SOFT WEAK BAIRE SPACES

It has been noticed that among the different fields of theoretical mathematics where soft sets theory is applied extensively, the most of papers are related to topology. In this paper, the concept of soft weak baire spaces, i.e., the soft generalization of the concept of weak baire spaces as defined by Renukadevi and Muthulakshmi [1] has been presented. Finally, the basic properties of such spaces and accordingly the defined continuous soft functions between these spaces have been studied.


Introduction
Many real life problems have uncertainties and so many theories have emerged to address it like probability, fuzzy and rough sets and interval mathematics theory.But these are not enough to represent the continuously appeared uncertainties in many branches, such as medicine, economics, engineering, social sciences, etc.
The Russian mathematician Molodtsov [2] introduced the concept of soft set as an extension of crisp sets.He successfully applied it to many areas such as probability theory, operation researches, game theory, smoothness of functions, Perron integration, theory of measurement and so on.Since then, the theory of soft sets has been widely and intensively discussed.
Császár continued to try to find a more general structure from general topology, generalized topology, and minimal structure.In 2010, Császár [11] introduced the notion of weak structures and proved that it can replace the already defined structures.A sub-collection w ⊂ P (X) is said to be a weak structure on X if and only if it contains the empty set.Its properties have been investigated intensively (see [16][17][18][19][20][21][22]).
Recently, Zakari et al. [23] integrated the soft sets theory with weak structures to define the soft weak structures.Moreover, they discussed and verified the separation axioms and compactness of soft weak structures.Their results are an extension to the corresponding notions in weak structure, minimal structure, generalized topology, general topology and soft topology.However, Shi and Pang [24] demonstrated the redundancies of soft topologies that this can not be achieved on soft weak structures.Also, Zakari et al. [23] proved that many properties in weak structures can not be achieved in soft weak structures.
The applications of baire spaces are varied in complete metric spaces.To improve its applications, some spaces like hyperspace and Volterra space have been studied by some researchers [25,26].Later on, m-baire spaces are introduced by Chakrabarti and Dasgupta [27].The aim of this paper is to introduce and study soft weak baire spaces.Therefore, some characterizations and properties of soft weak baire spaces are investigated.Finally, the soft images and inverse soft images of such spaces are presented.

Preliminaries
In the sequel, X and E refers to a non-empty set and the parameters set, respectively.2 X denotes the power set of X and A be a non-empty subset of E. The function f A : A −→ 2 X is said to be a soft set [2] over X.For each e ∈ A, f A (e) is the set of e-approximate elements of the soft set f A .A soft set f A is called a finite (resp.countable) soft set if f A (e) is finite (resp.countable) for any e ∈ A. In 2008, Majumdar and Samanta [28] extended the soft set by re-defining it as a function From now on, XE refers to the collection of all soft sets over X with respect to the attributes set E. A null (resp.absolute) soft set f ∅ ∈ XE (resp.f E ∈ XE ), denoted by Φ (resp.X), defined by f ∅ (e) = ∅ (resp.f E (e) = X) for all e ∈ E. For any two soft sets f A , g B ∈ XE , we have: (5) If f A ∈ XE , then the complement of f A , given by f A , is given by f [29] The soft point (x 0 ) A , where A ⊂ E, is a soft set defined by (x 0 ) A (e) = {x 0 } for each e ∈ A and some x 0 ∈ X, and for each e ∈ A.
Aygünoglu and Aygün [29] proved that (x 0 ) A ∈g B h C does not lead to (x 0 ) A ∈g B or (x 0 ) A ∈h C , while this leads to the existence of two soft points (x 0 ) A1 ∈g B and (x 0 ) A2 ∈h C such that (x 0 ) A = (x 0 ) A1 (x 0 ) A2 .Definition 2. [30] Let XE and ỸF be two collections of soft sets, and let α : X −→ Y , β : E −→ F be two functions.The image and preimage of a soft set under the soft function ϕ βα : XE −→ ỸF are given by for each f A ∈ XE and g B ∈ ỸF .A soft function ϕ βα is said to be injective (resp.surjective) if both α and β are injective (resp.surjective).
Definition 3. [23] Let XE be a collection of soft sets on X with respect to the set of attributes E.Then, the collection σ ⊂ XE is called a soft weak structure iff Φ ∈ σ.The triplet (X, E, σ) is said to be a soft weak structure.
A soft weak structure (X, E, σ) have the property Moreover, σ is said to be a strong soft weak structure if X ∈ σ.Proposition 1. [23] For any soft weak structure (X, E, σ), σ e = {f A (e)|f A ∈ σ} is a weak structure on X for any e ∈ E. Definition 4. [23] For any soft weak structure (X, E, σ) and Y ⊂ X, the relative soft weak structure or the soft weak substructure For any soft weak structure (X, E, σ) and f A ∈ XE , Zakari et al. [23] defined f A as the soft intersection of all σ-closed soft supsets of f A and f A as the soft union of all σ-open soft subsets of f A .The following theorem presents the main properties of the introduced operators: Theorem 1. [23] For any soft weak structure (X, E, σ) and soft sets f A , g B ∈ XE , we have the following properties: For any soft weak structure (X, E, σ) and The converse need not be true in general as shown in [23] . By G, we refer to the collection of all σ * -open soft sets with respect to the soft weak structure (X, E, σ).It is easily to verify that G is a generalized soft topology.Proposition 2. [23] Let (X, E, σ) be a soft weak structure and g B ∈ XE .Then: Proposition 4. [23] Let σ Y be a soft weak substructure of σ and f A , g B ∈ XE .Then the following statements hold: Where g A Y and f A Y denote the relative closure and relative interior of g B and f A , respectively.
Theorem 2. Let (X, E, σ) be a soft weak structure and (Y, E, σ Y ) be open soft weak substructure of (X, E, σ) having the Since σ has the property I, σ Y has the same property and so (x (2) σ-nowhere dense or σ-rare soft set if f A = Φ.Lemma 1.Let (X, E, σ) be a soft weak structure and f A ∈ XE .Then the following statements hold: and so f A is σ-dense soft set. ( and so f A is σ-rare soft set. The converse Lemma 1 (1) is not true in general.Moreover, we can not drop the condition of σ * -openness in (2) as shown by following example: Then σ is a soft weak structure.If the soft set l E is defined by Definition 6.Let (X, E, σ) be a soft weak structure.The soft set f A is called: (1) σ-first category (briefly, σ-FCat) soft set if f A can be written as a countable soft union of σ-rare soft sets.
Now, let ϕ βα be a f σc-soft function and f A ∈ XE be a σ 1 -dense soft set.Then f A = Φ and hence which implies that ϕ βα (f A ) = Ỹ .By the hypothesis, ϕ −1 βα (ϕ βα (f A ) ) = X which implies that f A = X so that f A = X and hence f A = Φ, which is a contradiction.Therefore, ϕ βα (f A ) = Φ.Now, suppose that ϕ βα is f σo and g B ∈ ỸF is σ 2 -dense soft set.Then g B = Ỹ and hence g B = Φ and so Theorem 5. Every σ-continuous soft function is f σc.
Proof.Let (X, E, σ 1 ) and (Y, F, σ 2 ) be two soft weak structures and ϕ βα : XE −→ ỸF be σ-continuous soft function.If Proof.Let (X, E, σ 1 ) and (Y, F, σ 2 ) be two soft weak structures, and ϕ βα : XE −→ ỸF be almost σ-open soft function.For The converse of the above two theorems is not true in general as shown by the following example: and let the soft function ϕ βα : XE −→ ỸF defined by:

Soft weak baire space
In this section, we present the baire property of soft weak structures.We have called those soft weak structures that have baire property by soft weak baire spaces.Definition 9. A soft weak structure ( XE , σ) is said to be a soft weak baire space if the soft intersection of each countable collection of σ-dense σ * -open soft sets is σ-dense.g Bn is σ-dense.Therefore, (X, E, σ) is a soft weak baire space.
Conversely, let (X, E, σ) be a soft weak baire space and Ỹ is σ-open soft set.Then Ỹ is soft weak baire space and hence Ỹ is σ-SCat.

Conclusion
In this paper, we studied soft weak baire spaces as an extension to weak baire spaces given in [1].Further, we presented some soft functions between it.Also, the relation with the second category axiom is introduced and discussed.A soft weak baire space is a parameterized family of weak baire spaces.So, we think that our results will play, in fact, an important role in soft version of analysis, topology and mathematical logic.Therefore, we suppose that this is an extra justification for the results studied in this paper.In future, we intend to introduce the soft hereditary classes [32] and modify soft weak baire spaces based on it and the future research will be performed in the same direction.
where h C (e) = f A (e) ∪ g B (e) for any e ∈ E and C = A ∪ B.(4) f A g B = k D , where k D (e) = f A (e) ∩ g B (e) for any e ∈ E and D = A ∩ B.

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An = Φ for each countable family {f An : n ∈ N} of σ-dense and σ * -open soft sets.Proof.(1) Suppose that ∞ d n=1 An = Φ for each countable family {f An : n ∈ N} of σ-dense soft sets, then ∞ n=1 An = X where f An is a σ-rare soft set.Therefore, An is a soft intersection of σ-dense soft sets, which contradicts with the hypothesis.Thus ∞ n=1 An = X for each collection of σ-rare soft sets.Hence (X, E, σ) is of the σ-SCat.

( 2 )fff
If {f An : n ∈ N} is a countable collection of σ-dense σ * -open soft sets.Then {f An : n ∈ N} is a countable collection of σ-rare soft sets.Then ∞ n=1 An is of the σ-FCat.But (X, E, σ) is of the σ-SCat, then An = Φ.Corollary 1.A soft weak structure (X, E, σ) is of σ-Scat in itself if ∞ d n=1 An = Φ for each countable collection {f An : n ∈ N} of σ-dense and σ * -open soft sets.

Theorem 7 .ffffggffn=1gn=1gggg
Every soft weak baire space (X, E, σ) is of σ-SCat.Proof.Let {f An : n ∈ N} be a countable family of σ-dense and σ * -open soft sets, then ∞ d n=1 An = X and so ∞ d n=1 An = Φ.Therefore, (X, E, σ) is of σ-SCat.Theorem 8.If (X, E, σ) is a soft weak structure, then (X, E, σ) is a soft weak baire space if and only if ∞ d n=1 An = Φ implies ∞ n=1 An = Φ, for each countable family {f An : n ∈ N} of σ * -closed soft sets.Proof.Let {f An : n ∈ N} be a countable family of σ * -closed soft sets such that ∞ d n=1 f An = Φ and let g Bn = (f An ) .Then g Bn is σ * -open soft set such that g Bn = f An = f An = X, i.e., g Bn is σ-dense soft set for each n ∈ N, i.e., {f An : n ∈ N} is a countable collection of σ-dense σ * -open soft set.Therefore, ∞ d n=1 Bn is σ-dense soft set and so g Bn = X which implies that ( g Bn ) = Φ and so ∞ d n= g Bn = Φ, i.e., ∞ n=1 Bn = Φ and so ∞ n=1 An = Φ.Now, let {f An : n ∈ N} be a countable collection of σ-dense and σ * -open soft sets.Then {f An : n ∈ N} is a countable collection of σ * -closed soft sets such that f An = Φ.Then by hypothesis, An is σ-dense soft set and so (X, E, σ) is a soft weak baire space.Theorem 9.For any soft weak structure (X, E, σ) with the property I, (X, E, σ) is a soft weak baire space if and only if every soft point has a σ-open soft set which is soft weak baire space.Proof.Let {g Bn : n ∈ N} be a countable family of σ-dense and σ * -open soft sets, and (x 0 ) A ∈ Ỹ where Ỹ ∈ σ.Then Ỹ = Ỹ g Bn and Ỹ g Bn Y = Ỹ Ỹ g Bn for each n ∈ N. Therefore, Ỹ g Bn Y = Ỹ g Bn = Ỹ and Ỹ g Bn is σ-dense soft set with respect to Ỹ for each n ∈ N. Now, Ỹ g Bn Y Ỹ Ỹ g Bn = Ỹ Ỹ g Bn = Ỹ g Bn and so Ỹ g Bn Y = Ỹ g Bn so that Ỹ g Bn is σ * -open soft set for each n ∈ N. By the hypothesis, Ỹ is soft weak baire space and so it is of the σ-SCat.Hence ∞ d n=1 Ỹ g Bn = Φ and so Ỹ ∞ d Bn = Φ.Therefore, (x 0 ) A ∈ ∞ d Bn , which implies that d ∞ n=1 g Bn is σ-dense soft set.Therefore (X, E, σ) is a soft weak baire space.Conversely, if (X, E, σ) is soft weak baire space and (x 0 ) A ∈ Ỹ where Ỹ ∈ σ, then Ỹ is a soft weak baire space.Hence, each soft point has a σ-open soft set which is a soft weak baire space.Theorem 10.For any soft weak structure (X, E, σ) has the I property, (X, E, σ) is a soft weak baire space if and only if every nonempty σ-open soft set is of σ-SCat.Proof.Let {g Bn : n ∈ N} be a countable family of σ-dense and σ * -open soft set and let (x 0 ) A ˜ ∈ ∞ d n=1 Bn .Then, there exists a σ-open soft set Ỹ such that (x 0 ) A ∈ Ỹ and Ỹ ∞ d n=1 Bn = Φ and so ∞ d n=1 Ỹ g Bn = Φ.Since Ỹ ∈ σ and g Bn is σ-dense for each n ∈ N, then Ỹ g Bn is σ-dense soft set with respect to Ỹ .Moreover, Ỹ g Bn Ỹ Ỹ g Bn = Ỹ Ỹ g Bn = Ỹ g Bn and so Ỹ g Bn is σ * -open soft set.By the hypothesis, Ỹ is of σ-SCat and so ∞ d n=1 Ỹ g Bn = Φ, which is a contradiction.Hence (x 0 ) A ∈ ∞ d n=1 Bn and so ∞ d n=1