<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619093pt" id="M1" height="9.70824pt" version="1.1" viewBox="-0.0657574 -9.24633 19.974 9.70824" width="19.974pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-240" d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"/></g><g transform="matrix(.017,0,0,-0.017,9.698,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg>Algebra and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619093pt" id="M2" height="9.70824pt" version="1.1" viewBox="-0.0657574 -9.24633 19.974 9.70824" width="19.974pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g113-240"/></g><g transform="matrix(.017,0,0,-0.017,9.698,0)"><use xlink:href="#g117-33"/></g></svg>Baire in Fuzzy Soft Setting

Khalil, Shuker Mahmood; Ulrazaq, Mayadah; Abdul-Ghani, Samaher; Al-Musawi, Abu Firas

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

Advances in Fuzzy Systems

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Research Article | Open Access

Volume 2018 | Article ID 5731682 | https://doi.org/10.1155/2018/5731682

Algebra and Baire in Fuzzy Soft Setting

Shuker Mahmood Khalil,¹Mayadah Ulrazaq,¹Samaher Abdul-Ghani,¹and Abu Firas Al-Musawi¹

Academic Editor: Katsuhiro Honda

Received30 Jan 2018

Revised14 May 2018

Accepted29 May 2018

Published02 Jul 2018

Abstract

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 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 are introduced. Furthermore, several examples are shown as well.

1. Introduction

The 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 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.

2. Preliminaries

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

Definition 1 ([20]). Assume is an initial universe set and is a set of parameters. Let refer to the family of all fuzzy soft sets (FSSs) in and . The multivalued map defined by is said to be a fuzzy soft set (FSS) over , where if and if . We refer to family of all (FSSs) over by .

Definition 2 ([20]). We say is null (FSS) and we refer to it by , if , is the null (FSS) of , where .

Definition 3 ([20]). Assume and , where . We say is absolute (FSS) and we refer to it by .

Definition 4 ([20]). We say is a fuzzy soft subset of a (FSS) over a common universe if and ; i.e., if and and denoted by .

Definition 5 ([20]). Assume and are (FSSs) over a common universe . We say they are fuzzy soft equal if and .

Definition 6 ([20]). Assume and are (FSSs) over a common universe . Their union is the (FSS) , defined by , where . For this case, we write .

Definition 7 ([20]). Assume and are (FSSs) over a common universe . Their intersection is a (FSS) , defined by , where . For this case, we write .

Definition 8 ([27]). Assume is a (FSS). We refer to its complement by , and it is defined as

Remark 9. Let be a family of fuzzy soft sets. The union (intersection) of any number of family is defined by , where .

Definition 10 ([28]). Assume is the family of (FSSs) over . We say is a fuzzy soft topology on if the following axioms hold:
(i) belong to .
(ii) The union of any number of (FSSs) in belongs to .
(iii) The intersection of any two (FSSs) in belongs to .
We say is a fuzzy soft topological space (FSTS) over . Each member in is called (FSOS) (FSOS) in and its complement is called fuzzy soft closed set (FSCS) in , where .

Definition 11 ([28]). The union of all fuzzy soft open subsets of over is called the interior of and is denoted by .

Proposition 12 ([28]). .

Definition 13 ([28]). Let be a (FSS). Then the intersection of all closed sets, each containing , is called the closure of and is denoted by .

Remark 14. (1) For any (FSS) in a (FSTS) , it is easy to see that and .
(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.

3. Fuzzy Soft 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) in a (FSTS) is called fuzzy soft dense if there exists no (FSCS) in such that .

Definition 16. A (FSS) in a (FSTS) is called fuzzy soft nowhere dense if there exists no nonempty (FSOS) in such that . That is, .

Definition 17. A (FSS) in a (FSTS) is called fuzzy soft meager if is a countable union of fuzzy soft nowhere dense sets [i.e., if , where ’s are fuzzy soft nowhere dense sets in , ]. Otherwise, will be called a fuzzy soft second category set.

Definition 18. A (FSTS) is called fuzzy soft meager or (fuzzy soft first category) space if the (FSS) is a fuzzy soft meager set in . That is, , where ’s are fuzzy soft nowhere dense sets in . Otherwise, will be called a fuzzy soft second category space.

Definition 19. Assume is a fuzzy soft meager set in . We say is a fuzzy soft residual set in .

Definition 20. Assume is a (FSTS). We say is a fuzzy soft Baire space if each sequence of fuzzy soft nowhere dense sets in such that .

Definition 21. A (FSS) in a (FSTS) is called fuzzy soft set in if , where .

Remark 22. Definition 20 can be written in other equivalent forms as follows:
We say is a fuzzy soft Baire space if every countable fuzzy soft closed cover of , the set is fuzzy soft dense in .
We say is a fuzzy soft Baire space if every sequence of (FSOSs) with the same closure , we have .
We say is a fuzzy soft Baire space if every fuzzy soft meager and fuzzy soft set in is fuzzy soft nowhere dense.

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 . 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 , and .

Definition 24. A (FSS) in a (FSTS) is called fuzzy soft set in if , where .

Definition 25. A (FSS) in a (FSTS) is called fuzzy soft nowhere dense set if is a fuzzy soft set in such that .

Example 26. As an illustration, consider the following example. Suppose the (FSSs) describe attractiveness of the cars with respect to the given parameters, which my friends are going to buy. which is the set of all cars under consideration. Let be the collection of all fuzzy subsets of . Also, let , where stand for the attributes of cheap, qualification, and colorful, respectively. Let be defined as follows:Then, is clearly a fuzzy soft topology on . Now consider the (FSS)in . Then is a fuzzy soft set in and and hence is a fuzzy soft nowhere dense set in . The (FSS)is a fuzzy soft set in and and hence is not a fuzzy soft nowhere dense set in .

Definition 27. A (FSS) in a (FSTS) is called fuzzy soft meager if , where ’s are fuzzy soft nowhere dense sets in . Any other (FSS) in is said to be of fuzzy soft second category.

Definition 28. A (FSTS) is called fuzzy soft meager space if the (FSS) is a fuzzy soft meager set in . That is, , where ’s are fuzzy soft nowhere dense sets in . Otherwise, will be called a fuzzy soft second category space.

Definition 29. Assume is a fuzzy soft meager set in a (FSTS) . We say is a fuzzy soft residual set in .

Proposition 30. Let be a (FSTS). Then the following are equivalent:
(1) is a fuzzy soft Baire space.
(2) for every fuzzy soft meager set in .
(3) for every fuzzy soft residual set in .

Proof. (1) (2). Let be a fuzzy soft meager set in . Then , where ’s, are fuzzy soft nowhere dense sets in . Then, we have . Since is a fuzzy soft Baire space, . Hence, for any fuzzy soft meager set in .
(2) (3). Let be a fuzzy soft residual set in . Then is a fuzzy soft meager set in . By hypothesis, . Then . Hence, for any fuzzy soft residual set in .
(3) (1). Let be a fuzzy soft meager set in . Then , where ’s are fuzzy soft nowhere dense sets in . Now is a fuzzy soft meager set in implying that is a fuzzy soft residual set in . By hypothesis, we have . Then . Hence, . That is, , where ’s are fuzzy soft nowhere dense sets in . Hence, is a fuzzy soft Baire space.

Proposition 31. If is a fuzzy soft dense and fuzzy soft set in a (FSTS) , then is a fuzzy soft meager set in .

Proof. Since is a fuzzy soft set in , , where and since is a fuzzy soft dense set in , . Then . But . Hence, . That is, . Then we have for each and hence which implies that and hence . Therefore, is a fuzzy soft nowhere dense set in . Now . Therefore, , where ’s are fuzzy soft nowhere dense sets in . Hence, is a fuzzy soft meager set in .

Lemma 32. If is a fuzzy soft dense and fuzzy soft set in a (FSTS) , then is a fuzzy soft residual set in .

Proof. Since is a fuzzy soft dense and fuzzy soft set in , by Proposition 31, we have that is a fuzzy soft meager set in and hence is a fuzzy soft residual set in .

Proposition 33. In a (FSTS) , a (FSS) is a fuzzy soft nowhere dense set in if and only if is a fuzzy soft dense and fuzzy soft set in .

Proof. Let be a fuzzy soft nowhere dense set in . Then , where , and . Then implies that . Also , where , for . Hence, we have is a fuzzy soft dense and fuzzy soft set in .
Conversely, let be a fuzzy soft dense and fuzzy soft set in . Then , where for . Now . Hence, is a set in and [since is a fuzzy soft dense]. Therefore, is a fuzzy soft nowhere dense set in .

Definition 34. Assume is a (FSTS). We say is a fuzzy soft Baire space if each sequence of fuzzy soft nowhere dense sets in such that .

Example 35. Let be the set of three flats and (), modern (), security services be the set of parameters. Then, we consider that is a fuzzy soft topology on defined as follows:, , , , , , , , , , , , , and Now consider the (FSSs)andin . Then and are fuzzy soft sets in and , , and .Then and are fuzzy soft nowhere dense sets in . Moreover, and also and therefore is a fuzzy soft Baire space.

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

Example 37. Assume is a set of soldiers under consideration and courageous (); strong (); smart() is a set of parameters framed to choose the best soldier. Then, we consider that is a fuzzy soft topology on defined as follows:, , , , , , , , and Now are fuzzy soft nowhere dense sets in . . Therefore, is a fuzzy soft meager set in . . Hence, is not a fuzzy soft Baire space. Now consider the (FSSs) , and in and also , . Then and are fuzzy soft nowhere dense sets in . Now the (FSS) is a fuzzy soft meager set in and . Hence, is a fuzzy soft Baire space.

Proposition 38. Let be a (FSTS). Then the following are equivalent:
(1) is a fuzzy soft Baire space.
(2) for every fuzzy soft meager set in .
(3) for every fuzzy soft residual set in .

Proof. (1) (2). Let be a fuzzy soft meager set in . Then , where (’s, are fuzzy soft nowhere dense sets in . Then, we have . Since is a fuzzy soft Baire space, . Hence, for any fuzzy soft meager set in .
(2) (3). Let be a fuzzy soft residual set in . Then is a fuzzy soft meager set in . By hypothesis, . Then . Hence, for any fuzzy soft residual set in .
(3) (1). Let be a fuzzy soft meager set in . Then , where ’s are fuzzy soft nowhere dense sets in . Now is a fuzzy soft meager set in implying that is a fuzzy soft residual set in . By hypothesis, we have . Then . Hence, . That is, , where ’s are fuzzy soft nowhere dense sets in . Hence, is a fuzzy soft Baire space.

Proposition 39. If the (FSTS) is a fuzzy soft Baire space, then , where the (FSSs) ’s are fuzzy soft dense and fuzzy soft sets in .

Proof. Let ’s, be fuzzy soft dense and fuzzy soft sets in . By Proposition 33, ’s are fuzzy soft nowhere dense sets in . Then the (FSS) is a fuzzy soft meager set in . Now . Since is a fuzzy soft Baire space, by Proposition 38, we have . Then . This implies that .

Proposition 40. If the (FSTS) is a fuzzy soft Baire space, then , where the (FSSs) sets ’s, are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft sets in .

Proof. Let the (FSTS) be a fuzzy soft Baire space and the (FSSs) ’s, be fuzzy soft dense and fuzzy soft sets in . By Proposition 39, . Then . This implies that . Also by Proposition 31, ’s are fuzzy soft meager sets in . Hence, , where the (FSSs) ’s, are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft sets in .

Proposition 41. If the fuzzy soft meager sets are formed from the fuzzy soft dense and fuzzy soft sets in a fuzzy soft Baire space , then is a fuzzy soft Baire space.

Proof. Let the (FSTS) be a fuzzy soft Baire space. By Proposition 40, , where the (FSSs) ’s, are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft sets in . Now . Then we have . This implies that , where is a fuzzy soft meager set in . By Proposition 30, is a fuzzy soft Baire space.

Proposition 42. If the (FSTS) is a fuzzy soft meager space, then is not a fuzzy soft Baire space.

Proof. Let the (FSTS) be a fuzzy soft meager space. Then , where ’s are fuzzy soft nowhere dense sets in . Now . Hence, by definition, is not a fuzzy soft Baire space.

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

Example 44. Let there be three houses in the universe given by and let (); steel (); and brick 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 is a fuzzy soft topology defined as follows:, , , , , , , , , Now consider the (FSSs) , and in . Then and are fuzzy soft sets in and . Then and are fuzzy soft nowhere dense sets in . Now . Therefore, is a fuzzy soft second category space. But and therefore is not a fuzzy soft Baire space.

Proposition 45. If , where the (FSSs) ’s are fuzzy soft dense and fuzzy soft sets in a (FSTS) , then is a fuzzy soft second category space.

Proof. Let ’s, be fuzzy soft dense and fuzzy soft sets in . By Proposition 33, ’s are fuzzy soft nowhere dense sets in . Now implies that . Then . Hence, is not a fuzzy soft meager space and therefore is a fuzzy soft second category space.

Proposition 46. If is a fuzzy soft meager set in , then there is a fuzzy soft set in such that .

Proof. Let be a fuzzy soft meager set in . Thus, , where ’s are fuzzy soft nowhere dense sets in . Now ’s are (FSOSs) in . Then is a fuzzy soft set in and . Now . That is, and is a fuzzy soft set in . Let . Hence, if is a fuzzy soft meager set in , then there is a fuzzy soft set in such that .

Proposition 47. If is a fuzzy soft residual set in a (FSTS) such that in such that , where is a fuzzy soft dense and fuzzy soft set in , then is a fuzzy soft Baire space.

Proof. Let be a fuzzy soft residual set in a (FSTS) . Thus, is a fuzzy soft meager set in . Now by Proposition 46, there is a fuzzy soft set in such that . This implies that . Let . Then is a fuzzy soft set in and implies that . If , then we have . Hence, by Proposition 30, is a fuzzy soft Baire space.

Proposition 48. If the (FSTS) is a fuzzy soft Baire space and if , then there exists at least one set such that .

Proof. Suppose that , (), where ’s are fuzzy soft nowhere dense sets in . Then , implying that , a contradiction to being a fuzzy soft Baire space. Hence, , for at least one set in .

Proposition 49. If the (FSTS) is a fuzzy soft Baire space, then no nonempty (FSOS) is a fuzzy soft meager set in .

Proof. Let be nonempty (FSOS) in a fuzzy soft Baire space . Suppose that , where the (FSSs) ’s are fuzzy soft nowhere dense sets in . Then . Since is a fuzzy soft Baire space, . This implies that . Then we will have , which is a contradiction, since implies that . Hence, no nonempty (FSOS) is a fuzzy soft meager set in .

Definition 50. A (FSTS) is called a fuzzy soft submaximal space if for each (FSS) in such that ; then in .

Proposition 51. If the (FSTS) is a fuzzy soft submaximal space and if is a fuzzy soft meager set in , then is a fuzzy soft meager set in .

Proof. Let be a fuzzy soft meager set in , where the (FSSs) ’s, are fuzzy soft nowhere dense sets in . Then we have and ’s, are fuzzy soft sets in . Now , implying that and hence . Since is a fuzzy soft submaximal space, the fuzzy soft dense sets ’s are (FSOSs) in and hence ’s are (FSCSs) in . Then and imply that . That is, ’s are fuzzy soft nowhere dense sets in . Therefore, is a fuzzy soft meager set in .

Proposition 52. If the (FSTS) is a fuzzy soft Baire space and fuzzy soft submaximal space, then is a fuzzy soft Baire space.

Proof. Let be a fuzzy soft meager set in . Since is a fuzzy soft submaximal space, by Proposition 51, is a fuzzy soft meager set in . Since is a fuzzy soft Baire space, by Proposition 38, . Hence, for the fuzzy soft meager set in , we have . Therefore, by Proposition 30, is a fuzzy soft Baire space.

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.

Proof. Let the (FSTS) be a fuzzy soft Baire space. Then, by Proposition 39, , where the (FSSs) ’s, are fuzzy soft dense and fuzzy soft sets in . Now from , we have . This implies that . Since the (FSSs) ’s are fuzzy soft dense in , . Then we have . This implies that . Also, since is a fuzzy soft space, the non−empty fuzzy soft sets ’s in are fuzzy soft open in . Then ’s are (FSCSs) in . Then and imply that . That is, ’s are fuzzy soft nowhere dense sets in . Therefore, we have , where ’s are fuzzy soft nowhere dense sets in . Hence, by Proposition 30, is a fuzzy soft Baire space.

Definition 55. A (FSTS) is called a fuzzy soft almost resolvable space if , where the (FSSs) ’s in are such that . Otherwise, is called a fuzzy soft almost irresolvable space.

Proposition 56. If the (FSTS) is a fuzzy soft almost irresolvable space, then is a fuzzy soft second category space.

Proof. Let ’s, be the fuzzy soft dense and fuzzy soft sets in . Now implies that . That is, . Since is a fuzzy soft almost irresolvable space, , where the (FSSs) ’s in are such that . Now implies that . Hence, we have , where the (FSSs) ’s are fuzzy soft dense and fuzzy soft sets in a (FSTS) . Thus, by Proposition 45, is a fuzzy soft second category space.

Definition 57. A (FSTS) is called a fuzzy soft hyperconnected space if every (FSOS) is fuzzy soft dense in . That is, .

Proposition 58. If , where ’s are fuzzy soft dense and fuzzy soft sets in , then is a fuzzy soft Baire space.

Proof. The proof is obvious.

Proposition 59. If , where the (FSSs) ’s are fuzzy soft sets in a fuzzy soft hyperconnected and fuzzy soft space , then is a fuzzy soft Baire space.

Proof. Let ’s, be the fuzzy soft sets in such that . Since is a fuzzy soft space, the fuzzy soft sets ’s in are fuzzy soft open in . Also since is a fuzzy soft hyperconnected space, the (FSOSs) ’s in are fuzzy soft dense sets in . Hence, the (FSSs) ’s, are fuzzy soft dense and fuzzy soft sets in and . Hence, by Proposition 58, is a fuzzy soft Baire space.

Definition 60. Let be a fuzzy soft subset of a fuzzy soft space . Then is said to be fuzzy soft embedded in if each fuzzy soft subset of which is contained in is fuzzy soft nowhere dense in (i.e., ).

Proposition 61. Let be a fuzzy soft dense subspace of a fuzzy soft Baire space . If is fuzzy soft embedded in , then is a fuzzy soft Baire space.

Proof. Observe that if is not a fuzzy soft Baire space; then there is a sequence of fuzzy soft open subsets of such that each is fuzzy soft dense in and yet . Then there is a sequence of fuzzy soft open subsets of such that and . Each is fuzzy soft dense in and is a fuzzy soft Baire space. Hence, is fuzzy soft dense in and therefore in . Since is not fuzzy soft embedded in .

Proposition 62. Let be a fuzzy soft dense subspace of a fuzzy soft Baire space . If is dense in , then is a fuzzy soft Baire space if and only if is fuzzy soft embedded in .

Proof. Assume that is not fuzzy soft embedded in . Let be a fuzzy soft subset of which is contained in and which is fuzzy soft dense in some relatively (FSOS) of . Let be an fuzzy soft open subset of with . Then is a subset of which is fuzzy soft dense in and which is contained in . Let , where each is open in and . The (FSSs) are fuzzy soft open and fuzzy soft dense subsets of and yet . It follows that is not a fuzzy soft Baire space. Consequently, is not a fuzzy soft Baire space. Conversely, assume that is fuzzy soft embedded in and, by Proposition 61, then is a fuzzy soft Baire space.

Lemma 63. Let be a fuzzy soft Baire space. If is a nonempty fuzzy soft nowhere dens set of , where is a fuzzy soft open subset of , then for every nonempty fuzzy soft open subset of there is such that .

Proof. Let be a nonempty fuzzy soft open subset of . Then, is a nonempty fuzzy soft open subset of ; hence, is also fuzzy soft Baire. Since and each is a fuzzy soft closed subset of , by Remark 22, then there is such that .

Proposition 64. Let be a fuzzy soft Baire space and let be fuzzy soft dense. Then is a fuzzy soft Baire space if and only if every fuzzy soft set in contained in is fuzzy soft nowhere dense.

Proof. Necessity. Let , where is a fuzzy soft open subset of for each , which is contained in . Then, . In virtue of Remark 22, is fuzzy soft dense in . Suppose that . Then, there is such that . On the other hand, we know that . Hence, which implies that , but this is impossible.
Sufficiency. Assume that is no fuzzy soft Baire. According to Remark 22, there is a countable fuzzy soft closed cover of such that is not dense in . For each , choose a fuzzy soft closed subset of such that , for each . Let which is a set of contained in . If , then would be a fuzzy soft closed cover of and, by Remark 22, then would be fuzzy soft dense in which is not possible. So, . Choose a nonempty fuzzy soft open subset of such that , . By Lemma 63, we can find such that . Hence, , but this is a contradiction. Thus, 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 is fuzzy soft Baire if every fuzzy soft dense subspace of is fuzzy soft Baire.

An immediate consequence of Proposition 64 is the following.

Corollary 66. 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.

Corollary 68. Every fuzzy soft Baire and fuzzy soft almost space is fuzzy soft 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 be a (FSTS). Then, the class is the fuzzy soft algebra in generated by all (FSOSs) and all fuzzy soft nowhere dense sets.

Remark 71. (1) For a (FSTS) , the collection of all fuzzy soft Borel sets on forms a fuzzy soft algebra.
(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.
(1) is fuzzy soft Baire.
(2) is fuzzy soft Baire and every fuzzy soft set with empty interior is fuzzy soft nowhere dense.
(3) Every fuzzy soft meager subset 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.

Proof. (1)(2). This is Corollary 66.
(2) (3). Let be a fuzzy soft meager set. Assume , where is fuzzy soft nowhere dense . Therefore, is a set in and is fuzzy soft dense in because its complement is a fuzzy soft meager set and is fuzzy soft Baire. Let . The (FSS) clearly has empty interior. Hence, is a fuzzy soft set with empty interior; by hypothesis, is fuzzy soft nowhere dense. Also is a fuzzy soft nowhere dense set. Therefore, is a fuzzy soft nowhere dense set as well. On the other hand, is a fuzzy soft meager set. Since is fuzzy soft Baire, . Therefore, and 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 is fuzzy soft nowhere dense; i.e., has empty interior. Therefore, is a fuzzy soft open dense subspace of .
(4) (2). Let be a set with empty. First observe that . Since is an set with empty interior, is a fuzzy soft dense set of By assumption, is also fuzzy soft dense in . That is, is fuzzy soft dense in . Hence, and so .
(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 and clauses (2) and (3).
(5) (6). This implication is obvious because the fuzzy soft algebra of fuzzy soft Borel sets is contained in the class .
(6) (1). It is enough to observe that (6) (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 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?

Data Availability

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

Conflicts of Interest

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

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Copyright © 2018 Shuker Mahmood Khalil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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