363 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License ̃ - open sets and ̃ - continuous in soft topological spaces Nehmat K. Ahmed 1 and Shaymaa M. Mohammed 2* 1,2 Department of Mathematics, College of Education, Salahaddin University, Erbil, Iraq. *Corresponding Author. Received: 15 December 2024 Accepted: 16 June 2025 Published: 20 October 2025 doi.org/10.30526/38.4.4071 Abstract Our work introduces the notion of -open set. This class of ̃ - open sets is not directly comparable to the categories of soft open, -closed. We demonstrate that the category of ̃ -open sets lies strictly between the class of ̃ -open sets and ̃ -open sets. Additionally, we investigate the connections that exist between ̃ -open sets along with other kinds of soft sets. Moreover, we provide several criteria that are sufficient for determining the equivalence between ̃ -open sets and every one of ̃ -open sets and ̃ -open sets. Also, according to our findings, the family of ̃ -open sets is a supra soft topology. Furthermore, we make it clear the correlation that exists between the respective categories of ̃ -open sets in a soft topological space and in its soft topological subspace. Finally, the class of ̃ -continuous function is introduced, it is main properties are studied and derive some of the properties of these soft functions under the soft composition of soft functions. Keywords: - open, soft soft semi-closed, -continuous function. 1. Introduction The concept of first using soft set theory by (1) was presented as a novel tool in mathematics for dealing with several forms of ambiguity in complicated scientific problems and solving these problems. Soft sets research and its characteristics were used to several areas of mathematics, including theory, probability, operations research, algebra, mathematical analysis, etc. In 2011, the main concepts of soft topologies were investigated by (2). They presented the ideas of a soft open set and showed that a soft topological space produces a parameterized family of topological spaces, soft closed set, soft interior point, soft closure and soft separation axioms. A multitude of contributions has been made to the research of topological principles in soft environments after soft topology was developed (3- 6). Athar Kharal, A. and Ahmad, B. (7) explained what soft mapping is in the context of soft classes and investigated several properties of soft set images and inverse images. Chen (8) introduced and investigated soft semi-open sets and their properties. Mahanta and Das (9) also introduced and defined soft semi-open sets and soft semi-continuous functions. The authors of (10) introduced the concept of soft -open subsets. However, this work introduces a novel kind of soft open sets (respectively soft functions) called ̃ - open sets (resp.; ̃ -continuous functions), which are strictly placed between the https://orcid.org/0000-0002-7134-3348 mailto:nehmat.ahmed@su.edu.krd https://orcid.org/0009-0006-7522-2953 mailto:shaymaa.mohammed@su.edu.krd https://orcid.org/0000-0002-7134-3348 mailto:nehmat.ahmed@su.edu.krd https://orcid.org/0009-0006-7522-2953 mailto:shaymaa.mohammed@su.edu.krd https://orcid.org/0000-0002-7134-3348 mailto:nehmat.ahmed@su.edu.krd https://orcid.org/0009-0006-7522-2953 mailto:shaymaa.mohammed@su.edu.krd https://orcid.org/0000-0002-7134-3348 mailto:nehmat.ahmed@su.edu.krd https://orcid.org/0009-0006-7522-2953 mailto:shaymaa.mohammed@su.edu.krd https://orcid.org/0000-0002-7134-3348 mailto:nehmat.ahmed@su.edu.krd https://orcid.org/0009-0006-7522-2953 mailto:shaymaa.mohammed@su.edu.krd https://orcid.org/0000-0002-7134-3348 mailto:nehmat.ahmed@su.edu.krd https://orcid.org/0009-0006-7522-2953 mailto:shaymaa.mohammed@su.edu.krd IHJPAS. 2025,38(4) 364 soft classes of ̃ - open sets and ̃ - open sets ̃ -continuous functions and soft - continuous functions). Some of ̃ -continuous functions basic properties and relationships with some other types of soft functions are given. Throughout this work, will be a nonempty initial universal set and will be a set of parameters. ( ̃ ̃ ) ̃ ̃ ) or basically ̃ and ̃ show soft topological spaces (simply, STS), the family of all soft clopen sets (resp.; semi closed) in ̃ is denoted by ̃ ( ̃) ̃ ̃)). A pair ( ) is known as a soft set over , in which is a function Therefore ) The collection of soft sets ( ) over a universal set with the parameter set is indicated by ̃ ̃) . 2.Preliminaries 2.1. Definition (11) For any two soft subsets ) ) over a common universe , we say that ) is a soft subset of ) indicated as ) ̃ ), if ) ) . 2.2. Definition (12) The soft complement of a soft set ) is indicated by ) or ̃ ) and is described as ) ) in which ) is a function Provided by ) ), 2.3. Definition (13) a soft set ) is said to be. 1. A null soft set indicated by ̃ if ., ) . 2. An absolute soft set indicated by ̃ if ) . 3. A soft point indicated by , if ) { We say that ) if ) 2.4. Definition (13) Let ) ) 1. ) ) ̃ ) ) ) ) . 2. ) ) ̃ ) ) ) ) . 2.5. Definition (14) Let ̃ ̃ ̃ ̃) , then ̃ is described as a soft topology on ̃ if 1. ̃ ̃ ̃ ̃. 2. ) ) ̃ ̃, then ) ̃ ) ̃ ̃. 3. If ) ̃ ̃ , then ̃ ) ̃ ̃. The triple ( ̃ ̃ )) (simply, ̃ ) is known as a STS over ̃ . ) is soft open set if ) ̃ ̃. ) is soft closed set if ) ̃ ̃ . The triple ( ̃ ̃ ̃ )is a soft subspace of ( ̃ ̃ ) where W , ̃ ) ̃ ̃ ) ) ̃ is known as ―the soft relative topology‖ on ̃, and ) ̃ ̃ ) . 2.6. Definition (14) Let ̃ be a collection of soft subsets over a ̃ .Then ̃ is described as a soft supra topology on ̃ if 1. ̃ ̃ ̃ ̃. 2. The union of any number of soft sets in ̃ belonges to ̃. The triple ( ̃ ̃ ) is called supra soft topological space and the elements of ̃ are called supra open sets. The soft complement of any soft supra open set is called supra soft closed. 2.7.Definition ̃ ) is a soft interior of the soft subset ) and is defined as follows. ̃ ) ̃ ) ) ̃ ) ) ̃ ̃}. IHJPAS. 2025,38(4) 365 2.8. Definition ̃ ) is a soft closure of the soft subset ) and is defined as follows. ̃ ) ̃ ) ) ̃ ) ) ̃ ̃ }. 2.9. Definition (15) A soft subset ) of a STS ( ̃ ̃ ) become a soft pre-open (resp., soft semi-open ,soft α-open, soft -open, soft b-open ,and soft regular-open) if ) ̃ ̃ ))(resp., ) ̃ ( ̃ )) ) ̃ ( ̃ ( ̃ ))) ) ̃ ( ̃ ( ̃ ))) ) ̃ ̃ )) ̃ ̃ ( ̃ )) ) ̃ ̃ ))) The family of all soft pre-open (resp., soft semi-open ,soft α-open, soft - open, soft b-open, and soft regular-open) subsets of ̃ is indicated by ̃ ̃)(resp., ̃ ( ̃) ̃ ( ̃) ̃ ( ̃) ̃ ( ̃) ̃ ̃)) 2.10. Definition A soft subset ) of STS ( ̃ ̃ ) is recognized as a soft -open (13) (resp., soft open (14), soft -open (14),soft -open (15), soft -open (16) and soft -open (17)) set, if ) ̃ ̃ ̃) (resp., ̃ ̃ ̃) ̃ ̃) ̃ ̃), and ̃ ̃) ) and ̃ ) a soft closed (resp., soft semi-closed, soft regular open, soft closed, soft closed and soft closed) subset ) of ̃ s.t. ̃ ) ̃ ). The family of all soft - open (resp., soft -open, soft -open ,soft -open, soft -open and soft -open) subsets of ̃ is indicated by ̃ ̃) (resp., ̃ ( ̃) ̃ ( ̃) ̃ ̃) ̃ ̃) and ̃ ̃) ). 2.11. Definition (15,16) A STS ( ̃ ̃ ) is recognized as. 1. Soft extremally disconnected space (simply, SEDS), if ̃ ) ̃ ̃ ) ̃ ̃. 2. Soft locally indiscrete, if ) ̃ ̃ ,then ) ̃ ̃ . 3. Soft -space , if ̃ ̃ ̃) such that , there are soft open sets ) and ) s.t ̃ ) ̃ ) and ̃ ) ̃ ). 2.12. Definition (15) Let ̃ ̃ ) be a STS. The soft -interior of a soft subset ) ̃ ̃) is the union of all soft open sets over ̃ whose soft closures are located within ), and is indicated by ̃ ). The soft subset ) is known as soft - open if ̃ ) ). The complement of a soft -open set is called soft -closed. 2.13. Definition (14) Suppose that ̃ ̃) and ̃ ̃) be soft categories. Suppose and be functions. Next, a soft function ̃ ̃) ̃ ̃) is described as the following. 1. For a soft set ) ̃ ̃) , ) ) ) is a soft set in ̃ ̃) provided by ) ) { ( ) ( ))) ) For . ) ) is called a soft image of a soft set ). If , Afterward, we’ll write ) ) as ) 2. For a soft set ) in ̃ ̃) where ) ) ) is a soft set in ̃ ̃) provided by ) ) { ( ( ))) ) for , ) ) is known as a soft inverse image of a soft set ). We’ll write ) ) as ). The soft function is called surjective if and are surjective. The soft function is called injective if and are injective functions. IHJPAS. 2025,38(4) 366 2.14. Definition (15) Suppose that ( ̃ ̃ ) ̃ ̃ ) be a soft mapping and . Then the restriction of to ̃ ̃) is the soft mapping ̃ ̃) from ̃ ̃) to ̃ ̃) which is defined by the functions and where is the restriction of to . 2.15. Definition (18)A soft function ̃ ( ̃ ̃ ) ̃ ̃ ) is known as. 1. Soft continuous, if ̃ ) ̃ ̃ ) ̃ ̃. 2. Soft perfectly continuous, if ̃ ) is a soft clopen set in ̃ ) ̃ ̃. 3. Soft -continuous, if ̃ ) ̃ ̃ ̃) ) ̃ ̃. 4. Soft regular continuous, if ̃ ) ̃ ̃ ( ̃) ) ̃ ̃ 5. ̃ -continuous, if ̃ ) ̃ ̃ ̃) ) ̃ ̃. 6. Soft -continuous, if ̃ ) ̃ ̃ ̃) ) ̃ ̃. 7. Soft -continuous, if ̃ ) ̃ ̃ ̃) ) ̃ ̃. 8. ̃ -continuous, if ̃ ) ̃ ̃ ̃) ) ̃ ̃. 9. Soft irresolute, ̃ ) ̃ ̃ ̃) ) ̃ ̃ ̃). 10. Soft open. if ̃ ) ̃ ̃ ) ̃ ̃. 2.16. Proposition (16) A STS ( ̃ ̃ ) is -space iff ̃ ̃ ̃ ̃) is soft closed set. 2.17. Proposition (16) Let ( ̃ ̃ ) be a STS, then ̃ ̃) forms a soft supra topology. 2.18. Proposition (19) If ( ̃ ̃ ) be a SEDS, then 1. ̃ ( ̃) ̃ ̃ ( ̃) 2. ̃ ( ̃) ̃ ̃ ( ̃) 2.19. Proposition (10) If ) ̃ ̃ ( ) ̃ ̃ ( ̃) in a STS ( ̃ ̃ ) , then ) ̃ ( ) ̃ ̃ ( ̃). 2.20. Proposition (14) Let ( ̃ ̃ ) be a STS, ̃ ̃ ( ̃) iff ̃ ̃ ( ̃) 2.21. Proposition (15) Let ( ̃ ̃ ) be a soft regular space iff ̃ and ̃ ), ) ̃ ̃ s.t. ̃ ) ̃ ̃ ) ̃ ) 2.22. Proposition (18) Let ̃ ̃ ) be a soft subspace of ( ̃ ̃ ) and ) ̃ ̃ then, ̃ ̃ ) ̃ ) if ̃ ̃ ̃ 2.23. Proposition (16) Let ̃ ̃ ) be a soft subspace of ( ̃ ̃ ) and ) ̃ ̃ s.t. ̃ ̃ ̃ ( ̃) ) ̃ ̃ ( ̃), then ) ̃ ̃ ( ̃). 3. ̃ - Open Sets. In this section, we study a new type of soft sets called ̃ - open set in soft topological spaces. Some properties of this type of soft set are given, and the relationships with a few other kinds of soft sets are introduced. 3.1. Definition A soft set ) of STS ( ̃ ̃ ) is called ̃ -open if ) ̃ ̃ ̃) ̃ ) ( ) ̃ ̃ ̃) s.t. ̃ ) ̃ ) The family of all ̃ -open subsets of ( ̃ ̃ ) is indicated by ̃ ̃ ̃ ) or ̃ ̃ ). 3.2. Proposition A soft set ) of STS ( ̃ ̃ ) is ̃ -open iff ) ( ) where ) ̃ ̃ ̃) and ) ̃ ̃ ̃) ̃ Proof: Obvious. Corollary A soft set ) of STS ( ̃ ̃ ) is ̃ -open. If ) ̃ ̃ ( ̃) ̃ ̃ ( ̃) Proof: directly from definition 3.1. IHJPAS. 2025,38(4) 367 Remark 1- ̃ ( ̃) ̃ ̃ ( ̃) But ̃ ( ̃) ̃ ̃ ( ̃) is not true in general. 2- ̃ ( ̃) ̃ ( ̃) 3- ̃ ̃ ( ̃) 4- ̃ ̃ ( ̃) The examples that follow demonstrate the earlier point. Example Consider , with the two soft topological ̃ { ̃ ̃ ( ) ( )} ̃ { ̃ ̃ ( ) ( ) )} where ̃ ) ) ̃ ) ) , ) ) ) ) ) ) ) ) ) ( ) ) ) ( ) ) ) ) ) ) ,( ) ) ) ( ) ) ) ( ) ) ) ( ) ) ) ( ) ) ) ( ) ) ) ( ) ) ) ( ) ) ) ̃. ( ̃ ̃ ) and ( ̃ ̃ ) are two STSs over . 1- ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ ̃ ̃ ) 2- ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ ( ̃ ̃ ) 3- ( ) ̃ ̃ ( ) ̃ ̃ ̃ ̃ ) ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ 4- ( ) ̃ ̃ ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ ( ̃ ̃ ) ( ) ̃ ̃ The following diagram shows the relations between ̃ ( ̃) ̃ ( ̃) ̃ ( ̃) ̃ ( ̃), ̃ ( ̃) ̃ ̃ ) ̃ ( ̃) ̃ ( ̃) ̃ ( ̃) ̃ ̃) ̃ ( ̃) ̃ ( ̃) 3.3. Proposition For any soft set ) of STS ( ̃ ̃ ) 1. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 2. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃). 3. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 4. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 5. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 6. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 7. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 8. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) 9. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) IHJPAS. 2025,38(4) 368 10. ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) The following example illustrates above Proposition. Example Consider the STS ( ̃ ̃ ) in Example, let ( ) ̃ ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) ( ) ̃ ̃ ( ̃ ̃ ) , but ( ) ̃ ̃ ( ̃ ̃ ), ̃ ( ̃ ̃ ) ̃ ( ̃ ̃ ) and ̃ ( ̃ ̃ ). 3.4.Proposition If a space ̃ ̃ ) is a soft T1-space, subsequently ̃ ( ̃) ̃ ̃) ̃ ̃) Proof: ) ̃ ̃ ̃) If ) ̃ then ) ̃ ̃ ̃) If ( ) ̃, then for each ̃ ) ̃ ̃ . Hence, ̃ ̃ ) Therefore, ) ̃ ̃ ( ̃) since ̃ ̃ ̃ ( ̃)then ) ̃ ̃ ( ̃) Hence ̃ ̃) ̃ ̃ ̃) ̃ ̃ ̃) but ̃ ̃) ̃ ̃ ̃) and ̃ ̃) ̃ ̃ ̃) generally, therefore ̃ ( ̃) ̃ ̃) ̃ ̃) Corollary If ( ̃ ̃ )is soft T1-space, then ̃ ̃ ̃ ( ̃) ̃ ( ̃) ̃ ̃ ( ̃) The following findings demonstrate that any union of ̃ -open sets of ( ̃ ̃ ) is ̃ . 3.5. Proposition Arbitrary soft union of soft -open sets in a STS ( ̃ ̃ ) is a soft -open set. Proof: Let ) -open sets in STS ( ̃ ̃ ) Then ) ̃ ̃ ̃) and by Proposition 2.14, ̃ ) } ̃ ̃ ̃). Let ̃ ̃ ) , hence ) for some λ Λ. Since ) ̃ ̃ ( ̃) λ, so there is ) ̃ ̃ ( ̃) s.t. ̃ ) ̃ ) ̃ ̃ ) so ̃ ) ̃ ̃ ) Therefore, ̃ ) ̃ ̃ O( ̃). The soft intersection of two soft -open sets need not be a soft -open sets in general as an exam: Example If ),( ) ̃ ̃ O( ̃) then ) ( ) ̃ ( ̃ ̃ ) let ), ) ̃ ̃ O( ̃ ̃ ). Now we have ) ) = ) It is clear that ) ̃ ̃ O( ̃ ̃ ) and hence ) ) ̃ ̃ O( ̃ ̃ ) 3.6. Proposition If ̃ O ̃)is a soft topology on ̃, then ̃ O ̃) is also a soft topology on ̃. Proof: This is sufficient evidence to demonstrate whether the intersection of two ̃ open subsets is ̃ open. Suppose ),( ) ̃ ̃ O ̃), then ), ) ̃ ̃ O ̃). Since ̃ ̃) is a soft topology on ̃, so ) ) ̃ ̃ O ̃). Let ̃ ) ) then ̃ ) and ̃ ) so there exist ) ) ̃ ̃ ̃). s.t. ̃ ) ̃ ) nd ̃ ) ̃ ) which implies that ̃ ) ) ̃ ) ) then ) ) ̃ ̃ ̃). Thus, ) ) ̃ ̃ ̃). 3.7. Proposition A STS ( ̃ ̃ ) is SEDS if ̃ ( ̃) ̃ ( ̃). Proof: By Remark 3.4(1) and Proposition 2.18(1),the prove only if part will follow. Conversely, let ) ̃ ̃. Then, ̃ ) ̃ ( ̃ )). That is, ̃ ) ̃ ̃ ( ̃), so by Proposition 3.6(6), ̃ ) ̃ ̃ ( ̃). By hypothesis, ̃ ) ̃ ̃ ( ̃). That is, IHJPAS. 2025,38(4) 369 ̃ ) ̃ ̃ ( ̃ ( ̃ ))) Then ̃ ) ̃ ̃ ̃ )),but ̃ ̃ )) ̃ ̃ ). Hence, ̃ ) ̃ ̃ )). This means that, ̃ ) ̃ ̃. Thus, ̃ ̃ ) is a SEDS. 3.8. Proposition A ̃ ̃ ) is SEDS iff ̃ ̃) ̃ ̃ ̃). 3.9. Proposition If a space ̃ ̃ ) is a soft locally indiscrete space, then ̃ ( ̃) ̃ ̃) ̃ ̃) Corollary Let ̃ ̃ ) is a soft locally indiscrete space, then. 1. ̃ ( ̃) ̃. 2. ̃ ( ̃) ̃ ( ̃) 3.10. Proposition A subset ) of ̃ ̃ ) is ̃ - open iff for each ̃ ), ) ̃ ̃ ( ̃) s.t. ̃ ) ̃ ) Proof: Assume that ) ̃ ̃ ( ̃). Then for each ̃ ) ) ) ̃ ̃ ( ̃) containing s.t. ̃ ) ̃ ). Conversely, suppose that ̃ ) here exists ) ̃ ̃ ( ̃) s.t. ̃ ) ̃ ) Thus ) ̃ ) ̃ ) for each ̃ ) this implies that ) ) therefore ) is ̃ -open set. 3.11.Proposition If ̃ ̃ ) is soft regular space, then ̃ ̃ ̃ O( ̃). Proof: Let ) ̃ ̃, then ) ̃ ̃ ( ̃). Since ̃ is regular, then by Proposition 2.21, for each ̃ ), ) ̃ ̃ s.t. ̃ ) ̃ ̃ ) ̃ ) So that, ̃ ̃ ) ̃ ) .Since ̃ ) ̃ ̃ ̃) , then ( ) ̃ ̃ ( ̃). 3.12.Proposition Let ̃ ̃ ) be a SEDS. If ) ̃ ̃ O( ̃) and ) ̃ ̃ O( ̃) . Then ) ) ̃ ̃ ( ̃). Proof: Let ) ̃ ̃ O( ̃) and ) ̃ ̃ O( ̃) then, ) ̃ ̃ then by Proposition 2.19 ) ) ̃ ̃ ( ̃). If ̃ ) ) then, since ) ̃ ̃ ( ̃), ) ̃ ̃ ( ̃) s.t. ̃ ) ̃ ) and so, ̃ ) ) ̃ ) ). Since ) ̃ ̃ ( ̃) in a SEDS so by Proposition 2.18(2), (G, ) ̃ ̃ ( ̃) then it is soft semi-closed and hence ) ) ̃ ̃ ( ̃). Therefore, ) ) ̃ ̃ ( ̃) 3.13.Proposition Let ̃ be a SEDS. If ) ̃ ̃ O( ̃) and ) ̃ ̃ O( ̃) . Then ) ) ̃ ̃ ( ̃). Proof: by Proposition 2.20 and Proposition 3.19. 3.14. Proposition Let ( ̃ ̃ ) be a STS and ) ) ̃ ̃ ̃) . If ) ̃ ̃ ( ̃ ) and ) ̃ ̃ ( ̃), then ) ) ̃ ̃ ̃) Proof: Let ) ̃ ̃ ̃) and ) ̃ ̃ ( ̃), then ) ̃ ̃ ( ̃) Then by Proposition 2.19 ) ) ̃ ̃ ̃). After that, let's ̃ ) ), therefore ̃ ) and ̃ ) so ) ̃ ̃ ( ̃) such that ̃ ) ̃ ). Since ) ̃ ̃ ( ̃) then ) ̃ ̃ ( ̃) implying that (K, )∩(G, ) ̃ ̃ ( ̃). Therefore, ̃ ) ) ̃ ) ) Thus, ) ) is ̃ ) Corollary Suppose ̃ be a STS, let ) ) ̃ ̃. If ) ̃ ̃ ̃) , ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃), then ) ) ̃ ̃ ̃ ) 3.15. Proposition Let ( ̃ ̃ )be a soft subspace of a soft space ( ̃ ̃ ) ̃ ̃ ̃ . If ) ̃ ̃ ̃), then ) ̃ ̃ ̃ ̃ ̃). IHJPAS. 2025,38(4) 370 Proof: Since ̃ ̃ ̃ and by Proposition 2.22 then ̃ ̃ ) ̃ ) for all ) ̃ ̃. Hence, we obtain ̃ ̃ ̃ ̃ ) ̃ ̃)) ̃ ̃ ) ̃ ̃) ̃ ̃ ̃ ̃ ̃ ) ̃ ̃ ̃)) ̃ ̃ ̃ ̃ ( ̃ )) ̃ ̃ ( ̃ ( ̃)) ̃ ̃ ̃ ̃ ( ̃ )) ̃ ̃ ) ̃ ̃ ( ̃ ) ̃ ( ̃ )) ̃ ) ̃ ̃ ( ̃ ̃ ( ) ̃ ̃)) ̃ ) ̃ ̃ ) ̃ ̃ ̃ ̃ ̃ ). 3.16.Proposition Let ( ̃ ̃ ) be a soft subspace of a soft space ( ̃ ̃ ) ̃ ̃ ̃ ) . If ) ̃ ̃ ̃ ) ,then ) ̃ ̃ ̃ ̃ ̃ ). Proof: since ̃ ̃ ̃ ) ̃ ̃ ̃) by Proposition 2.19 ) ̃ ̃ ̃ ̃ ̃) Since ) ̃ ̃ ̃ ̃and ̃ ̃ ̃ then ̃ ̃ ̃ ̃) then by Proposition 2.20 ) ̃ ̃ ̃ ̃ ̃ ).But ) ̃ ̃ ̃ ) then ̃ ) (G, ) ̃ ̃ ( ̃) s.t. ̃ ) ̃ ). Hence ̃ ) ̃ ̃ ̃ ) ̃ ̃ ,since ̃ ̃ ̃ and (G, ) ̃ ̃ ̃ ) ,then by Proposition 3.23 (G, ) ̃ ̃ ̃ ̃ ̃ ) therefor ) ̃ ̃ ̃ ̃ ̃ ). Shown by the example following, the converse of Proposition 3.24 is incorrect, if ̃ ̃ ̃. Example Examine ,and }. We consider ̃ ) ̃ ̃ where ) { ) { ) { ) { ) { ) { Then ̃ ) ̃ and ̃ ̃ ̃ ̃ ) where ) { ) { ) { ) { ) { ) { So ) ̃ ̃ ̃) but ) ̃ ̃ ̃) 3.17. Proposition Let ̃ ̃ ) be a subspace of a space ̃ ̃ ) and ) ̃ ̃. If ) ̃ ̃ ̃) and ̃ ̃ ̃ ̃), then ) ̃ ̃ ̃) . Proof: Let ( ) ̃ ̃ ̃), then ) ̃ ̃ ̃) and for ̃ ), there is ) ̃ ̃ ̃) s.t. ̃ ) ̃ ) Since ̃ ̃ ̃, and ) ̃ ̃ ) ̃ ̃ ̃ ̃ ̃ ̃ ̃ ))) ̃ ̃ ̃ ))), ) ̃ ̃ ̃).since ) ̃ ̃ ( ̃) ) ̃ ̃ ( ̃) s.t. ) ) ̃ ̃ but ̃ ̃ ̃ then it is soft semi-closed therefore ) ̃ ̃ ̃) The following corollary is derived directly from Proposition 3.26. and Proposition 3.24. Corollary Let ̃ ̃ ) be a STS, and ) ̃ be soft subsets of ̃ s.t. ) ̃ ̃ ̃ ̃ and ̃ ̃ ̃ ( ̃). Therefore ) ̃ ̃ ̃ ) iff ) ̃ ̃ ̃ ). Corollary Let ) ̃ ̃ ( ̃) ) ̃ ̃and ̃ ̃ ̃ ( ̃), then ) ̃ ̃ ̃ ̃) IHJPAS. 2025,38(4) 371 Proof: Let ) ̃ ̃ ̃), then ) ̃ ̃ ̃). Since ̃ ̃ ̃, by Proposition 2.19, ) ̃ ̃ ̃ ̃). Since ̃ ̃ ̃ ( ̃) ) ) ̃ ̃ ̃ ̃). And ̃ ), there is ) ̃ ̃ ( ̃) ̃ ) ̃ ) Hence, ̃ ) ̃ ̃ ) ̃ by Proposition 3.23. ) ̃ ̃ ̃ ( ̃) and therefore, ) ̃ ̃ ̃ ̃ ). 4. ̃ -Continuity In this section we define the concept of soft ̃ -continuous function by using soft -open sets. We then explain various properties of this concept as well as compare it to several other forms of soft continuous functions. 4.1.Definition 4.1. A soft function ( ̃ ̃ ) ̃ ̃ ) is called ̃ -continuous at a soft point ̃ ̃ ̃) , if ) ̃ ̃ containing ), ) ̃ ̃ O( ̃) containing s.t. ) ̃ ) And it is called ̃ continuous function, if is ̃ continuous at every soft point of ̃. 4.2. Proposition 4.2. A soft function ( ̃ ̃ ) ̃ ̃ ) is ̃ -continuous iff the inverse image ) ̃ ̃ O( ̃) , ) ̃ ̃ Proof: Suppose is ̃ -continuous and let ) ̃ ̃. We show that )) ̃ ̃ O( ̃) If )) ̃, then )) ̃ ̃ O( ̃) If )) ̃, then ̃ )) which implies that ) ̃ ). Since is ̃ continuous, ̃ ) ̃ ̃ O( ̃) s.t. )) ̃ ). This implies that ̃ ) ̃ )). This shows that )) ̃ ̃ O( ̃) Conversely, let ̃ ̃ ̃) ) ̃ ) ̃ ̃, Then ̃ )) ̃ ̃ ( ̃) ) )) s.t. )) ))) ̃ ). Hence, is ̃ -continuous. 4.3. Proposition Let ( ̃ ̃ ) ̃ ̃ ) 1. is ̃ continuous, if is a ̃ -continuous. 2. is ̃ -continuous, if is a ̃ (resp. ̃ -continuous, ̃ ̃ ̃ C ̃ ̃ ̃ and soft perfectly continuous) Proof: By ̃ ( ̃) ̃ ̃ ( ̃) and Proposition 4.2. we get prove (1). And by Proposition 3.6. and Proposition 4.2. Prove (2) will follow. The following examples illustrate the previous remark. Example Consider the soft topology ̃ { ̃ ̃ ( ) ( )} in Example 3.5. Then the identity function ( ̃ ̃ ) ̃ ̃ ) is ̃ -continuous but not ̃ -continuous since ( ) ̃ ̃ , but ( ) ( ) ̃ ̃ O( ̃). 1. Consider the soft topology ̃ { ̃ ̃ ( ) ( ) )} in Example 3.5. Then the identity function ( ̃ ̃ ) ̃ ̃ ) is ̃ -continuous but is not ̃ , ̃ - continuous, ̃ ̃ ̃ C ̃ and soft perfectly continuous. Since ( ) ̃ ̃,then ( ) ̃ ̃ O( ̃)but ( ) is not ̃ , ̃ -open, ̃ ̃ ̃ ̃ C ̃ and soft clopen. IHJPAS. 2025,38(4) 372 2. Consider the soft topology ̃ { ̃ ̃ ( ) ( )} ̃ { ̃ ̃ ( ) ( ) )} in Example 3.5. then ( ̃ ̃ ) ( ̃ ̃ ) is ̃ -continuous but is not ̃ ̃ since ( ) ̃ ̃, but ( ) ( ) is not ̃ and ̃ open. 4.4.Proposition A soft function ( ̃ ̃ ) ̃ ̃ ) is ̃ - continuous iff is a soft -continuous and ̃ ̃ and ) ̃ ̃ containing ), ) ̃ ̃ ̃) s.t )) ̃ ). Proof: let ( ̃ ̃ ) ̃ ̃ ) be a ̃ continuous function and also let ̃ ̃ ̃) and ) ̃ ̃ containing ). By assumption, ̃ ) ̃ ̃ ̃) s.t. )) ̃ ). Since ) ̃ ̃ ̃) ) ̃ ̃ ̃) s.t ̃ ) ̃ ). This implies that ) ̃ ). Therefore, is ̃ - continuous. Then is soft - continuous. Conversely, suppose ) ̃ ̃. We have to show that )) ̃ ̃ ̃). Since is soft -continuous, then )) ̃ ̃ ̃) . Let ̃ )). Then ) ̃ ). By hypothesis ̃ ) ̃ ̃ ̃) s.t. )) ̃ ), which implies that ̃ ) ̃ )), therefore )) ̃ ̃ ̃) .Hence by Proposition 4.2 is ̃ -continuous. 4.5. Proposition If ̃ is soft -space in ( ̃ ̃ ) ̃ ̃ ) The following properties are equivalent. 1. is ̃ -continuous. 2. is ̃ -continuous. 3. is ̃ -continuous. Proof: by Proposition 3.8 4.6.Proposition If a soft function ( ̃ ̃ ) ̃ ̃ ), is soft continuous, and ̃ is soft - space, then is ̃ -continuous. 4.7.Proposition If ( ̃ ̃ ) ̃ ̃ ), is soft continuous, and ̃ is soft regular space, then is ̃ -continuous. Proof: By Proposition 4.2 and Proposition 3.18 we get the prove. 4.8. Proposition If ( ̃ ̃ ) ̃ ̃ ), is soft continuous, and ̃ is SEDS, then is ̃ -continuous (resp., ̃ continuous). Proof: By Proposition 4.2 and Proposition 3.13(resp., Proposition 3.14) we get the prove. 4.9.Proposition If ̃ is soft locally indiscrete space in ( ̃ ̃ ) ̃ ̃ ) the following properties are equivalent. 1. is ̃ -continuous. 2. is ̃ -continuous. 3. is ̃ -continuous. Proof: by Proposition 3.15. Corollary Let ( ̃ ̃ ) ̃ ̃ ) be a soft function s.t. ( ̃ ̃ ) is a soft locally indiscrete space. Then (1) is a soft -continuous function, iff is a soft continuous function. (2) is a soft -continuous function, iff is a soft -continuous function. Proof: (1) and (2) direct by using corollary 3.16 and by Proposition 4.2. IHJPAS. 2025,38(4) 373 5. Properties and Comparisons 5.1.Proposition Let ( ̃ ̃ ) ̃ ̃ ) be ̃ continuous function. If ̃ ̃ ̃ (resp., soft clopen), then ̃ ̃) ̃ ̃ ) ̃ ̃ ) is ̃ -continuous function in the subspace ̃. Proof: Let ) ̃ ̃. Since is ̃ -continuous function, then by Proposition 4.2 ) is ̃ O( ̃). Since ̃ ̃ ̃, then by Proposition 3.24 | ̃ ( ̃) ) ) )) ̃ ̃ ̃ O( ̃). This shows that ̃ ̃) ̃ ̃ ) ̃ ̃ ) is ̃ - continuous function. 5.2.Proposition Let ( ̃ ̃ ) ̃ ̃ ) be ̃ -continuous function. If ̃ ̃ ̃ O( ̃)then ̃ ( ̃) ) ̃ ̃ ) ̃ ̃ ) is ̃ -continuous function in the subspace ̃. Proof: Since every soft regular open is soft open, this is a direct result of Proposition 4.2 and Proposition 5.1. 5.3.Proposition Let ( ̃ ̃ ) ̃ ̃ ) and ̃ ̃ ) ( ̃ ̃ ) be two soft functions, then the following properties hold. (1) If is a ̃ continuous and is a soft continuous, then is a ̃ continuous. (2) If is a ̃ irresolute and is a ̃ continuous, then is a ̃ continuous. Proof: (1) Let ) ̃ ̃. Then ) ̃ ̃ by continuity of . Since is a ̃ - continuous, ( )) ̃ ̃ ̃) and hence, ( ) ) ( )) ̃ ̃ ̃). Therefore, is a ̃ -continuous. (2) Let ) ̃ ̃. Since is a ̃ -continuous, ) ̃ ̃ ̃) . Since is a ̃ - irresolute, ( )) ̃ ̃ ̃) and hence, ( ) ) ( )) ̃ ̃ ̃). Therefore, is a ̃ -continuous. 5.4.Proposition Let ( ̃ ̃ ) ̃ ̃ ) and ̃ ̃ ) ( ̃ ̃ ) be two soft functions. If is a ̃ -open and surjective and is a ̃ -continuous, then is a ̃ -continuous. Proof: Let ) ̃ ̃. Since is a ̃ -continuous, ( ) ) ( )) ̃ ̃ ̃). Since is a ̃ -open and surjective, then ( ( ))) ) ̃ ̃ ̃) . Hence, is a ̃ -continuous. 5.5.Proposition A function ( ̃ ̃ ) ̃ ̃ ) is ̃ -continuous. If ̃ ̃ ̃) , a soft clopen set ) of ̃ containing s.t. ) ) ̃ is ̃ -continuous. Proof: Let ̃ ̃ ̃) , then by hypothesis, ) ̃ ̃ ̃) containing s.t. ) ) ̃ is ̃ -continuous. Let ) ̃ ̃) containing ), a ̃ - open subset ) of ) containing s.t. ( )) ) ̃ ). Since ) ̃ ̃ ̃). By Proposition 3.26 ) ̃ ̃ ̃) and hence )) ̃ ). This implies that is ̃ -continuous. 5.6.Proposition If ̃ ) ) , where ) ) ̃ ̃ ̃) and ( ̃ ̃ ) ̃ ̃ ) is a function s.t. both ) and ) are both ̃ -continuous, then is ̃ -continuous. IHJPAS. 2025,38(4) 374 Proof: Let ) ̃ ̃.Then )) )) )) )) )). Since ) and ) are ̃ -continuous. Then by Proposition 4.2 )) )) and )) )) are ̃ -open sets in ) and ) respectively. Since ) ) ̃ ̃ ̃), then by Corollary 3.26 )) )), )) )) ̃ ̃ ̃). Since the union of two ̃ -open sets are ̃ - open. Hence )) ̃ ̃ ̃). Therefore, by Proposition 4.2 is ̃ - continuous. In general, if ̃ ) ̃ , where each ) is a soft clopen set and ( ̃ ̃ ) ̃ ̃ ) is a function s.t. ) is ̃ -continuous for each , then is ̃ -continuous. 5.7.Proposition Let ̃ ) ) , where ) ) ̃ ̃ ( ̃). Let ) ̃ ̃ ) and ) ̃ ̃ ) be a ̃ -continuous. If ) ), ̃ ) ). Then the function ) ) ̃ s.t. ) { ) ̃ ) ) ̃ ) Is ̃ -continuous. Proof: Let ) be a soft open set of ̃. Now ( )) )) )). Since is ̃ -continuous, then by Proposition 4.2, ) is ̃ -open set in ). But ) is soft clopen set in ̃. Then by Corollary 3.26, ) is ̃ -open set in ̃. Similarly, ) is ̃ -open set in ), and hence ̃ -open set in ̃. Since the union of two ̃ -open set is ̃ -open. Therefore, ( )) ) ) is ̃ - open set in ̃. Hence by Proposition 4.2 is ̃ -continuous. 5.8. Proposition Let ( ̃ ̃ ) ̃ ̃ ) be ̃ -continuous. If ̃ is a soft clopen subset of a STS ̃, then ( ̃ ̃ ) ( ̃ ̃ ) is ̃ -continuous. Proof: Suppose ̃ ̃ ̃) and ) be any soft open set of ̃ containing ), then ) ̃ is a soft open set in ̃. But ) ̃ ̃ ̃ ) ̃ ̃ ̃) , then ) ̃ ) ̃. Since ( ̃ ̃ ) ̃ ̃ ) is ̃ -continuous, then a ̃ -open set ) containing s.t. )) ̃ ( ) ̃) ̃ ). Therefore ( ̃ ̃ ) ( ̃ ̃ ) is ̃ -continuous. 6. Conclusion Through the current research work, we have continued to investigate the properties of soft beta open sets in soft topological spaces. ̃ -open sets is defined as a new kind of soft sets which is a stronger form of soft -open sets and weaker than each of ̃ -open, soft clopen sets and some other. We presented and examined ̃ -continuous functions in a STS. We also identified some characterizations and basic properties of these soft functions as well as their relationships to certain other categories of soft functions and documented with some illustrative examples. Acknowledgments No thanks. Conflict of Interest The publication fee for this article was funded by the University of Salahaddin - Erbil. IHJPAS. 2025,38(4) 375 Funding No funding. References 1. Molodtsov D. Soft Set Theory—First Results. Comput Math Appl. 1999;37:19–31. https.//doi.org/10.1016/S0898-1221(99)00056-5 2. Shabir M, Naz M. On soft topological spaces. Comput Math Appl. 2011;61(7):1786–1799. https.//doi.org/10.1016/j.camwa.2011.02.006 3. Al-shami TM, Mhemdi A, Abu-Gdairi R. A novel framework for generalizations of soft open sets and its applications via soft topologies. Mathematics. 2023;11: 840. https.//doi.org/10.3390/math11040840 4. Noori S, Yousif YY. Soft Simply Compact Spaces. Iraqi Journal of Science. 2020;SI(1):14. https.//doi.org/10.24996/ijs.2020.SI.1.14 5. Yousif YY, Hussain LA, Hussain MA. Fibrewise Soft Bitopological Spaces. J Interdiscip Math. 2021;24:1925–1934. https.//doi.org/10.1080/09720502.2021.1966948 6. Al Ghour S. On some weaker forms of soft continuity and their decomposition theorems. J Math Comput Sci. 2023;29:317–328. https.//doi.org/10.22436/jmcs.029.04.02 7. Kharal A, Ahmad B. Mappings on Soft Classes. New Math Nat Comput. 2011;7(3):471–481. https.//doi.org/10.1142/S1793005711002025 8. Chen B. Soft Semi-open sets and related properties in soft topological spaces. Applied Mathematics & Information Sciences. 2013;7(1):287–294. 9. Mahanta J, Das PK. On soft topological space via semiopen and semiclosed soft sets. Kyungpook Math J. 2014;54(2):221–236. https.//doi.org/10.5666/KMJ.2014.54.2.221 10. Yumak Y, Kaymakci AK. Soft Beta-open sets and their applications. J New Theory. 2015;4:80–89. URL. https.//arxiv.org/pdf/1312.6964 11. Hammood A A, Esmaeel RB . Some Games with Soft -ᶅ -Pre-Generalized Open Sets. Ibn AL- Haitham Journal for Pure and Applied Sciences. 2021;34(4):45–57. https.//doi.org/10.30526/34.4.2702 12. Mohammad R, Esmaeel R. New Games via soft-I-Semi-g-Separation axioms. Ibn AL-Haitham Journal for Pure and Applied Sciences. 2020;33(4):122–136. https.//doi.org/10.30526/33.4.2517 13. Mussa YS, Khalaf BA. SS-c-Open sets in Soft Topological Spaces. J Garmian Univ. 2015;1:1–22. 14. Mohammed RA, Ameen RS. Soft ξ-Open Sets in Soft Topological Spaces. Sci J Univ. 2019;7(3):108–119. https.//doi.org/10.25271/sjuoz.2019.7.3.598 15. Ahmed NK, amko . s pc-open sets and s pc-continuity in Soft Topological Spaces. ZANCO J Pure Appl Sci. 2018;30(6):72–84. 16. Fayad E, Mahdi . Soft βc-open sets and soft βc-continuity. Int Math Forum 2017;12(1):9–26. http.//dx.doi.org/10.12988/imf.2017.611150 17. Hameed SZ, Hussein AK. On Soft bc-Open Sets in Soft Topological Spaces. Iraqi J of Sci. 2020;1:238–242. http.//dx.doi.org/10.24996/ijs.2020.SI.1.32 18. Payman MM, Hardi AS, Halgurd MD. Soft Sp-Continuous Functions. Iraqi J Sci. 2024;65(8):4441–4459. https.//doi.org/10.24996/ijs.2024.65.8.26 19. Asaad AB. Results on Soft Extremally Disconnectedness of Soft Topological Spaces. J Math Comput Sci. 2017;17(4):448–464. http.//dx.doi.org/10.22436/jmcs.017.04.02 20. Aydın T, Enginoğlu S. Some results on soft topological notions. J New Results Sci. 2021;10(1):65– 75. URL. https.//dergipark.org.tr/tr/pub/jnrs/issue/62194/910337 https://doi.org/10.1016/S0898-1221(99)00056-5 https://doi.org/10.1016/j.camwa.2011.02.006 https://doi.org/10.3390/math11040840 https://doi.org/10.24996/ijs.2020.SI.1.14 https://doi.org/10.1080/09720502.2021.1966948 https://doi.org/10.22436/jmcs.029.04.02 https://doi.org/10.1142/S1793005711002025 https://doi.org/10.5666/KMJ.2014.54.2.221 https://arxiv.org/pdf/1312.6964 https://doi.org/10.30526/34.4.2702 https://doi.org/10.30526/33.4.2517 https://doi.org/10.25271/sjuoz.2019.7.3.598 http://dx.doi.org/10.12988/imf.2017.611150 http://dx.doi.org/10.24996/ijs.2020.SI.1.32 https://doi.org/10.24996/ijs.2024.65.8.26 http://dx.doi.org/10.22436/jmcs.017.04.02 https://dergipark.org.tr/tr/pub/jnrs/issue/62194/910337