𝜎− Algebra and 𝜎− Baire in Fuzzy Soft Setting

. We first introduce some new notions of Baireness in fuzzy soft topological space (FSTS). Next, their characterizations and basic properties are investigated in this work. The notions of fuzzy soft dense, fuzzy soft nowhere dense, fuzzy soft meager, fuzzy soft second category, fuzzy soft residual, fuzzy soft Baire, fuzzy soft 𝛿− sets, fuzzy soft 𝜆 𝜎 − sets, fuzzy soft 𝜎− nowhere dense, fuzzy soft 𝜎− meager, fuzzy soft 𝜎− residual, fuzzy soft 𝜎− Baire, fuzzy soft 𝜎− second category, fuzzy soft 𝜎− 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 𝜎− algebra are introduced. Furthermore, several examples are shown as well.


Introduction
The concepts of Baire spaces have been studied and discussed extensively in general topology in [1][2][3][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][10][11][12][13][14][15][16][17][18][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][21][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][25][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 −sets, fuzzy soft   −sets, fuzzy soft −nowhere dense, fuzzy soft −meager, fuzzy soft −residual, fuzzy soft −Baire, fuzzy soft −second category, fuzzy soft −residual, fuzzy, fuzzy soft submaximal space, fuzzy soft −space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft −embedded, fuzzy soft −Baire, fuzzy soft almost −spaces, fuzzy soft Borel, and fuzzy soft −algebra.Moreover, several examples are given to illustrate the notions introduced in this work.

Definition 10 ([28]
). Assume  is the family of (FSSs) over .We say  is a fuzzy soft topology on  if  the following axioms hold: (i) Φ,  belong to .
(ii) The union of any number of (FSSs) in  belongs to .
(2) For any fuzzy soft   subset of a (FSTS) (, , ), we define the fuzzy soft subspace topology    on   by   ∈    if   =   ∩  for some   ∈ .
(3) For any fuzzy soft   in   fuzzy soft subspace of a (FSTS), we denote the interior and closure of   in   by     (  ) and     (  ), respectively.

Fuzzy Soft 𝜎−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.Remark 23.We say (  ) is a family of all soft sets over a universe set  and the parameter set .Moreover, the cardinality of (  ) is given by ((  )) = 2 ()× () .Therefore, in this paper for each (FSS)   over (, ) we can define   by using matrix form as follows: The order of this matrix is given by  × , where  = (),  = (), and   =  .8.12.9 ) .

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  is fuzzy soft −Baire if every fuzzy soft dense subspace of  is fuzzy soft Baire.Proof.This is a consequence of Proposition 64 and Definition 67.

An immediate consequence of
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  be a (FSTS).Then, the class () is the fuzzy soft −algebra in  generated by all (FSOSs) and all fuzzy soft nowhere dense sets.
(2) The fuzzy soft −algebra of fuzzy soft Borel sets is contained in the class ().
(3) It is clear to show that   ⊆ belongs to the class () if and only if   may be expressed in the form   =   ∪  , where   is a fuzzy soft −set and   is fuzzy soft meager.
Theorem 72.The following seven conditions on a space (, , ) are equivalent.
(2) (, , ) is fuzzy soft Baire and every fuzzy soft −set with empty interior is fuzzy soft nowhere dense.
(4) (, , ) is fuzzy soft Baire and every fuzzy soft dense −set has fuzzy soft dense interior.
(5) (, , ) is fuzzy soft Baire and every set in the class () with empty interior is fuzzy soft nowhere dense.
(6) (, , ) is fuzzy soft Baire and every fuzzy soft Borel set with empty interior is fuzzy soft nowhere dense.(7) (, , ) is fuzzy soft Baire and the union of a fuzzy soft −set with empty interior and a fuzzy soft meager set of  is fuzzy soft nowhere dense.
(3) ⇒ (4).It follows from Remark 22 that  is a fuzzy soft Baire space.Let   ⊆ be a fuzzy soft dense −set of .Since  \   is a fuzzy soft meager set, the hypothesis implies that  \  M is fuzzy soft nowhere dense; i.e.,   ( \   ) has empty interior.Therefore,   =  \   ( \   ) =   (  ) is a fuzzy soft open dense subspace of .
(5) ⇒ (6).This implication is obvious because the fuzzy soft −algebra of fuzzy soft Borel sets is contained in the class ().

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  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 (, , ) is fuzzy soft −Baire if every fuzzy soft set in (, , ) 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 −Baire and each concept of our notions that are given in this work?

Definition 53 .
A fuzzy soft −space is a ()(, , ) with the property that states that if countable intersection of fuzzy soft open sets in (, , ) is fuzzy soft open.That is, every non−empty fuzzy soft −set in (, , ) is fuzzy soft open in (, , ).Proposition 54.If the (FSTS) (, , ) is a fuzzy soft −Baire space and fuzzy soft −space, then (, , ) is a fuzzy soft Baire space.
Proposition 64 is the following.Suppose that  is a fuzzy soft Baire space.Then, every fuzzy soft −set in  with empty interior is fuzzy soft nowhere dense iff  is fuzzy soft −Baire Proof.It follows from Proposition 64 and Definition 65.Definition 67.We say that a (FSTS)  is a fuzzy soft almost −space if every non−empty fuzzy soft −set in  has a nonempty interior.Every fuzzy soft Baire and fuzzy soft almost −space is fuzzy soft −Baire.