# [sigma]-Algebra and [sigma]-Baire in Fuzzy Soft Setting.

1. IntroductionThe concepts of Baire spaces have been studied and discussed extensively in general topology in [1-4]. Thangaraj and Balasubramanian [5] studied the notion of somewhat fuzzy continuous functions. Next, Thangaraj and Anjalmose investigated and discussed the notion of Baire space in fuzzy topology [6]. After that, they introduced the notion of fuzzy Baire space [7].

Soft sets theory was introduced by Molodtsov [8]. It explains new type of mathematical tool of soft sets and it deals with vagueness when solving problems in practice as in engineering, environment, social science, and economics, which cannot be handled as classical mathematical tools. Also, other authors such as Maji et al. [9-19] have further studied the theory of soft sets and used this theory in pure mathematics to solve some decision making problems. Next, the notion of fuzzy soft set is investigated and discussed [20-22]. Since then much attention has been used to generalize the basic notions of fuzzy topology in soft setting. In other words, the modern theories of fuzzy soft topology have been developed.

In recent years, fuzzy soft topology has been found to be very useful in solving many practical problems [23]. Also, rough fuzzy and other applications are studied [24-26]. The main purpose of this work is to introduce new concepts of fuzzy soft Baireness in fuzzy soft topological spaces. In section three, we introduce fuzzy soft dense sets, fuzzy soft nowhere dense sets, fuzzy soft meager sets, fuzzy soft second category sets, fuzzy soft meager spaces, fuzzy soft second category spaces, fuzzy soft residual sets, fuzzy soft Baire spaces, fuzzy soft [delta]-sets, fuzzy soft [[lambda].sub.[sigma]]-sets, fuzzy soft [sigma]-nowhere dense, fuzzy soft [sigma]-meager, fuzzy soft [sigma]-residual, fuzzy soft [sigma]-Baire, fuzzy soft [sigma]-second category, fuzzy soft [sigma]-residual, fuzzy, fuzzy soft submaximal space, fuzzy soft P-space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft A-embedded, fuzzy soft D-Baire, fuzzy soft almost P-spaces, fuzzy soft Borel, and fuzzy soft o-algebra. Moreover, several examples are given to illustrate the notions introduced in this work.

2. Preliminaries

In this section, we give few definitions and properties regarding fuzzy soft sets.

Definition 1 ([20]). Assume U is an initial universe set and E is a set of parameters. Let [I.sup.U] refer to the family of all fuzzy soft sets (FSSs) in U and A [subset or equal to] E. The multivalued map [F.sub.A] : A [right arrow] [I.sup.U] defined by [mathematical expression not reproducible] is said to be a fuzzy soft set (FSS) over (U, E), where [sigma] if e [member of] A and [sigma] if e [member of] E \ A. We refer to family of all (FSSs) over (U, E) by FS(U, E).

Definition 2 ([20]). We say [F.sub.[phi]] [ FS(U, E) is null (FSS) and we refer to it by [PHI], if [for all]e [member of] E, F(e) is the null (FSS) [bar.0] of U, where [bar.0](x) = 0 [for all]x [member of] U.

Definition 3 ([20]). Assume [F.sub.E] [member of] FS(U,E) and [F.sub.E](e) = [bar.1] [for all]e [member of] E, where [bar.1](x) = [1.bar] [for all]x [member of] U. We say [F.sub.E] is absolute (FSS) and we refer to it by [bar.E].

Definition 4 ([20]). We say [F.sub.A] is a fuzzy soft subset of a (FSS) [G.sub.B] over a common universe U if A [subset or equal to] B and [F.sub.A](e) [subset or equal to] [G.sub.B](e) [for all]e [member of] A; i.e., if [mathematical expression not reproducible] and [for all]e [member of] E and denoted by [F.sub.A] [??] [G.sub.B].

Definition 5 ([20]). Assume [F.sub.A] and [G.sub.B] are (FSSs) over a common universe U. We say they are fuzzy soft equal if [F.sub.A] [??] [G.sub.B] and [G.sub.B] [??] [F.sub.A].

Definition 6 ([20]). Assume [F.sub.A] and [G.sub.B] are (FSSs) over a common universe U. Their union is the (FSS) [H.sub.C], defined by [mathematical expression not reproducible], where C = A [??] B. For this case, we write [H.sub.C] = [F.sub.A] [??] [G.sub.B].

Definition 7 ([20]). Assume [F.sub.A] and [G.sub.B] are (FSSs) over a common universe U. Their intersection is a (FSS) [H.sub.C], defined by [mathematical expression not reproducible], where C = A [??] B. For this case, we write [H.sub.C] = [F.sub.A] [??] [G.sub.B].

Definition 8 ([27]). Assume [F.sub.A] [member of] FS(U, F) is a (FSS). We refer to its complement by [F.sub.A.sup.c], and it is defined as

[mathematical expression not reproducible]. (1)

Remark 9. Let [mathematical expression not reproducible] be a family of fuzzy soft sets. The union (intersection) of any number (J [subset or equal to] I) of family K is defined by [mathematical expression not reproducible].

Definition 10 ([28]). Assume [psi] is the family of (FSSs) over U. We say [psi] is a fuzzy soft topology on U if [psi] the following axioms hold:

(i) [PHI], [bar.E] belong to [psi].

(ii) The union of any number of (FSSs) in [psi] belongs to [psi].

(iii) The intersection of any two (FSSs) in [psi] belongs to [psi].

We say (U, E, [psi]) is a fuzzy soft topological space (FSTS) over U. Each member in [psi] is called (FSOS) (FSOS) in U and its complement is called fuzzy soft closed set (FSCS) in U, where [mathematical expression not reproducible].

Definition 11 ([28]). The union of all fuzzy soft open subsets of [F.sub.A] over (U, E) is called the interior of [F.sub.A] and is denoted by [int.sup.fs] ([F.sub.A]).

Proposition 12 ([28]). [int.sup.fs] ([F.sub.A] [??] [G.sub.B]) = [int.sup.fs]([F.sub.A]) [??] [int.sup.fs]([G.sub.B]).

Definition 13 ([28]). Let [F.sub.A] [member of] FS(U, E) be a (FSS). Then the intersection of all closed sets, each containing [F.sub.A], is called the closure of [F.sub.A] and is denoted by [cl.sup.fs]([F.sub.A]).

Remark 14. (1) For any (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]), it is easy to see that [([cl.sup.fs]([F.sub.A])).sup.c] = [int.sup.fs]([F.sub.A.sup.c]) and [([int.sup.fs]([F.sub.A])).sup.c] [cl.sup.fs] ([F.sub.A.sup.c]).

(2) For any fuzzy soft [F.sub.A] subset of a (FSTS) (U, E, [psi]), we define the fuzzy soft subspace topology [mathematical expression not reproducible] on [F.sub.A] by [mathematical expression not reproducible] if [K.sub.D] = [F.sub.A] [??] [G.sub.B] for some [G.sub.B] [member of] [psi].

(3) For any fuzzy soft [H.sub.C] in [F.sub.A] fuzzy soft subspace of a (FSTS), we denote the interior and closure of [H.sub.C] in [F.sub.A] by [mathematical expression not reproducible] and [mathematical expression not reproducible], respectively

3. Fuzzy Soft [sigma]-Baire Spaces

In this section, we introduce new notions of (FSTSs) using new classes of (FSSs) which are introduced in this section and obtained their properties.

Definition 15. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft dense if there exists no (FSCS) [G.sub.B] in (U, E, [psi]) such that [F.sub.A] [??] [G.sub.B] [??] [bar.E].

Definition 16. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft nowhere dense if there exists no nonempty (FSOS) [G.sub.B] in (U, E, [psi]) such that [G.sub.B] [??] [cl.sup.fs]([F.sub.A]). That is, [int.sup.fs]([cl.sup.fs]([F.sub.A])) = [PHI].

Definition 17. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft meager if [F.sub.A] is a countable union of fuzzy soft nowhere dense sets [i.e., if [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft nowhere dense sets in (U, E, [psi]), [for all]i [member of] I [subset or equal to] N]. Otherwise, [F.sub.A] will be called a fuzzy soft second category set.

Definition 18. A (FSTS) (U, E, [psi]) is called fuzzy soft meager or (fuzzy soft first category) space if the (FSS) [bar.E] is a fuzzy soft meager set in (U, E, [psi]). That is, [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft nowhere dense sets in (U, E, [psi]). Otherwise, (U, E, [psi]) will be called a fuzzy soft second category space.

Definition 19. Assume [F.sub.A] is a fuzzy soft meager set in (U, E, [psi]). We say [F.sub.A.sup.c] is a fuzzy soft residual set in (U, E, [psi]).

Definition 20. Assume (U, E, [psi]) is a (FSTS). We say (U, E, [psi]) is a fuzzy soft Baire space if each sequence [mathematical expression not reproducible] of fuzzy soft nowhere dense sets in (U, E, [psi]) such that [mathematical expression not reproducible].

Definition 21. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft [delta]-set in (U, E, [psi]) if [mathematical expression not reproducible].

Remark 22. Definition 20 can be written in other equivalent forms as follows:

(i) We say (U, E, [psi]) is a fuzzy soft Baire space if every countable fuzzy soft closed cover [mathematical expression not reproducible] of [bar.E], the set [mathematical expression not reproducible] is fuzzy soft dense in [bar.E].

(ii) We say (U, E, [psi]) is a fuzzy soft Baire space if every sequence [mathematical expression not reproducible] of (FSOSs) with the same closure [F.sub.A], we have [mathematical expression not reproducible].

(iii) We say (U, E, [psi]) is a fuzzy soft Baire space if every fuzzy soft meager and fuzzy soft [delta]-set in [bar.E] is fuzzy soft nowhere dense.

Remark 23. We say SS([U.sub.E]) is a family of all soft sets over a universe set U and the parameter set E. Moreover, the cardinality of SS([U.sub.E]) is given by n(SS([U.sub.A])) = [2.sup.n(U)xn(E)]. Therefore, in this paper for each (FSS) [F.sub.E] over (U, E) we can define [F.sub.E] by using matrix form as follows:

[mathematical expression not reproducible] (2)

The order of this matrix is given by m x k, where m = n(E), k = n(U), and [mathematical expression not reproducible] and 1 [less than or equal to] j [less than or equal to] k.

Definition 24. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft [[lambda].sub.[sigma]]-set in (U, E, [psi]) if [mathematical expression not reproducible].

Definition 25. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft [sigma]-nowhere dense set if [F.sub.A] is a fuzzy soft [[lambda].sub.[sigma]]-set in (U, E, [psi]) such that [int.sup.fs]([F.sub.A]) = [phi].

Example 26. As an illustration, consider the following example. Suppose the (FSSs) [F.sub.E], [G.sub.E], [H.sub.E], [J.sub.E], [K.sub.E], [L.sub.E], [T.sub.E] describe attractiveness of the cars with respect to the given parameters, which my friends are going to buy. U = {[x.sub.1], [x.sub.2], [x.sub.3]} which is the set of all cars under consideration. Let [I.sup.U] be the collection of all fuzzy subsets of U. Also, let E = {[e.sub.1], [e.sub.2], [e.sub.3]}, where [e.sub.1], [e.sub.2], [e.sub.3] stand for the attributes of cheap, qualification, and colorful, respectively. Let [F.sub.E], [G.sub.E], [H.sub.E], [J.sub.E], [K.sub.E], [L.sub.E], [T.sub.E] be defined as follows:

[mathematical expression not reproducible]. (3)

Then, [psi] = {[PHI], [bar.E], [F.sub.E], [G.sub.E], [H.sub.E], [J.sub.E], [K.sub.E], [L.sub.E], [T.sub.E]} is clearly a fuzzy soft topology on [bar.E]. Now consider the (FSS)

[mathematical expression not reproducible] (4)

in (U, E, [psi]). Then [alpha] is a fuzzy soft [[lambda].sub.[sigma]]-set in (U, E, [psi]) and [int.sup.fs]([alpha]) = [PHI] and hence [alpha] is a fuzzy soft [sigma]-nowhere dense set in (U, E, [psi]). The (FSS)

[mathematical expression not reproducible] (5)

is a fuzzy soft [[lambda].sub.[sigma]]-set in (U, E, [psi]) and [int.sup.fs]([beta]) = [F.sub.E] [not equal to] [PHI] and hence [beta] is not a fuzzy soft [sigma]-nowhere dense set in (U, E, [psi]).

Definition 27. A (FSS) [F.sub.A] in a (FSTS) (U, E, [psi]) is called fuzzy soft [sigma]-meager if [mathematical expression not reproducible], are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Any other (FSS) in (U, E, [psi]) is said to be of fuzzy soft [sigma]-second category.

Definition 28. A (FSTS) (U, E, [psi]) is called fuzzy soft [sigma]-meager space if the (FSS) [bar.E] is a fuzzy soft [sigma]-meager set in (U, E, [psi]). That is, [mathematical expression not reproducible], are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Otherwise, (U, E, [psi]) will be called a fuzzy soft [sigma]-second category space.

Definition 29. Assume [F.sub.A] is a fuzzy soft [sigma]-meager set in a (FSTS) (U, E, [psi]). We say [F.sub.A.sup.c] is a fuzzy soft [sigma]-residual set in (U, E, [psi]).

Proposition 30. Let (U, E, [psi]) be a (FSTS). Then the following are equivalent:

(1) (U, E, [psi]) is a fuzzy soft Baire space.

(2) [int.sup.fs] ([F.sub.A]) = [PHI] for every fuzzy soft meager set [F.sub.A] in (U, E, [psi]).

(3) [cl.sup.fs]([G.sub.B]) = [bar.E] for every fuzzy soft residual set [G.sub.B] in (U, E, [psi]).

Proof. (1) [??] (2). Let [F.sub.A] be a fuzzy soft meager set in (U, E, [psi]). Then [mathematical expression not reproducible], i [member of] I are fuzzy soft nowhere dense sets in (U, E, [psi]). Then, we have [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft Baire space, [mathematical expression not reproducible]. Hence, [int.sup.fs]([F.sub.A]) = [PHI] for any fuzzy soft meager set [F.sub.A] in (U, E, [psi]).

(2) [??] (3). Let [G.sub.B] be a fuzzy soft residual set in (U, E, [psi]). Then [G.sub.B.sup.c] is a fuzzy soft [sigma]-meager set in (U, E, [psi]). By hypothesis, [int.sup.fs] ([G.sub.B.sup.c]) = [PHI]. Then [([cl.sup.fs] ([G.sub.B])).sup.c] = [PHI]. Hence, [cl.sup.fs]([G.sub.B]) = [bar.E] for any fuzzy soft residual set [G.sub.B] in (U, E, [psi]).

(3) [??] (1). Let [F.sub.A] be a fuzzy soft meager set in (U, E, [psi]). Then [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft nowhere dense sets in (U, E, [psi]). Now is a fuzzy soft meager set in (U, E, [psi]) implying that [F.sub.A.sup.c] is a fuzzy soft residual set in (U, E, [psi]). By hypothesis, we have [cl.sup.fs][([F.sub.A]).sup.c] = [bar.E]. Then [([int.tup.fs]([F.sub.A])).sup.c] = [bar.E]. Hence, [int.sup.fs]([F.sub.A]) = [PHI]. That is, [mathematical expression not reproducible], are fuzzy soft nowhere dense sets in (U, E, [psi]). Hence, (U, E, [psi]) is a fuzzy soft Baire space.

Proposition 31. If [F.sub.A] is a fuzzy soft dense and fuzzy soft [delta]-set in a (FSTS) (U, E, [psi]), then [F.sub.A.sup.c] is a fuzzy soft meager set in (U, E, [psi]).

Proof. Since [F.sub.A] is a fuzzy soft [delta]-set in (U, E, [psi]), [mathematical expression not reproducible], where [mathematical expression not reproducible] and since [F.sub.A] is a fuzzy soft dense set in (U, E, [psi]), [cl.sup.fs]([F.sub.A]) = [bar.E]. Then [mathematical expression not reproducible]. But [mathematical expression not reproducible]. Hence, [mathematical expression not reproducible]. That is, [mathematical expression not reproducible]. Then we have [mathematical expression not reproducible] for each [mathematical expression not reproducible] and hence [mathematical expression not reproducible] which implies that [mathematical expression not reproducible] and hence [mathematical expression not reproducible]. Therefore, [mathematical expression not reproducible] is a fuzzy soft nowhere dense set in (U, E, [psi]). Now [mathematical expression not reproducible]. Therefore, [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft nowhere dense sets in (U, E, [psi]). Hence, [F.sub.A.sup.c] is a fuzzy soft meager set in (U, E, [psi]).

Lemma 32. If [F.sub.A] is a fuzzy soft dense and fuzzy soft [delta]-set in a (FSTS) (U, E, [psi]), then [F.sub.A] is a fuzzy soft residual set in (U, E, [psi]).

Proof. Since [F.sub.A] is a fuzzy soft dense and fuzzy soft [delta]-set in (U, E, [psi]), by Proposition 31, we have that [F.sub.A.sup.c] is a fuzzy soft meager set in (U, E, [psi]) and hence [F.sub.A] is a fuzzy soft residual set in (U,E, [psi]).

Proposition 33. In a (FSTS) (U, E, [psi]), a (FSS) [F.sub.A] is a fuzzy soft [sigma]-nowhere dense set in (U,E, [psi]) if and only if [F.sub.A.sup.c] is a fuzzy soft dense and fuzzy soft [delta]-set in (U, E, [psi]).

Proof. Let [F.sub.A] be a fuzzy soft [sigma]-nowhere dense set in (U, E, [psi]). Then [mathematical expression not reproducible], where [mathematical expression not reproducible] and [int.sup.fs] ([F.sub.A]) = [PHI]. Then [mathematical expression not reproducible] implies that [cl.sup.fs]([F.sub.A.sup.c]) = [bar.E]. Also [mathematical expression not reproducible], where [mathematical expression not reproducible], for i [member of] I. Hence, we have [mathematical expression not reproducible] is a fuzzy soft dense and fuzzy soft [delta]-set in (U, E, [psi]).

Conversely, let [F.sub.A] be a fuzzy soft dense and fuzzy soft [delta]-set in (U, E, [psi]). Then [mathematical expression not reproducible], for i [member of] I. Now [mathematical expression not reproducible]. Hence, [F.sub.A.sup.c] is a [[lambda].sub.[sigma]]-set in (U, E, [psi]) and [int.sup.fs]([F.sub.A.sup.c]) = [([cl.sup.fs]([F.sub.A])).sup.c] = [([bar.E]).sup.c] = [PHI] [since [F.sub.A] is a fuzzy soft dense]. Therefore, [F.sub.A.sup.c] is a fuzzy soft [sigma]-nowhere dense set in (U, E, [psi]).

Definition 34. Assume (U, E, [psi]) is a (FSTS). We say (U, E, [psi]) is a fuzzy soft [sigma]-Baire space if each sequence [mathematical expression not reproducible] of fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]) such that [mathematical expression not reproducible].

Example 35. Let U = {[c.sub.1], [c.sub.2], [c.sub.3]} be the set of three flats and E = {costly ([e.sub.1]), modern ([e.sub.2]), security services ([e.sub.3])} be the set of parameters. Then, we consider that [psi] = {[PHI], [bar.E], [F.sub.E], [G.sub.E], [H.sub.E]}, [??] {[T.sup.i.sub.E] | i = 1, 2, 3, ..., 14} is a fuzzy soft topology on [bar.E] defined as follows:

[mathematical expression not reproducible]. (6)

[mathematical expression not reproducible]. Now consider the (FSSs)

[mathematical expression not reproducible], (7)

and

[mathematical expression not reproducible], (8)

in (U, E, [psi]). Then [alpha], [beta] and [theta] are fuzzy soft [[lambda].sub.[sigma]]-sets in (U, E, [psi]) and [int.sup.fs]([alpha]) = [PHI], [int.sup.fs]([beta]) = [PHI], and [int.sup.fs]([theta]) = [PHI]. Then [alpha], [beta] and [theta] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]).

Moreover, [([T.sup.2.sub.E]).sup.c] [??] [([T.sup.3.sub.E]).sup.c] [??] [([T.sup.7.sub.E]).sup.c] [??] [([T.sup.9.sub.E]).sup.c] [??] [([T.sup.11.sub.E]).sup.c] [??] [([T.sup.12.sub.E]).sup.c] [??] [([T.sup.14.sub.E]).sup.c] [??] = [theta] and also [int.sup.fs]([alpha] [??] [beta] [??] [theta]) = [int.sup.fs]([theta]) = [PHI] and therefore (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Remark 36. A fuzzy soft [sigma]-Baire space need not be a fuzzy soft Baire space. For, consider the following example.

Example 37. Assume X = {[c.sub.1], [c.sub.2], [c.sub.3]} is a set of soldiers under consideration and E = {courageous ([e.sub.1]); strong ([e.sub.2]); smart([e.sub.3])} is a set of parameters framed to choose the best soldier. Then, we consider that [psi] = {[PHI], [bar.E], [F.sub.E], [G.sub.E], [H.sub.E]} [??] {[T.sup.i.sub.E] | i = 1, 2, 3, ..., 9} is a fuzzy soft topology on [bar.E] defined as follows:

[mathematical expression not reproducible]. (9)

[mathematical expression not reproducible]. Now {[F.sub.E.sup.c], [G.sub.E.sup.c], [H.sub.E.sup.c], [([T.sup.1.sub.E]).sup.c], [([T.sup.2.sub.E]).sup.c], [([T.sup.3.sub.E]).sup.c], [([T.sup.5.sub.E]).sup.c], [([T.sup.7.sub.E]).sup.c], [([T.sup.8.sub.E]).sup.c], [([T.sup.1.sub.E]).sup.c]} are fuzzy soft nowhere dense sets in (U E, [psi]). [([T.sup.1.sub.E]).sup.c] = [F.sub.E].sup.c] [??] [G.sub.E].sup.c] [??] [H.sub.E].sup.c] [??] [([T.sup.2.sub.E]).sup.c] [??] [([T.sup.3.sub.E]).sup.c] [??] [([T.sup.5.sub.E]).sup.c] [??] [([T.sup.7.sub.E]).sup.c] [??] [([T.sup.8.sub.E]).sup.c] [??] [([T.sup.9.sub.E]).sup.c]. Therefore, [([T.sup.1.sub.E]).sup.c] is a fuzzy soft meager set in (U, E, [psi]). [int.sup.fs]([([T.sup.1.sub.E]).sup.c]) = [T.sup.1.sub.E] [not equal to] [PHI]. Hence, (U, E, [psi]) is not a fuzzy soft Baire space. Now consider the (FSSs) [alpha] = [([H.sub.E]).sup.c] [??] [([T.sup.5.sub.E]).sup.c] [??] [([T.sup.6.sub.E]).sup.c], and [beta] = [([F.sub.E]).sup.c] [??] [([T.sup.2.sub.E]).sup.c] [??] [([T.sup.4.sub.E]).sup.c] [??] [([T.sup.7.sub.E]).sup.c] [??] [([T.sup.8.sub.E]).sup.c] in (U, E, [psi]) and also [int.sup.fs]([alpha]) = [PHI], [int.sup.fs]([beta]) = [PHI]. Then [alpha] and [beta] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Now the (FSS) ([alpha] [??] [beta]) is a fuzzy soft [sigma]-meager set in (U, E, [psi]) and [int.sup.fs]([alpha] [??] [beta]) = [PHI]. Hence, (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Proposition 38. Let (U, E, [sigma]) be a (FSTS). Then the following are equivalent:

(1) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

(2) [int.sup.fs] ([F.sub.A]) = [PHI] for every fuzzy soft [sigma]-meager set [F.sub.A] in (U, E, [psi]).

(3) [cl.sup.fs] ([G.sub.B]) = [bar.E] for every fuzzy soft [sigma]-residual set [G.sub.B] in (U, E, [psi]).

Proof. (1) [??] (2). Let [F.sub.A] be a fuzzy soft [sigma]-meager set in (U, E, [psi]). Then [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Then, we have [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, [mathematical expression not reproducible]. Hence, [int.sup.fs]([F.sub.A]) = [PHI] for any fuzzy soft [sigma]-meager set [F.sub.A] in (U, E, [psi]).

(2) [??] (3). Let [G.sub.B] be a fuzzy soft [sigma]-residual set in (U, E, [psi]). Then [G.sub.B.sup.c] is a fuzzy soft [sigma]-meager set in (U, E, [psi]). By hypothesis, [int.sup.fs]([G.sup.c.sub.E]) = [PHI]. Then [([cl.sup.fs]([G.sup.c.sub.B])).sup.c] = [PHI]. Hence, [cl.sup.fs]([G.sub.B]) = [bar.E] for any fuzzy soft [sigma]-residual set [G.sub.B] in (U, E, [psi]).

(3) [??] (1). Let [F.sub.A] be a fuzzy soft [sigma]-meager set in (U, E, [psi]). Then [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Now [F.sub.A] is a fuzzy soft [sigma]-meager set in (U, E, [psi]) implying that [F.sub.A.sup.c] is a fuzzy soft [sigma]-residual set in (U, E, [psi]). By hypothesis, we have [cl.sup.fs][([F.sub.A]).sup.c] = [bar.E]. Then [([int.sup.fs]([F.sub.A])).sup.c] = [bar.E]. Hence, [int.sup.fs]([F.sub.A]) = [PHI]. That is, [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft [sigma]- nowhere dense sets in (U, E, [psi]). Hence, (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Proposition 39. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, then [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]).

Proof. Let [mathematical expression not reproducible] be fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]). By Proposition 33, [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Then the (FSS) [mathematical expression not reproducible] is a fuzzy soft [sigma]-meager set in (U, E, [psi]). Now [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, by Proposition 38, we have [int.sup.fs]([F.sub.A]) = [PHI]. Then [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible].

Proposition 40. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, then [mathematical expression not reproducible], where the (FSSs) sets [mathematical expression not reproducible], are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft [delta]-sets [mathematical expression not reproducible] in (U, E, [psi]).

Proof. Let the (FSTS) (U, E, [psi]) be a fuzzy soft [sigma]-Baire space and the (FSSs) [mathematical expression not reproducible] be fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]). By Proposition 39, [mathematical expression not reproducible]. Then [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible]. Also by Proposition 31, [mathematical expression not reproducible] are fuzzy soft meager sets in (U, E, [psi]). Hence, [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft [delta]-sets [mathematical expression not reproducible] in (U, E, [psi]).

Proposition 41. If the fuzzy soft meager sets are formed from the fuzzy soft dense and fuzzy soft [delta]-sets in a fuzzy soft [sigma]-Baire space (U, E, [psi]), then (U, E, [psi]) is a fuzzy soft Baire space.

Proof. Let the (FSTS) (U, E, [psi]) be a fuzzy soft [sigma]-Baire space. By Proposition 40, [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft [delta]-sets [mathematical expression not reproducible] in (U, E, [psi]). Now [mathematical expression not reproducible]. Then [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible], where [mathematical expression not reproducible] is a fuzzy soft meager set in (U, E, [psi]). By Proposition 30, (U, E, [psi]) is a fuzzy soft Baire space.

Proposition 42. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-meager space, then (U, E, [psi]) is not a fuzzy soft [sigma]-Baire space.

Proof. Let the (FSTS) (U, E, [psi]) be a fuzzy soft [sigma]-meager space. Then [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [sigma]). Now [mathematical expression not reproducible]. Hence, by definition, (U, E, [psi]) is not a fuzzy soft [sigma]-Baire space.

Remark 43. By Proposition 42, we consider that if the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, then (U, E, [psi]) is a fuzzy soft [sigma]-second category space. Moreover, the converse is not true in general. That means a fuzzy soft [sigma]-second category space need not be a fuzzy soft [sigma]-Baire space.

Example 44. Let there be three houses in the universe U given by U = {[s.sub.1], [s.sub.2], [s.sub.3]} and let E = {stone ([e.sub.1]); steel ([e.sub.2]); and brick ([e.sub.3])} be the set of parameters framed to choose one house to rent, where (brick) means the brick built houses, (steel) means the steel built houses, and (stone) means the stone built houses. Then, we consider [psi] = {[PHI], [bar.E], [F.sub.E], [G.sub.E], [H.sub.E]} [??] {[T.sup.i.sub.E] | i = 1, 2, 3, ..., 10} is a fuzzy soft topology defined as follows:

[mathematical expression not reproducible]. (10)

[mathematical expression not reproducible]. Now consider the (FSSs) [alpha] = [([G.sub.E]).sup.c] [??] [([T.sup.4.sub.E]).sup.c] [??] [([T.sup.5.sub.E]).sup.c], and [beta] = [([F.sub.E]).sup.c] [??] [([H.sub.E]).sup.c] in (U, E, [psi]). Then [alpha] and [beta] are fuzzy soft [[lambda].sub.[sigma]-sets in (U, E, [psi]) and [int.sup.fs] ([alpha]) = [PHI], [int.sup.fs]([beta]) = [PHI]. Then [alpha] and [beta] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Now ([alpha] [??] [beta]) [not equal to] [bar.E]. Therefore, (U, E, [psi]) is a fuzzy soft [sigma]-second category space. But [int.sup.fs]([alpha] [??] [beta]) [not equal to] [PHI] and therefore (U, E, [psi]) is not a fuzzy soft [sigma]-Baire space.

Proposition 45. If [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft dense and fuzzy soft [delta]-sets in a (FSTS) (U, E, [psi]), then (U, E, [psi]) is a fuzzy soft [sigma]-second category space.

Proof. Let [mathematical expression not reproducible] be fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]). By Proposition 33, [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Now [mathematical expression not reproducible] implies that [mathematical expression not reproducible]. Then [mathematical expression not reproducible]. Hence, (U, E, [psi]) is not a fuzzy soft [sigma]-meager space and therefore (U, E, [psi]) is a fuzzy soft [sigma]-second category space.

Proposition 46. If [F.sub.A] is a fuzzy soft [sigma]-meager set in (U, E, [psi]), then there is a fuzzy soft [[lambda].sub.[sigma]]-set [G.sub.B] in (U, E, [psi]) such that [F.sub.A] [??] [G.sub.B].

Proof. Let [F.sub.A] be a fuzzy soft [sigma]-meager set in (U, E, [psi]). Thus, [mathematical expression not reproducible], where [mathematical expression not reproducible] are fuzzy soft [sigma]- nowhere dense sets in (U, E, [psi]). Now [mathematical expression not reproducible] are (FSOSs) in (U, E, [psi]). Then [mathematical expression not reproducible] is a fuzzy soft [delta]-set in (U, E, [psi]) and [mathematical expression not reproducible]. Now [mathematical expression not reproducible]. That is, [F.sub.A] [??] [T.sup.c] and [T.sup.c] is a fuzzy soft [[lambda].sub.[sigma]]-set in (U, E, [psi]). Let [G.sub.B.sup.c] [??] T. Hence, if [F.sub.A] is a fuzzy soft [sigma]-meager set in (U, E, [psi]), then there is a fuzzy soft [[lambda].sub.[sigma]]-set [G.sub.B] in (U, E, [psi]) such that [F.sub.A] [??] [G.sub.B].

Proposition 47. If [G.sub.B] is a fuzzy soft [sigma]-residual set in a (FSTS) (U, E, [psi]) such that [G.sub.B] in (U, E, [psi]) such that [F.sub.A] [??] [G.sub.B], where [F.sub.A] is a fuzzy soft dense and fuzzy soft [delta]-set in (U, E, [psi]), then (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Proof. Let [G.sub.B] be a fuzzy soft [sigma]-residual set in a (FSTS) (U, E, [psi]). Thus, [G.sub.B.sup.c] is a fuzzy soft [sigma]-meager set in (U, E, [psi]). Now by Proposition 46, there is a fuzzy soft [[lambda].sub.[sigma]]-set T in (U, E, [psi]) such that [G.sub.B.sup.c] [??] T. This implies that [T.sup.c] [??] [G.sub.B]. Let [F.sub.A] = [T.sup.c]. Then [F.sub.A] is a fuzzy soft [delta]-set in (U, E, [psi]) and [F.sub.A] [??] [G.sub.B] implies that [cl.sup.fs]([F.sub.A]) [??] [cl.sup.fs]([G.sub.B]). If [cl.sup.fs]([F.sub.A]) = [bar.E], then we have [cl.sup.fs]([G.sub.B}) = [bar.E]. Hence, by Proposition 30, (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Proposition 48. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space and if [mathematical expression not reproducible], then there exists at least one [[lambda].sub.[sigma]]-set [mathematical expression not reproducible] such that [mathematical expression not reproducible].

Proof. Suppose that [int.sup.fs]([F.sub.A]) = [PHI], [for all] (i [member of] I), where [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Then [mathematical expression not reproducible], implying that [mathematical expression not reproducible], a contradiction to (U, E, [psi]) being a fuzzy soft [sigma]-Baire space. Hence, [mathematical expression not reproducible], for at least one [[lambda].sub.[sigma]]-set [mathematical expression not reproducible] in (U, E, [psi]).

Proposition 49. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, then no nonempty (FSOS) is a fuzzy soft [sigma]-meager set in (U, E, [psi]).

Proof. Let [F.sub.A] be nonempty (FSOS) in a fuzzy soft [sigma]-Baire space (U, E, [psi]). Suppose that [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Then [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, [mathematical expression not reproducible]. This implies that [int.sup.fs]([F.sub.A]) = [PHI]. Then we will have [F.sub.A] = [PHI], which is a contradiction, since [F.sub.A] [member of] [psi] implies that [int.sup.fs]([F.sub.A]) = [F.sub.A] [not equal to] [PHI]. Hence, no nonempty (FSOS) is a fuzzy soft [sigma]-meager set in (U, E, [psi]).

Definition 50. A (FSTS) (U, E, [psi]) is called a fuzzy soft submaximal space if for each (FSS) [F.sub.A] in (U, E, [psi]) such that [cl.sup.fs] ([F.sub.A]) = [bar.E]; then [F.sub.A] [member of] [psi] in (U, E, [psi]).

Proposition 51. If the (FSTS) (U, E, [psi]) is a fuzzy soft submaximal space and if [F.sub.A] is a fuzzy soft [sigma]-meager set in (U, E, [psi]), then [F.sub.A] is a fuzzy soft meager set in (U, E, [psi]).

Proof. Let [mathematical expression not reproducible] be a fuzzy soft [sigma]-meager set in (U, E, [psi]), where the (FSSs) [mathematical expression not reproducible] are fuzzy soft [sigma]-nowhere dense sets in (U, E, [psi]). Then we have [mathematical expression not reproducible] and [mathematical expression not reproducible] are fuzzy soft [[lambda].sub.[sigma]]-sets in (U, E, [psi]). Now [mathematical expression not reproducible], implying that [mathematical expression not reproducible] and hence [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft submaximal space, the fuzzy soft dense sets [mathematical expression not reproducible] are (FSOSs) in (U, E, [psi]) and hence [mathematical expression not reproducible] are (FSCSs) in (U, E, [psi]). Then [mathematical expression not reproducible] and [mathematical expression not reproducible] imply that [mathematical expression not reproducible]. That is, [mathematical expression not reproducible] are fuzzy soft nowhere dense sets in (U, E, [psi]). Therefore, [mathematical expression not reproducible] is a fuzzy soft meager set in (U, E, [psi]).

Proposition 52. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space and fuzzy soft submaximal space, then (U, E, [psi]) is a fuzzy soft Baire space.

Proof. Let [F.sub.A] be a fuzzy soft [sigma]-meager set in (U, E, [psi]). Since (U, E, [psi]) is a fuzzy soft submaximal space, by Proposition 51, [F.sub.A] is a fuzzy soft meager set in (U, E, [psi]). Since (U, E, [psi]) is a fuzzy soft [sigma]-Baire space, by Proposition 38, [int.sup.fs]([F.sub.A]) = [PHI]. Hence, for the fuzzy soft meager set FA in (U, E, [psi]), we have [int.sup.fs]([F.sub.A]) = [PHI]. Therefore, by Proposition 30, (U, E, [psi]) is a fuzzy soft Baire space.

Definition 53. A fuzzy soft P-space is a (FSTS)(U, E, [psi]) with the property that states that if countable intersection of fuzzy soft open sets in (U, E, [psi]) is fuzzy soft open. That is, every non-empty fuzzy soft [delta]-set in (U, E, [psi]) is fuzzy soft open in (U, E, [psi]).

Proposition 54. If the (FSTS) (U, E, [psi]) is a fuzzy soft [sigma]-Baire space and fuzzy soft P-space, then (U, E, [psi]) is a fuzzy soft Baire space.

Proof. Let the (FSTS) (U, E, [psi]) be a fuzzy soft [sigma]-Baire space. Then, by Proposition 39, [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]). Now from [mathematical expression not reproducible], we have [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible].

Since the (FSSs) [mathematical expression not reproducible] are fuzzy soft dense in (U, E, [psi]), [mathematical expression not reproducible]. Then we have [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible]. Also, since (U, E, [psi]) is a fuzzy soft P-space, the non-empty fuzzy soft [delta]-sets [mathematical expression not reproducible] in (U, E, [psi]) are fuzzy soft open in (U, E, [psi]). Then [mathematical expression not reproducible] are (FSCSs) in (U, E, [psi]). Then [mathematical expression not reproducible] imply that [mathematical expression not reproducible]. That is, [mathematical expression not reproducible] are fuzzy soft nowhere dense sets in (U, E, [psi]). Therefore, we have [mathematical expression not reproducible], where (F'AC)'s are fuzzy soft nowhere dense sets in (U, E, [psi]). Hence, by Proposition 30, (U, E, [psi]) is a fuzzy soft Baire space.

Definition 55. A (FSTS) (U, E, [psi]) is called a fuzzy soft almost resolvable space if [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] in (U, E, [psi]) are such that [mathematical expression not reproducible]. Otherwise, (U, E, [psi]) is called a fuzzy soft almost irresolvable space.

Proposition 56. If the (FSTS) (U, E, [psi]) is a fuzzy soft almost irresolvable space, then (U, E, [psi]) is a fuzzy soft [sigma]-second category space.

Proof. Let [mathematical expression not reproducible] be the fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]). Now [mathematical expression not reproducible] implies that [mathematical expression not reproducible]. That is, [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft almost irresolvable space, [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] in (U, E, [psi]) are such that [mathematical expression not reproducible]. Now [mathematical expression not reproducible] implies that [mathematical expression not reproducible]. Hence, we have [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft dense and fuzzy soft [delta]-sets in a (FSTS) (U, E, [psi]). Thus, by Proposition 45, (U, E, [psi]) is a fuzzy soft [sigma]-second category space.

Definition 57. A (FSTS) (U, E, [psi]) is called a fuzzy soft hyperconnected space if every (FSOS) [F.sub.A] is fuzzy soft dense in (U, E, [psi]). That is, [cl.sup.fs]([F.sub.A]) = [bar.E] [for all][PHI] = [F.sub.A] [member of] [psi].

Proposition 58. If [mathematical expression not reproducible], are fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]), then (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Proof. The proof is obvious.

Proposition 59. If [mathematical expression not reproducible], where the (FSSs) [mathematical expression not reproducible] are fuzzy soft [delta]-sets in a fuzzy soft hyperconnected and fuzzy soft P-space (U, E, [psi]), then (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Proof. Let [mathematical expression not reproducible] be the fuzzy soft [delta]-sets in (U, E, [psi]) such that [mathematical expression not reproducible]. Since (U, E, [psi]) is a fuzzy soft P-space, the fuzzy soft [delta]-sets [mathematical expression not reproducible] in (U, E, [psi]) are fuzzy soft open in (U, E, [psi]). Also since (U, E, [psi]) is a fuzzy soft hyperconnected space, the (FSOSs) [mathematical expression not reproducible] in (U, E, [psi]) are fuzzy soft dense sets in (U, E, [psi]). Hence, the (FSSs) [mathematical expression not reproducible], i [member of] I are fuzzy soft dense and fuzzy soft [delta]-sets in (U, E, [psi]) and [mathematical expression not reproducible]. Hence, by Proposition 58, (U, E, [psi]) is a fuzzy soft [sigma]-Baire space.

Definition 60. Let [F.sub.A] be a fuzzy soft subset of a fuzzy soft space [bar.E]. Then [F.sub.A] is said to be fuzzy soft A-embedded in [bar.E] if each fuzzy soft [delta]-subset [G.sub.B] of [bar.E] which is contained in [F.sub.A] is fuzzy soft nowhere dense in [F.sub.A] (i.e., [mathematical expression not reproducible].

Proposition 61. Let [F.sub.A] be a fuzzy soft dense subspace of a fuzzy soft Baire space [bar.E]. If [bar.E] \ [F.sub.A] is fuzzy soft A-embedded in [bar.E], then [F.sub.A] is a fuzzy soft Baire space.

Proof. Observe that if [F.sub.A] is not a fuzzy soft Baire space; then there is a sequence [mathematical expression not reproducible] ... of fuzzy soft open subsets of [F.sub.A] such that each [mathematical expression not reproducible] is fuzzy soft dense in [G.sub.B] and yet [mathematical expression not reproducible]. Then there is a sequence [mathematical expression not reproducible] ... of fuzzy soft open subsets of [bar.E] such that [G.sub.B] = [H.sub.C] [??] [F.sub.A] and [mathematical expression not reproducible]. Each [mathematical expression not reproducible] is fuzzy soft dense in [H.sub.C] and [H.sub.C] is a fuzzy soft Baire space. Hence, [mathematical expression not reproducible] is fuzzy soft dense in [H.sub.C] and therefore in [H.sub.C] [??] [bar.E]\ [F.sub.A]. Since [mathematical expression not reproducible] is not fuzzy soft A-embedded in [bar.E].

Proposition 62. Let [F.sub.A] be a fuzzy soft dense subspace of a fuzzy soft Baire space [bar.E]. If [bar.E]\[F.sub.A] is dense in [bar.E], then [F.sub.A] is a fuzzy soft Baire space if and only if [bar.E]\[F.sub.A] is fuzzy soft A-embedded in [bar.E].

Proof. Assume that [bar.E]\ [F.sub.A] is not fuzzy soft A-embedded in [bar.E]. Let [G.sub.B] be a fuzzy soft [delta]-subset of [bar.E] which is contained in [bar.E]\[F.sub.A] and which is fuzzy soft dense in some relatively (FSOS) [H.sub.C] of [bar.E]\ [F.sub.A]. Let [K.sub.D] be an fuzzy soft open subset of [bar.E] with [K.sub.D] [??] ([bar.E] [F.sub.A]) = [H.sub.C]. Then [T.sub.M] = [K.sub.D] [??] [G.sub.B] is a [delta]-subset of [bar.E] which is fuzzy soft dense in [K.sub.D] and which is contained in [H.sub.C]. Let [mathematical expression not reproducible], where each [mathematical expression not reproducible] is open in [bar.E] and [mathematical expression not reproducible]. The (FSSs) [mathematical expression not reproducible] are fuzzy soft open and fuzzy soft dense subsets of [K.sub.D] [??] [F.sub.A] and yet [mathematical expression not reproducible]. It follows that [K.sub.D] [??] [F.sub.A] is not a fuzzy soft Baire space. Consequently, [F.sub.A] is not a fuzzy soft Baire space. Conversely, assume that [bar.E] [F.sub.A] is fuzzy soft A-embedded in [bar.E] and, by Proposition 61, then [F.sub.A] is a fuzzy soft Baire space.

Lemma 63. Let [bar.E] be a fuzzy soft Baire space. If [mathematical expression not reproducible] is a nonempty fuzzy soft nowhere dens [delta]-set of [bar.E], where [mathematical expression not reproducible] is a fuzzy soft open subset of [bar.E] [for all]i [member of] I, then for every nonempty fuzzy soft open subset [T.sub.M] of [bar.E] there is i [member of] I such that [mathematical expression not reproducible].

Proof. Let [T.sub.M] be a nonempty fuzzy soft open subset of [bar.E]. Then, [T.sub.M] \ [cl.sup.fs]([G.sub.B]) is a nonempty fuzzy soft open subset of [bar.E]; hence, [T.sub.M] \ [cl.sup.fs]([G.sub.B]) is also fuzzy soft Baire. Since [mathematical expression not reproducible] and each [mathematical expression not reproducible] is a fuzzy soft closed subset of [bar.E], by Remark 22, then there is i [member of] I such that [mathematical expression not reproducible].

Proposition 64. Let [bar.E] be a fuzzy soft Baire space and let [F.sub.A] [??] [bar.E] be fuzzy soft dense. Then [F.sub.A] is a fuzzy soft Baire space if and only if every fuzzy soft [delta]-set in E contained in [bar.E]\ [F.sub.A] is fuzzy soft nowhere dense.

Proof.

Necessity. Let [mathematical expression not reproducible], is a fuzzy soft open subset of [bar.E] for each i [member of] I, which is contained in [bar.E] \ [F.sub.A]. Then, [mathematical expression not reproducible]. In virtue of Remark 22, [mathematical expression not reproducible] is fuzzy soft dense in [F.sub.A]. Suppose that [int.sub.fs]([cl.sup.fs]([G.sub.B])) [not equal to] [PHI]. Then, there is n [member of] I such that [mathematical expression not reproducible]. On the other hand, we know that [mathematical expression not reproducible]. Hence, [mathematical expression not reproducible] which implies that [mathematical expression not reproducible], but this is impossible.

Sufficiency. Assume that [F.sub.A] is no fuzzy soft Baire. According to Remark 22, there is a countable fuzzy soft closed cover [mathematical expression not reproducible] of [F.sub.A] such that [mathematical expression not reproducible] is not dense in [F.sub.A]. For each i [member of] I, choose a fuzzy soft closed subset [mathematical expression not reproducible] of [bar.E] such that [mathematical expression not reproducible], for each i [member of] I. Let [mathematical expression not reproducible] which is a [delta]-set of [bar.E] contained in [bar.E] \ [F.sub.A]. If [T.sub.M] = [PHI], then [mathematical expression not reproducible] would be a fuzzy soft closed cover of [bar.E] and, by Remark 22, then [mathematical expression not reproducible] would be fuzzy soft dense in [bar.E] which is not possible. So, [T.sub.M] [not equal to] [PHI]. Choose a nonempty fuzzy soft open subset [K.sub.D] of [bar.E] such that [mathematical expression not reproducible]. By Lemma 63, we can find r [member of] I such that [mathematical expression not reproducible]. Hence, [mathematical expression not reproducible], but this is a contradiction. Thus, [F.sub.A] is fuzzy soft Baire.

4. Baireness in Fuzzy Soft Setting

In this section, we shall study the new class of fuzzy soft Baire spaces.

Definition 65. We say a space [bar.E] is fuzzy soft D-Baire if every fuzzy soft dense subspace of [bar.E] is fuzzy soft Baire.

An immediate consequence of Proposition 64 is the following.

Corollary 66. Suppose that [bar.E] is a fuzzy soft Baire space. Then, every fuzzy soft [delta]-set in [bar.E] with empty interior is fuzzy soft nowhere dense iff [bar.E] is fuzzy soft D-Baire

Proof. It follows from Proposition 64 and Definition 65.

Definition 67. We say that a (FSTS) [bar.E] is a fuzzy soft almost P-space if every non-empty fuzzy soft [delta]-set in [bar.E] has a nonempty interior.

Corollary 68. Every fuzzy soft Baire and fuzzy soft almost P-space is fuzzy soft D-Baire.

Proof. This is a consequence of Proposition 64 and Definition 67.

Definition 69. A fuzzy soft Borel set is any (FSS) in a (FSTS) that can be formed from (FSOSs) (or, equivalently, from (FSCSs)) through the operations of countable union, countable intersection, and relative complement.

Definition 70. Let [bar.E] be a (FSTS). Then, the class FSPB([bar.E]) is the fuzzy soft [sigma]-algebra in [bar.E] generated by all (FSOSs) and all fuzzy soft nowhere dense sets.

Remark 71. (1) For a (FSTS) (U, E, [psi]), the collection of all fuzzy soft Borel sets on [bar.E] forms a fuzzy soft [sigma]-algebra.

(2) The fuzzy soft [sigma]-algebra of fuzzy soft Borel sets is contained in the class FSPB([bar.E]).

(3) It is clear to show that [F.sub.A] [??] [bar.E] belongs to the class FSPB([bar.E]) if and only if [F.sub.A] may be expressed in the form [F.sub.A] = [G.sub.B] [??] [H.sub.C], where [G.sub.B] is a fuzzy soft [delta]-set and [H.sub.C] is fuzzy soft meager.

Theorem 72. The following seven conditions on a space (U, E, [psi]) are equivalent.

(1) (U, E, [psi]) is fuzzy soft D-Baire.

(2) (U, E, [psi]) is fuzzy soft Baire and every fuzzy soft [delta]-set with empty interior is fuzzy soft nowhere dense.

(3) Every fuzzy soft meager subset [F.sub.A] [??] [bar.E] is fuzzy soft nowhere dense.

(4) (U, E, [psi]) is fuzzy soft Baire and every fuzzy soft dense [delta]-set has fuzzy soft dense interior.

(5) (U, E, [psi]) is fuzzy soft Baire and every set in the class FSPB([bar.E]) with empty interior is fuzzy soft nowhere dense.

(6) (U, E, [psi]) is fuzzy soft Baire and every fuzzy soft Borel set with empty interior is fuzzy soft nowhere dense.

(7) (U, E, [psi]) is fuzzy soft Baire and the union of a fuzzy soft [delta]-set with empty interior and a fuzzy soft meager set of [bar.E] is fuzzy soft nowhere dense.

Proof. (1) [??] (2). This is Corollary 66.

(2) [??](3). Let [F.sub.A] [??] [bar.E] be a fuzzy soft meager set. Assume [mathematical expression not reproducible], where [mathematical expression not reproducible] is fuzzy soft nowhere dense [for all]i [member of] I. Therefore, [mathematical expression not reproducible] is a [delta]-set in [bar.E] and [L.sub.M] is fuzzy soft dense in [bar.E] because its complement is a fuzzy soft meager set and [bar.E] is fuzzy soft Baire. Let [V.sub.D] = [int.sup.fs]([L.sub.M]). The (FSS) [L.sub.M] - [V.sub.D] clearly has empty interior. Hence, [L.sub.M] - [cl.sup.fs]([V.sub.D]) is a fuzzy soft [delta]-set with empty interior; by hypothesis, [L.sub.M] - [cl.sup.fs]([V.sub.D]) is fuzzy soft nowhere dense. Also [L.sub.M] [??] [Fr.sup.fs] ([V.sub.D]) is a fuzzy soft nowhere dense set. Therefore, [mathematical expression not reproducible] is a fuzzy soft nowhere dense set as well. On the other hand, [mathematical expression not reproducible] is a fuzzy soft meager set. Since [bar.E] is fuzzy soft Baire, [PHI] = [int.sup.fs]([bar.E] [V.sub.D]) = [bar.E] \ [cl.sup.fs]([V.sub.D]). Therefore, [cl.sup.fs] ([V.sub.D]) = [bar.E] and [F.sub.A] [??] [bar.E]\[V.sub.D] = [Fr.sup.fs] ([V.sub.D]) is fuzzy soft nowhere dense.

(3) [??] (4). It follows from Remark 22 that [bar.E] is a fuzzy soft Baire space. Let [L.sub.M] [??] [bar.E] be a fuzzy soft dense [delta]-set of [bar.E]. Since [bar.E] \ [L.sub.M] is a fuzzy soft meager set, the hypothesis implies that [bar.E] \ [L.sub.M] is fuzzy soft nowhere dense; i.e., [cl.sup.fs]([bar.E] \ [L.sub.M]) has empty interior. Therefore, [V.sub.D] = [bar.E] \ [cl.sup.fs]([bar.E] [L.sub.M]) = [int.sup.fs]([L.sub.M]) is a fuzzy soft open dense subspace of [bar.E].

(4) [??] (2). Let [G.sub.B] be a [delta]-set with empty. First observe that [int.sup.fs]([cl.sup.fs]([G.sub.B])) [??] [cl.sup.fs]([cl.sup.fs]([G.sub.B]) \ [G.sub.B]). Since [cl.sub.fs]([G.sub.B]) \ [G.sub.B] is an [[lambda].sub.[sigma]]-set with empty interior, [bar.E] \ ([cl.sup.fs]([G.sub.B]) \ [G.sub.B]) is a fuzzy soft dense [delta]-set of [bar.E] By assumption, [int.sup.fs]([bar.E]\([cl.sup.fs]([G.sub.B]) \ [G.sub.B])) is also fuzzy soft dense in [bar.E]. That is, [bar.E] [cl.sup.fs]([cl.sup.fs]([G.sub.B]) \ [G.sub.B]) is fuzzy soft dense in [bar.E]. Hence, [int.sup.fs]([cl.sup.fs]([cl.sup.fs]([G.sub.B]) [G.sub.B])) = [PHI] and so [int.sup.fs]([cl.sup.fs]([G.sub.B])) = [PHI].

(4) [??] (5). We have already established above the equivalence among clauses (1), (2), (3), and (4). The fifth clause follows directly from the properties of the class FSPB([bar.E]) and clauses (2) and (3).

(5) [??] (6). This implication is obvious because the fuzzy soft [delta]-algebra of fuzzy soft Borel sets is contained in the class FSPB([bar.E]).

(6) [??] (1). It is enough to observe that (6) [??] (2) [??] (1).

(1 [??] (7). We know the first six statements are equivalent to each other. Thus, clause (7) follows directly from clauses (2) and (3). (7) [??] (1). This is a consequence of Corollary 66.

5. Conclusion

In the present paper, we have introduced and discussed new notions of Baireness in fuzzy soft topological spaces. Furthermore, there are many problems and applications in algebra that deal with group theory and spaces. So, future work in this regard would be required to study some applications using the properties of [psi] in our new fuzzy soft spaces and new operations depend on fuzzy soft operations [??] and [??] to consider new fuzzy soft groups and fuzzy soft commutative rings. Also, let us say (U, E, [psi]) is fuzzy soft N-Baire if every fuzzy soft set in (U, E, [psi]) with empty interior is fuzzy soft nowhere dense. The question we are concerned with is as follows: what are the possible relationships considered between fuzzy soft N-Baire and each concept of our notions that are given in this work?

Data Availability

Data used to support the findings of this study are available from the corresponding author upon request.

https://doi.org/10.1155/2018/5731682

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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Shuker Mahmood Khalil (iD), Mayadah Ulrazaq, Samaher Abdul-Ghani, and Abu Firas Al-Musawi

Department of Mathematics, College of Science, University of Basrah, Basrah 61004, Iraq

Correspondence should be addressed to Shuker Mahmood Khalil; shuker.alsalem@gmail.com

Received 30 January 2018; Revised 14 May 2018; Accepted 29 May 2018; Published 2 July 2018

Academic Editor: Katsuhiro Honda

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Title Annotation: | Research Article |
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Author: | Khalil, Shuker Mahmood; Ulrazaq, Mayadah; Abdul-Ghani, Samaher; Al-Musawi, Abu Firas |

Publication: | Advances in Fuzzy Systems |

Date: | Jan 1, 2018 |

Words: | 10032 |

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