e′ ∈ E−{e}, and this fuzzy soft point is denoted by feA. Definition 14. [21] The fuzzy soft point feA is said to be belonging to the fuzzy soft set gB, denoted by feA∈̃gB, if for the element e ∈ A ∩B, µefA(x) ≤ µegB (x). Definition 15. [21] A fuzzy soft set gB in a fuzzy soft topological space (X,T, E) is called a fuzzy soft neighborhood of the fuzzy soft point feA if there exists a fuzzy open soft set hC such that feA∈̃hC v gB. A fuzzy soft set gB in a fuzzy soft topological space (X,T, E) is called a fuzzy soft neighborhood of the fuzzy soft set kD if there exists a fuzzy open soft set hC such that kD v hC v gB. The fuzzy soft neighborhood system of the fuzzy soft point feA, denoted by NT(feA), is the family of all its fuzzy soft neighborhoods. Definition 16. [21] Let (X,T, E) be a fuzzy soft topological space and Y ⊆ X. Let hYE be a fuzzy soft set over (Y,E) such that hYE : E → IY such that hYE(e) = µe hYE , where µe hYE (x) = { 1, x ∈ Y, 0, x 6∈ Y. Let TY = {hYE u gB : gB ∈ T}, then the fuzzy soft topology TY on (Y,E)is called fuzzy soft subspace topology for (Y,E) and (Y,TY , E) is called fuzzy soft subspace of (X,T, E). If hYE ∈ T (resp. hYE ∈ T c ), then (Y,TY , E) is called fuzzy open (resp. closed) soft subspace of (X,T, E). Definition 17. [8] Let FSS(X)E and FSS(Y )K be families of fuzzy soft sets over X and Y , respectively. Let u : X → Y and p : E → K be mappings. Then, the map fpu is called a fuzzy soft mapping from X to Y and denoted by fpu : FSS(X)E → FSS(Y )K such that, (1) If fA ∈ FSS(X)E. Then, the image of fA under the fuzzy soft mapping fpu is the fuzzy soft set over Y defined by fpu(fA), where ∀ k ∈ p(E), ∀ y ∈ Y , fpu(fA)(k)(y) = { ∨ u(x)=y [∨p(e)=k(fA(e))](x), x ∈ u−1(y), 0, otherwise. (2) If gB ∈ FSS(Y )K , then the pre-image of gB under the fuzzy soft mapping fpu is the fuzzy soft set over X defined by f−1 pu (gB), where ∀ e ∈ p−1(K), ∀ x ∈ X, f−1 pu (gB)(e)(x) = { gB(p(e))(u(x)), p(e) ∈ B, 0, otherwise. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1003 The fuzzy soft mapping fpu is called surjective (resp. injective) if p and u are surjective (resp. injective), also it is said to be constant if p and u are constant. Definition 18. Let (X,T1, E) and (Y,T2,K) be two fuzzy soft topological spaces and fpu : FSS(X)E → FSS(Y )K be a fuzzy soft mapping. Then, fpu is called (1) [8]Fuzzy soft continuous if f−1 pu (gB) ∈ T1 ∀ gB ∈ T2. (2) [15]Fuzzy open soft if fpu(gA) ∈ T2∀ gA ∈ T1. () [15]Fuzzy closed soft if fpu(hB) ∈ Tc 2∀ hB ∈ Tc 1. Theorem 1. [8] Let FSS(X)E and FSS(Y )K be two families of fuzzy soft sets. For the fuzzy soft function fpu : FSS(X)E → FSS(Y )K , the following statements hold, (a) f−1 pu ((gB)c) = (f−1 pu (gB))c∀ gB ∈ FSS(Y )K . (b) fpu(f−1 pu ((gB))) v gB∀ gB ∈ FSS(Y )K . If fpu is surjective, then the equality holds. (c) fA v f−1 pu (fpu(fA))∀ fA ∈ FSS(X)E. If fpu is injective, then the equality holds. (d) fpu(0̃E) = 0̃K , fpu(1̃E) v 1̃K . If fpu is surjective, then the equality holds. (e) f−1 pu (1̃K) = 1̃E and f−1 pu (0̃K) = 0̃E. (f) If fA v gA, then fpu(fA) v fpu(gA). (g) If fB v gB, then f−1 pu (fB) v f−1 pu (gB) ∀ fB, gB ∈ FSS(Y )K . 3. Fuzzy Supra Soft Topological Spaces Our aim of this section, is to introduce the notion of fuzzy supra soft topological spaces, which is a generalization to the notion of fuzzy soft topological spaces [21] and supra soft topological spaces [16]. We introduce and study the properties of the operations which will help researcher enhance and promote the further study on fuzzy supra soft topol- ogy.Furthermore, we list the main properties of the operations which give the deviations between these operations and that in fuzzy soft topological spaces. Definition 19. [29]. Let T be a collection of fuzzy soft sets over a universe X with a fixed set of parameters E, then T is called a fuzzy supra soft topology on X if (1) 1̃E , 0̃E ∈ T, where 0̃E(e) = 0 and 1̃E(e) = 1, ∀e ∈ E, (2) The union of any members of T, belongs to T. The triplet (X,T, E) is called a fuzzy supra soft topological space (fssts for short) over X. Also, each member of T is called a fuzzy supra open soft in (X,T, E). A fuzzy soft set fA over X is said to be fuzzy supra closed soft set in X, if its relative complement f cA is a fuzzy supra open soft set. We denote the set of all fuzzy supra open (closed) soft sets by FSOS(X) (FSCS(X)). A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1004 Remarks 1. Every fuzzy soft topological space is a fuzzy supra soft topological space, but the converse is not true in general as will shown in the following example. Example 1. Let X = {a, b, c} be the set of three cars under consideration and E = {e1(Modern Technology), e2(Luxurious), e3(Costly)}. Let A,B ⊆ E where A = {e1, e2} and B = {e2, e3}. Let T = {1̃E , 0̃E , f1A, f2B, f3E} where f1A, f2B, f3E are fuzzy soft sets over X representing the ”attractiveness of the cars” which Mr. R, Mr. S and Mr. T are going to buy, respectively which defined as follows: µe1f1A = {a0.6, b0.75, c0.3}, µe2f1A = {a0.5, b0.8, c0.7}, µe2f2B = {a0.4, b0.6, c0.3}, µe3f2B = {a0.3, b0.35, c0.45}, µe1f3E = {a0.6, b0.75, c0.3}, µe2f3E = {a0.5, b0.8, c0.7}, µe2f3E = {a0.3, b0.35, c0.45}. Then, T is a fuzzy supra soft topology on X, but not fuzzy soft topology, where f1Auf2B 6∈ T. Definition 20. Let (X,T∗, E) be a fuzzy soft topological space and (X,T, E) be a fssts. We say that T is a sfsst associated with T∗ if T∗ ⊂ T. Proposition 1. Let (X,T, E) be a fssts, then it is a parameterized collection of fuzzy supra topologies on X.i.e Te = {fA(e) : fA ∈ T} defines a fuzzy supra topology on X [7] for each e ∈ E. The following example supports our claim. Example 2. Let X = {a, b, c, d} be the set of four jobs under consideration and E = {e1(Salary), e2(Position)}. Let T = {1̃E , 0̃E , f1E , f2E , f3E , f4E , f5E , f6E , f7E , f8E , f9E , f10E , f11E , f12E}, where f1E , f2E , f3E , f4E , f5E , f6E , f7E , f8E , f9E , f10E , f11E , f12E are fuzzy soft sets over X representing the ”the income of the jobs” which some persons are going to work, respectively which defined as follows: µe1f1E = {a0.5, b0.4, c0, d0}, µe2f1E = {a0, b0.3, c0, d0.5}, µe1f2E = {a0, b0.4, c0, d0.5}, µe2f2E = {a0.4, b0, c0.6, d0}, µe1f3E = {a0.5, b0, c0.6, d0}, µe2f3E = {a0.4, b0, c0.6, d0.5}, µe1f4E = {a1, b1, c1, d1}, µe2f4E = {a0.4, b0, c0.6, d0.5}, µe1f5E = {a0.5, b0, c0.6, d0}, µe2f5E = {a1, b1, c1, d1}, µe1f6E = {a0.5, b0.4, c0, d0.5}, µe2f6E = {a0.4, b0.3, c0.6, d0.5}, µe1f7E = {a0.5, b0.4, c0.6, d0.5}, µe2f7E = {a0.4, b0.3, c0.6, d0.5}, µe1f8E = {a1, b1, c1, d1}, µe2f8E = {a0.4, b0.3, c0.6, d0.5}, µe1f9E = {a0.5, b0.4, c0.6, d0}, µe2f9E = {a1, b1, c1, d1}, µe1f10E = {a0.5, b0.4, c0.6, d0.5}, µe2f10E = {a0.4, b0, c0.6, d0.5}, µe1f11E = {a1, b1, c1, d1}, µe2f11E = {a0.4, b0, c0.6, d0.5}, µe1f12E = {a0.5, b0, c0.6, d0.5}, µe2f12E = {a1, b1, c1, d1}. Then, T defines a fuzzy supra soft topology on X. Then it is clear that Te1 and Te2 are fuzzy supra topologies on X, where Te1 = {1, 0, f1E(e1), f2E(e1), ......, f12E(e1)}, and Te2 = {1, 0, f1E(e2), f2E(e2), ......, f12E(e2)}. Now we show that the converse of Proposition 1 does not hold in general by giving the following counterexample. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1005 Example 3. Let X = {a, b, c, d} be the set of four houses under consideration and E = {e1(Wooden), e2(Luxurious)}. Define the fuzzy soft sets fi : E → IX on X, 1 ≤ i ≤ 5, representing the ”the goodness of the houses” which some persons are going to buy, respectively which defined as follows: µe1f1E = {a0.5, b0.4, c0, d0}, µe2f1E = {a0, b0.3, c0, d0.5}, µe1f2E = {a0, b0.4, c0, d0.5}, µe2f2E = {a0.4, b0, c0.6, d0}, µe1f3E = {a0.5, b0, c0.6, d0}, µe2f3E = {a0.4, b0, c0.6, d0.5}, µe1f4E = {a1, b1, c1, d1}, µe2f4E = {a0.4, b0, c0.6, d0.5}, µe1f5E = {a0.5, b0, c0.6, d0}, µe2f5E = {a1, b1, c1, d1}. Then, Te1 = {1, 0, f1E(e1), f2E(e1), ......, f12E(e1)}, and Te2 = {1, 0, f1E(e2), f2E(e2), ......, f12E(e2)}. are fuzzy supra topologies on X, at the time that the collection T = {1̃E , 0̃E , f1E , f2E , f3E , f4E , f5E} is not fssts, where f1E t f2E , f2E t f3E , f3E t f5E , f1E t f3E , f1E t f5E , f2E t f5E 6∈ T. Proposition 2. Let (X,T1, E) and (X,T2, E) be two fssts over the same universe X, then (X,T1 ∧ T2, E) is a fssts on X. Proof. (1) Since 1̃E , 0̃E ∈ T1 and 1̃E , 0̃E ∈ T2, 1̃E , 0̃E ∈ T1 ∧ T2. (2) Consider the collection of fuzzy soft sets {fiE : i ∈ I} ∈ T1 ∧T2. Then {fiE : i ∈ I} ∈ T1 and {fiE : i ∈ I} ∈ T2. Hence, ⊔ i∈I fiE ∈ T1 and ⊔ i∈I fiE ∈ T2. It is follows,⊔ i∈I fiE ∈ T1 ∧ T2. Therefore, T1 ∧ T2 is a fssts on X. Remarks 2. The union of fssts need not to be a fssts in general, as will shown in the following example. Example 4. Let X = {a, b, c} be the set of four watches under consideration and E = {e1(Expensive), e2(Luxurious)}. Let A,B ⊆ E where A = {e1, e2} and B = {e2, e3}. Let T1 = {1̃E , 0̃E , f1A, f2B, f3E} and T2 = {1̃E , 0̃E , g1E , g2B, g3E} where f1A, f2B, f3E , g1E , g2B, g3E are fuzzy soft sets over X, representing the ”the compositions of the watches” which some persons are going to buy, respectively which defined as follows: µe1f1A = {a0.65, b0.7, c0.35}, µe2f1A = {a0.5, b0.3, c0.9}, µe2f2B = {a0.4, b0.7, c0.25}, µe3f2B = {a0.3, b0.35, c0.45}, µe1f3E = {a0.65, b0.7, c0.35}, µe2f3E = {a0.5, b0.7, c0.9}, µe2f3E = {a0.3, b0.35, c0.45}, µe1g1E = {a0.3, b0.5, c0}, µe2g1E = {a0.4, b0.7, c0.1}, µe3g1E = {a0.3, b0.2, c0.95}, µe2g2B = {a0.5, b0.6, c1}, µe3g2B = {a0.8, b0, c0.6}, µe1g3E = {a0.3, b0.5, c0}, µe2g3E = {a0.5, b0.7, c1}, µe2g3E = {a0.8, b0.2, c0.95}. Then T1 and T2 defines fssts on X. But, T1 ∨ T2 = {1̃E , 0̃E , f1A, f2B, f3E , g1E , g2B, g3E} is not fssts on X, where f1A t g1E 6∈ T1 ∨ T2. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1006 Definition 21. A fuzzy soft set gB in a fssts (X,T, E) is called fuzzy supra soft neigh- borhood (briefly: fss neighborhood) of the fuzzy soft point feA∈̃XE if there exists a fuzzy supra open soft set hC such that feA∈̃hC v gB. The fuzzy supra soft neighborhood sys- tem of a fuzzy soft point feA, denoted by NT(feA), is the family of all its fuzzy supra soft neighborhoods. Example 5. Consider the fuzzy supra soft topological space (X,T, E) In Example 1. The fuzzy soft set gA, where µe1gA = {a0.7, b0.8, c0.4}, µe2gA = {a0.6, b1, c0.8} is a fss-neighborhood of the fuzzy soft point ke1A = {a0.5, b0.4, c0.1}. Theorem 2. A fuzzy soft set in a fssts is fuzzy supra soft open if and only if it is a fss-neighborhood of each of its fuzzy soft points. Proof. Obvious. Theorem 3. The fuzzy supra soft neighborhood system NT(feA) in a fssts (X,T, E) has the the following properties: (1) For all fuzzy soft points feA, NT(feA) 6= 0̃E, (2) If kB ∈ NT(feA) ,then feA∈̃kB, (3) If kD v gB and kD ∈ NT(feA), then kd ∈ NT(feA), (4) If e ∈ E is the support of the fuzzy supra soft set fA, then NT(feA) = ∩{NT(fe ′ A ) : 0 < fA(e′) < fA(e)}. (5) If kB ∈ NT(feA), then there exists hC ∈ NT(feA) such that hC v KB and hC ∈ NT(heC). Proof. (1) Since 1̃E is a fss-neighborhood of each of its fuzzy soft points feA from Theorem 2, 1̃E ∈ NT(feA). Then, NT(feA) 6= 0̃E (2) Let kB ∈ NT(feA). Then, there exists a fuzzy supra open soft set hC such that feA∈̃hC v kB. Hence, feA∈̃kB. (3) Let kD ∈ NT(feA). Then, there exists a fuzzy supra open soft set hC such that feA∈̃hC v kD. Since kD v gB, feA∈̃hC v kD v gB. Therefore, kD ∈ NT(feA). (4) Obvious. (5) Let kB ∈ NT(feA). Then, there exists a fuzzy supra open soft set sF such that feA∈̃sF v kB. But, sF is a fss-neighborhood of each of its fuzzy soft points from Theorem 2. Hence, sF ∈ NT(seF ). A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1007 Theorem 4. Let the collection N(geB be a fuzzy supra soft neighborhood system of the fuzzy soft point geB with the properties (1)-(5) in Theorem 3. Then, there exists a unique fssts T on X for which N(geB) coincides with the family of fss-neighborhood NT(geB) of geB with respect to T. Proof. Consider the collection T = {fA : fA ∈ N(geB) and geB∈̃fA}. We want to prove that T is a fssts on X. (1) Since 0̃E contains no fuzzy soft point, 0̃E ∈ T. Also, from condition (1), NT(feA) 6= 0̃E , so there exist some neighborhood of every fuzzy soft point in X, which is a superset of each of the fss-neighborhood s. Hence, 1̃E ∈ T from condition (3). (2) Let {(fA)j : j ∈ J} ⊆ T. Then, (fA)j ∈ N(geB), j ∈ J . By condition (3), tj∈J(fA)j ∈ N(geB). Therefore, T is a fssts on X. Now, we prove that NT(geB) = N(geB). Let hA ∈ N(geB), then there exists kC ∈ NT(geB) such that kC v KB and kC ∈ N(keC) from condition (5). It is follows, geB∈̃kC from condition (2). Hence, kC ∈ T from Theorem 2. This means, hA is T-fss-neighborhood of geB. Thus, NT(geB) ⊆ N(geB). (1) Conversely, Let nY ∈ NT(geB), then there exists a fuzzy supra open soft set mZ such that geB∈̃mZ v nY . It is follows, mZ ∈ N(geB) from Theorem 2. But, mZ v nY , then nY ∈ N(geB) from condition (3). Hence, N(geB) ⊆ NT(geB). (2) from (3.1) and (3.2) we have NT(geB) = N(geB). Definition 22. Let (X,T, E) be a fuzzy supra soft topological space over and gB ∈ SS(X)E. Then the fuzzy supra soft interior of gB, denoted by Fints(gB) is defined as Fints(gB) = t{hA : hA is fuzzy supra open soft set and hA v gB}. (3) Also, the fuzzy supra soft closure of gB, denoted by Fcls(gB) is defined as Fcls(gB) = u{hA : hA is fuzzy supra closed soft set and gB v hA}. (4) Definition 23. Let (X,T, E) be a fssts over X and gB ∈ SS(X)E. Then, feA ∈ SS(X)E is called fuzzy supra limit soft point of fA if (gB − feA)u hC 6= 0̃E ∀ hC ∈ FSOS(X). The set of all fuzzy supra limit soft points of gB is called the fuzzy supra soft derived of gB and denoted by dsf (gB). Theorem 5. Let (X,T, E) be a supra soft topological space and fA, gB ∈ SS(X)E. Then (1) Fints(1̃E) = 1̃E and Fints(0̃E) = 0̃E. (2) Fcls(1̃E) = 1̃E and Fcls(0̃E) = 0̃E. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1008 (3) fA is fuzzy supra open soft if and only if Fints(fA) = fA. (4) Fints(Fints(fA)) = Fints(fA) and Fints(fA) v fA. (5) fA is fuzzy supra closed soft set if and only if Fcls(fA) = fA. (6) Fcls(Fcls(fA)) = Fcls(fA) and fA v Fcls(fA). (7) If fA v gB, then Fints(fA) v Fints(gB) and Fcls(fA) v Fcls(gB). (8) Fcls(fA) = [Fints(f cA)]c. Proof. Obvious. Theorem 6. Let (X,T, E) be a fssts on X and fA ∈ SS(X)E. Then: (1) Fints(f cA) = 1̃E − [Fcls(fA)]. (2) Fcls(f cA) = 1̃E − [Fints(fA)]. Proof. (1) Since Fcls(fA) = u{hD : hD ∈ FSCS(X), fA v hD}, 1̃E − Fcls(fA) = t{hc D : h c D ∈ FSOS(X), h c D v f c A} = Fints(f cA). (2) By a similar way. In the next theorem, we list the main properties of the operations which give the deviations between these operations and that in fuzzy soft topological spaces. Theorem 7. Let (X,T, E) be a supra soft topological space and fA, gB ∈ SS(X)E. Then (1) Fcls(fA) t Fcls(gB) v Fcls(fA t gB). (2) dsf (fA) t dsf (gB) v dsf (fA t gB). (3) Fints(fA u gB) v Fints(fA) u Fints(gB). Proof. Immediate. Remarks 3. The equality of each part in Theorem 7 is not true in general as will shown in the following examples. Example 6. (1) Consider the fssts in Example 2 and consider the two fuzzy soft sets gE and hE over X defined by; µe1gE = {a0.5, b0.4, c0, d0}, µe2gE = {a0, b0.3, c0, d0.5} and µe1hE = {a0.5, b0.6, c1, d1}, µe2hE = {a1, b0.7, c1, d0.5}. It is follows, Fcls(gE t hE) = 1̃E and Fcls(gE) t Fcls(hE) = kE, where KE is a fuzzy soft set over X defined by: µe1kE = {a0.5, b0.6, c1, d1}, µe2kE = {a1, b0.7, c1, d0.5}. Therefore, Fcls(gE t hE) 6v Fcls(gE) t Fcls(hE). A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1009 (2) In (1), dsf (gE t hE) 6v dsf (gE) t dsf (hE) (3) Consider the fssts in Example 2 and consider the two fuzzy soft sets mE and nE over X defined by; µe1mE = {a0, b0.4, c0, d0.5}, µe2mE = {a0.4, b0, c0.6, d0}, µe1nE = {a0.5, b0, c0.6, d0}, µe2nE = {a0.4, b0, c0.6, d0.5}. It is follows, Fints(mE unE) = 0̃E and Fints(mE)uFints(nE) = lE, where lE is a fuzzy soft set over X defined by: µe1lE = {a0, b0, c0, d0.5}, µe2lE = {a0.4, b0, c0.6, d0.5}. Therefore, Fints(mE) u Fints(nE) 6v Fints(mE u nE). Definition 24. Let (X,T, E) be a fssts and Y ⊆ X. Let yE be a fuzzy soft set over (Y,E) defined by: yE : E → IY such that yE(e) = µeyE , where µeyE (x) = { 1, x ∈ Y, 0, x 6∈ Y. Then, the fssts TyE = {yEugB : gB ∈ T} is called the fuzzy supra soft subspace topology for yE and (Y,TyE , E) is called fuzzy supra soft subspace of (X,T, E). Example 7. Consider the fssts in Example 1, let Y = {a, b} ⊆ X. We consider the fuzzy soft set yE over (Y,E) defined as follows: µe1yE = {a1, b1, c0}, µe2yE = {a1, b1, c0}, µe3yE = {a1, b1, c0}. Then TyE = {yE u zE : zE ∈ T} where yE u 0̃E = 0̃E, yE u 1̃E = yE, yE u f1A = hA, where µe1hA = {a0.6, b0.75, c0}, µe2hA = {a0.5, b0.8, c0}, yE u f2B = hB, where µe2hB = {a0.4, b0.6, c0}, µe2hB = {a0.3, b0.35, c0}, yE u f3E = hE, where µe1hE = {a0.6, b0.75, c0}, µe2hE = {a0.5, b0.8, c0}, µe3hE = {a0.3, b0.35, c0}. Thus, the collection TyE = {yE u zE : zE ∈ T} is a fuzzy supra soft subspace of T. Proposition 3. Let (Y,TyE , E) be a fuzzy supra soft subspace of a fssts (X,T, E) and gB ∈ FSS(X)E. Then, gB is TyE -fuzzy supra closed soft if and only if gB = yE u kc for some T-fuzzy supra closed soft set kC . Proof. Immediate. Proposition 4. Let (Y,TyE , E) be a fuzzy supra soft subspace of a fssts (X,T, E) and gB ∈ FSS(X)E. Then, TyE -fuzzy supra soft closure of gB, denoted by FclsTyE , where FclsTyE (gB) = yE u Fcls(gB). Proof. Immediate. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1010 4. Fuzzy Supra Soft Continuous Mappings In this section, we consider the notion of fuzzy supra soft continuity as a generalization to fuzzy soft continuity [8], fuzzy semi-soft continuity [19], fuzzy pre-soft continuity [1], fuzzy α-soft continuity [2], fuzzy β-soft continuity [3] and fuzzy b-soft continuity [6], sup- ported by examples and counterexamples. We also introduce and study the concepts of fuzzy supra open (resp. closed) soft functions a generalization to fuzzy open (resp. closed) soft functions [30]. Definition 25. Let (X,T∗1, E), (Y,T∗2,K) be two fuzzy soft topological spaces, T1 be an associated fssts with T∗1 and fpu : FSS(X)E → FSS(Y )K be a soft function. Then, fpu is called fuzzy supra soft continuous (fss-continuous, in short) if f−1 pu (gB) ∈ T1 ∀ gB ∈ T∗2. Theorem 8. Let (X,T∗1, E), (Y,T∗2,K) be two fuzzy soft topological spaces, T1, T2 be associated fssts with T∗1, T∗2, respectively. Let fpu : FSS(X)E → FSS(Y )K be a soft function. Then, the following are equivalent: (1) fpu is fss-continuous. (2) f−1 pu (hB) ∈ Tc 1 ∀ hB ∈ T∗c2. (3) fpu(Fcls(gA)) v Fcls(fpu(gA)) ∀ gA ∈ FSS(X)E. (4) Fcls(f−1 pu (hB)) v f−1 pu (Fcls(hB)) ∀ hB ∈ FSS(Y )K . (5) f−1 pu (Fints(hB)) v Fints(f−1 pu (hB)) ∀ hB ∈ FSS(Y )K . Proof. (1) ⇒ (2) Let hB ∈ T∗c2. Then, hcB ∈ T∗2 and f−1 pu (hcB) ∈ T1 from Definition 25. Since f−1 pu (hcB) = (f−1 pu (hB))c from Theorem 1, f−1 pu (hB) ∈ Tc 1. (2) ⇒ (3) Let gA ∈ FSS(X)E . Since gA v f−1 pu (fpu(gA)) v f−1 pu (Fcls(fpu(gA))) ∈ Tc 1 from (2) and Theorem 1. Then, gA v Fcls(gA) v f−1 pu (Fcls(fpu(gA))). Hence, fpu(Fcls(gA)) v fpu(f−1 pu (Fcls(fpu(gA)))) v Fcls(fpu(gA))) from Theorem 1. Thus, fpu(Fcls(gA)) v Fcls(fpu(gA)). (3) ⇒ (4) Let hB ∈ FSS(Y )K and gA = f−1 pu (hB). Applying (3), fpu(Fclsf−1 pu (hB)) v Fcls(fpu(f−1 pu (hB))). Hence, Fcls(f−1 pu (hB)) v f−1 pu (fpu(Fcls(f−1 pu (hB)))) v f−1 pu (Fcls(fpu (f−1 pu (hB)))) v f−1 pu (Fcls(hB)) from Theorem 1. Thus, Fcls(f−1 pu (hB)) v f−1 pu (Fcls(hB)). (4) ⇒ (2) Let hB ∈ T∗c2. Then, Fcls(hB) = hB and Fcls(f−1 pu (hB)) v f−1 pu (Fcls(hB)) = f−1 pu (hB) from (4). But, we have f−1 pu (hB) v Fcls(f−1 pu (hB)). It is follows, f−1 pu (hB) = Fcls(f−1 pu (hB)), and consequently f−1 pu (hB) ∈ Tc 1. (1) ⇒ (5) Let hB ∈ FSS(Y )K . Since Fints(hB) ∈ T∗2, f−1 pu (Fints(hB)) ∈ T1 from (1). Hence, f−1 pu (Fints(hB)) = Fints(f−1 pu Fint s(hB)) v Fints(f−1 pu (hB)). Thus, f−1 pu (Fints(hB)) v Fints(f−1 pu (hB)). A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1011 (5) ⇒ (1) Let hB ∈ T∗2. Then Fints(hB) = hB and f−1 pu (Fints(hB)) = f−1 pu ((hB)) v Fints(f−1 pu (hB)) from (5). But, we have Fints(f−1 pu (hB)) v f−1 pu (hB). It is follows, Fints(f−1 pu (hB)) = f−1 pu (hB) ∈ T1. Hence, fpu is a fss-continuous. Theorem 9. Every fuzzy soft continuous function [8] is a fss-continuous. Proof. Immediate from Definition 18 (1) and Definition 25. Remarks 4. The converse of Theorem 9 is not true in general, as will shown in the following example. Example 8. Let X = {a, b, c}, Y = {l,m, n}, E = {e1, e2, e3} and K = {k1, k2, k3}. Define u : X → Y and p : A→ B as follows: u(a) = {n}, u(b) = {m} , u(c) = {m} and p(e1) = {k1}, p(e2) = {k1}, p(e3) = {k3}. Let (X,T∗1, E) be a fuzzy soft topological space over X, where T∗1 = {1̃E , 0̃E , fE}, where fE is a fuzzy soft set over X defined as follows: µe1fE = {a0.1, b0.5, c0.7}, µe2fE = {a0.2, b0.7, c0.5}, µe3fE = {a0.7, b0.4, c0.3}. Consider the associated fssts T1 with T∗1, where T1 = {1̃E , 0̃E , fE , hE , kE}, where fE , hE , kE are fuzzy soft sets over X defined as follows: µe1hE = {a0.2, b0.6, c0.6}, µe2hE = {a0.2, b0.6, c0.6}, µe3hE = {a0.9, b0.1, c0.1}, µe1kE = {a0.2, b0.6, c0.7}, µe2kE = {a0.2, b0.7, c0.5}, µe3kE = {a0.9, b0.4, c0.3}. Let (X,T∗2,K) be a fuzzy soft topological space over Y , where T∗2 = {1̃K , 0̃K , gK}, where gK is a fuzzy soft set over Y defined by: µk1gK = {a0.5, b0.6, c0.2}, µk2gK = {a0.8, b0.3, c0.4}, µk3gK = {a0.7, b0.1, c0.9}. Let fpu : (X,T∗1, E) → (Y,T∗2,K) be a soft function. Next, for p(ei) ∈ K, i = 1, 2, 3 we calculate f−1 pu (gK) as follows: f−1 pu (gK)(e1)(a) = gK(p(e1))(u(a)) = gK(k1)(n) = ({a0.5, b0.6, c0.2})(n) = 0.2, f−1 pu (gK)(e1)(b) = gK(p(e1))(u(b)) = gK(k1)(m) = ({a0.5, b0.6, c0.2})(m) = 0.6, f−1 pu (gK)(e1)(c) = gK(p(e1))(u(c)) = gK(k1)(m) = ({a0.5, b0.6, c0.2})(m) = 0.6, f−1 pu (gK)(e2)(a) = gK(p(e2))(u(a)) = gK(k1)(n) = ({a0.5, b0.6, c0.2})(n) = 0.2, f−1 pu (gK)(e2)(b) = gK(p(e2))(u(b)) = gK(k1)(m) = ({a0.5, b0.6, c0.2})(m) = 0.6, f−1 pu (gK)(e2)(c) = gK(p(e2))(u(c)) = gK(k1)(m) = ({a0.5, b0.6, c0.2})(m) = 0.6, f−1 pu (gK)(e3)(a) = gK(p(e3))(u(a)) = gK(k3)(n) = ({a0.9, b0.1, c0.1)(n) = 0.9, f−1 pu (gK)(e3)(b) = gK(p(e3))(u(b)) = gK(k3)(m) = ({a0.9, b0.1, c0.1)(m) = 0.1, f−1 pu (gK)(e3)(c) = gK(p(e3))(u(c)) = gK(k3)(m) = ({a0.9, b0.1, c0.1)(m) = 0.1. Hence, f−1 pu (gK) = {(e1, {a0.2, b0.6, c0.6}), (e2, {a0.2, b0.6, c0.6}), (e3, {a0.9, b0.1, c0.1)}. It is follows, f−1 pu (gK) ∈ T1 and f−1 pu (gK) 6∈ T∗1. Therefore, fpu is a fss-continuous but not fuzzy soft continuous. Definition 26. Let (X,T∗1, E), (Y,T∗2,K) be two fuzzy soft topological spaces, T2 be an associated fssts with T∗2 and fpu : FSS(X)E → FSS(Y )K be a soft function. Then, (1) fpu is called fuzzy supra open soft if fpu(gE) ∈ T2 ∀ gE ∈ T∗1. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1012 (2) fpu is called fuzzy supra closed soft if fpu(gE) ∈ Tc 2 ∀ gE ∈ T∗c1. Theorem 10. Every fuzzy open (resp. closed) soft function [15] is a fuzzy supra open (resp. closed) soft. Proof. Immediate from Definition 25 (2), (3) and Definition 26. Remarks 5. The converse of Theorem 10 is not true in general, as will shown in the following example. Example 9. In Example 8, let (X,T∗1, E) be a fuzzy soft topological space over X where, T∗1 = {1̃E , 0̃E , zE}, where zE is a fuzzy soft set over X defined as follows: µe1zE = {a0.5, b0, c0.8}, µe2zE = {a0.1, b0.9, c0.5}, µe3zE = {a0.4, b0.3, c0.6}. Let (X,T∗2,K) be a fuzzy soft topological space over Y where, T∗2 = {1̃K , 0̃K , gK}, where gK is a fuzzy soft set over Y defined by: µk1gK = {a0.3, b0.2, c0.7}, µk2gK = {a1, b0.3, c0.5}, µk3gK = {a0.2, b0.9, c0.1}. Consider the associated fssts T2 with T∗2, where T2 = {1̃K , 0̃K , gK , hK , iK}, where gK , hK , iK are fuzzy soft sets over X defined as follows: µk1hK = {a0, b0.9, c0.5}, µk2hK = {a0, b0.6, c0.4}, µk3hK = {a0, b0, c0}, µk1iK = {a0.3, b0.9, c0.7}, µk2iK = {a1, b0.6, c0.5}, µk3iK = {a0.2, b0.9, c0.1}. Let fpu : (X,T∗1, E) → (Y,T∗2,K) be a soft function. Hence, fpu(zE) = {(k1, {a0, b0.9, c0.5}), (k2, {a0, b0.6, c0.4}), (k3, {a0, b0, c0)}. It is follows, fpu(zE) ∈ T2 and fpu(zE) 6∈ T∗2. There- fore, fpu is a fuzzy supra open (resp. closed) soft function but not fuzzy open (resp. closed) soft. Theorem 11. Let (X,T∗1, E), (Y,T∗2,K) be two fuzzy soft topological spaces, T2 be an associated fssts with T∗2 and fpu : FSS(X)E → FSS(Y )K be a soft function. Then, (1) fpu is a fuzzy supra open soft mapping if and only if fpu(Fints(gE)) v Fints(fpu(gE)) ∀ gE ∈ FSS(X)E. (2) fpu is a fuzzy supra closed soft mapping if and only if Fcls(fpu(gE)) v fpu(Fcls(gE)) ∀ gE ∈ FSS(X)E. Proof. (1) (⇒:) Let gE ∈ FSS(X)E . Since Fints(gE) v fA, fpu(Fints(gE)) v fpu(gE) from Theorem 1. Since fpu is fuzzy supra open soft mapping, Fints(fpu(Fints(gE))) = fpu(Fints(gE)) v Fints(fpu(fA)). (:⇐) Let gE ∈ T∗1. Then, Fints(gE) = gE . By hypothesis, fpu(Fints(gE)) v Fints(fpu(gE)). Hence, fpu(gE) v Fints(fpu(gE)) v fpu(gE). It is follows, fpu(gE) = Fints(fpu(gE)). Thus, fpu(gE) ∈ T2. This completes the proof. (2) By a similar way. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1013 Definition 27. Let (X,T∗1, E), (Y,T∗2,K) be two fuzzy soft topological spaces, T1, T2 be associated fssts with T∗1, T∗2, respectively. Let fpu : FSS(X)E → FSS(Y )K be a soft function. Then, fpu is fuzzy supra soft homeomorphism if it is bijection, fss-continuous and f−1 pu is fss-continuous. Theorem 12. Let (X,T∗1, E), (Y,T∗2,K) be two fuzzy soft topological spaces, T2 be an associated fssts with T∗2 and fpu : FSS(X)E → FSS(Y )K be a bijective soft function. Then, the following are equivalent: (1) fpu is a fuzzy supra soft homeomorphism. (2) fpu is a fss-continuous and fuzzy supra closed soft mapping. (3) fpu is a fss-continuous and fuzzy supra open soft mapping. Proof. Immediate. 5. Fuzzy Supra Soft Compact Spaces In this section, we introduce the notion of fuzzy supra soft compact (resp. fuzzy supra soft lindelöf) spaces as a generalization to such introduced in [11, 12, 17, 24, 27, 30]. We also introduce some basic definitions of fuzzy supra soft compact spaces and theorems of the concept. Definition 28. Let (X,T, E) be a fssts. A family Ψ = {uiE : i∈Λ} of fuzzy soft sets is said to be a fuzzy supra open soft cover, if each member of Ψ is fuzzy supra open soft set. Definition 29. A fuzzy soft subset fA of the space (X,T, E) is said to be fuzzy supra soft compact (resp. fuzzy supra soft lindelöf), if every fuzzy supra open soft cover {uiE : i∈Λ} of fA has a finite (resp. countable) subfamily Λo of Λ such that fA v t̃i∈ΛouiE. The space (X,T, E) is said to be a fuzzy supra soft compact if 1̃E is a fuzzy supra soft compact as a fuzzy soft subset. Example 10. Let (X,T1, E) and (X,T2, E) be two fssts such that T1 ⊆ T2. If T2 is a fuzzy supra soft compact, then also T1. Proposition 5. If X is finite (resp. countable), then (X,T, E) is fuzzy supra soft compact (resp. fuzzy supra soft lindelöf) for any fuzzy supra soft topology T on X. Proof. It is easy to be obtained. Theorem 13. Every fuzzy supra closed soft subspace of a fuzzy supra soft compact space is a fuzzy supra soft compact. A. M. Abd El-latif / Eur. J. Pure Appl. Math, 12 (3) (2019), 999-1017 1014 Proof. Let zA be a fuzzy supra closed soft subspace of fuzzy supra soft compact space (X,T, E) and {uiE : i∈Λ} be a fuzzy supra open soft cover of zA. Hence, {uiE : i∈Λ}t zcA is a fuzzy supra open soft cover of 1̃E , and for 1̃E is fuzzy supra soft compact, there exists a finite subcover {uiE : i∈Λo} t zcA) for 1̃E . Now, [{uiE : i∈Λo} t (zcA)] − (zcA) is a finite subcover of {uiE : i∈Λ} for zA. So, zA is a fuzzy supra soft compact. Theorem 14. Every supra closed soft subspace of fuzzy supra soft lindelöf space is a fuzzy supra soft lindelöf. Proof. It is similar to the proof of Theorem 13. Theorem 15. Every fuzzy supra soft subspace of fssts (X,T, E) is a fuzzy supra soft compact if and only if every fuzzy supra open soft subspace of 1̃E is a fuzzy supra soft compact. Proof. Let (Y,TyE , E) be a fuzzy supra open soft subspace of a fssts (X,T, E) and {uαE : α∈Λ} be a fuzzy supra open soft cover of (Y,TyE , E). Assume that vC = tα∈ΛuαE . Hence vC is a fuzzy supra open soft subspace of 1̃E . By hypothesis, vC is a fuzzy supra soft compact. So, {uαE : α ∈ Λo,Λo is finite} is a finite subcover of vC . It is follows, vC v tα∈ΛouαE . Thus, yE v vC v tα∈ΛouαE . Therefore, (Y,TyE , E) is a fuzzy supra soft compact. For the necessity, it is clear. Theorem 16. Every fuzzy supra soft subspace of fssts (X,T, E) is a fuzzy supra soft lindelöf if and only if every fuzzy supra open soft subspace of 1̃E is a fuzzy supra soft lindelöf. Proof. It similar to the proof of Theorem 15. Theorem 17. Let (X,T1, E) and (Y,T2,K) be two fssts and let fpu:(X,T1, E)→(Y,T2,K) be a surjective and fss-continuous function. If (X,T1, E) is fuzzy supra soft compact, then (Y,T2,K) is also fuzzy supra soft compact. Proof. Let {uiK : i∈Λ} be a fuzzy supra open soft cover of 1̃K . Since fpu is a fss- continuous, {f−1 pu (uiK) : i∈Λ} is a fuzzy supra open soft cover of 1̃E , and for 1̃E is a fuzzy supra soft compact, there exists a finite subfamily Λo of Λ such that {f−1 pu (uiK) : i∈Λo} also forms a fuzzy supra open soft cover of 1̃E . Since fpu is surjective, {fpu(f−1 pu (uiK)) : i∈Λo} = {uiK : i∈Λo} forms a finite fuzzy supra open soft cover of 1̃K . This competes the proof. Theorem 18. Let (X,T1, E) and (Y,T2,K) be two fssts and let fpu:(X,T1, E)→(Y,T2,K) be a surjective and fss-continuous function. If (X,T1, E) is fuzzy supra soft lindelöf, then (Y,T2,K) is also fuzzy supra soft lindelöf. Proof. It is similar to the proof of Theorem 17. REFERENCES 1015 6. Conclusion In this paper, we introduce and study the notion of fuzzy supra soft topological spaces. We introduce the some new concepts in fuzzy supra soft topological spaces. 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