IHJPAS. 36(1)2023 333 This work is licensed under a Creative Commons Attribution 4.0 International License Ϣ −Semi-p Open Set Abstract Csaszar introduced the concept of generalized topological space and a new open set in a generalized topological space called Ϣ-preopen in 2002 and 2005, respectively. Definitions of Ϣ-preinterior and Ϣ-preclosuer were given. Successively, several studies have appeared to give many generalizations for an open set. The object of our paper is to give a new type of generalization of an open set in a generalized topological space called Ϣ-semi-p-open set. We present the definition of this set with its equivalent. We give definition of Ϣ-semi-p-interior and Ϣ-semi-p-closure of a set and discuss their properties. Also the properties of Ϣ-preinterior and Ϣ-preclosuer are discussed. In addition, we give a new type of continuous function in a generalized topological space as (Ϣ1, Ϣ2)-semi-p-continuous function and (Ϣ1, Ϣ2)-semi-p- irresolute function. The relationship between them is showed. We prove that every Ϣ-open (Ϣ- preopen) set is an Ϣ-semi-p-open set, but not conversely. Every (Ϣ1, Ϣ2)-semi-p-irresolute function is an (Ϣ1, Ϣ2)-semi-p-continuous function, but not conversely. Also we show that the union of any family of Ϣ-semi-p-open sets is an Ϣ-semi-p-open set, but the intersection of two Ϣ-semi-p-open sets need not to be an Ϣ-semi-p-open set. Keywords:Ϣ-semi-p-open , Ϣ-semi-p-interior , Ϣ-semi-p-closure, (Ϣ1, Ϣ2)-semi-p-irresolute and (Ϣ1, Ϣ2)-semi-p-continuous. 1.Introduction and Preliminaries In this paper, we denote a topological space by (Z, Ӽ) and the closure (interior) of a subset Ħ of Z by cl(Ħ)(int(Ħ)), respectively. 1. The interior of Ħ is the set int(Ħ) = ⋃{Ɯ: Ɯ ∈ Ӽ 𝑎𝑛𝑑 Ɯ ⊆ Ħ }. doi.org/10.30526/36.1.2969 Article history: Received 3 Augest 2022, Accepted 24 Augest 2022, Published in January 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Muna L. Abd Ul Ridha Department of Mathematics , College of Education for Pure Sciences,Ibn Al –Haitham, University of Baghdad, Baghdad ,Iraq. mona.laith1203a@ihcoedu.uobaghdad.edu.iq Suaad G. Gasim Department of Mathematics , College of Education for Pure Sciences,Ibn Al –Haitham, University of Baghdad, Baghdad ,Iraq. suaad.gedaan@yahoo.com https://creativecommons.org/licenses/by/4.0/ mailto:mona.laith1203a@ihcoedu.uobaghdad.edu.iq mailto:suaad.gedaan@yahoo.com IHJPAS. 36(1)2023 334 2. The closure of Ħ is the set cl(Ħ) = ⋂{Ƒ: Ƒ ∈ Ӽ′ 𝑎𝑛𝑑 Ħ ⊆ Ƒ } [1], where Ӽ′ symbolizes the family of closed subsets of Z. The term "preopen” was introduced for the first time in 1984 [2]. A subset A of a topological space (Z, Ӽ) is called a preopen set if A  Int(clA). The complement of a preopen set is called a preclosed set. The family of all preopen sets of Z is denoted by PO(Z). The family of all preclosed sets of Z is denoted by PC(Z). In 2000, Navalagi used "preopen" term to define a "Semi-p-open set" [3]. A subset A of a topological space (Z, Ӽ) is said to be semi-p-open set if there exists a preopen set U in Z such that U ⊆ A ⊆ pre-cl U. The family of all semi-p-open sets of Z is denoted by S-PO(Z). The complement of a semi-p-open set is called semi-p-closed set. The family of all semi-p-closed sets of Z is denoted by S-PC(Z). A function f: (Z1, Ӽ1) → (Z2, Ӽ2) is said to be a continuous function if the inverse image of any open set in Z2 is an open set in Z1 [4]. Navalagi used the term "preopen" to introduce new types of a continuous function "pre-irresolute function" and "pre-continuous function". A function f: (Z1, Ӽ1) → (Z2, Ӽ2) is called pre-irresolute(pre-continuous) function if the inverse image of any pre-open set in Z2 is a pre-open set inZ1 (the inverse image of any open set in Z2 is a pre-open set Z1). In [5], Al-Khazraji used the term of "Semi-p-open set" to define new types of continuous functions "semi-p-irresolute" and "semi-p-continuous" function. A function f: (Z1, Ӽ1) → (Z2, Ӽ2) is called a semi-p-irresolute (semi-p-continuous) function if the inverse image of any semi-p-open set in Z2 is a semi-p-open set in Z1(the inverse image of any open set in Z2 is a semi-p- open set in Z1). Let Z be a nonempty set, a collection Ϣ of subsets of Z is called a generalized topology (in brief, 𝐺𝑇) on Z if ∅ belongs to Ϣ and the arbitrary unions of elements of Ϣ is an element in Ϣ, (Z, Ϣ) is called generalized topological space (in brief, 𝐺𝑇𝑆) [6]. Every set in Ϣ is called Ϣ-open, while the complement of Ϣ-open is called Ϣ-closed; the family of all Ϣ-closed sets is denoted by Ϣ′. The union of all Ϣ-open set contained in a set Ħ is called the Ϣ- interior of Ħ and is denoted by int Ϣ(Ħ), whereas the intersection of all Ϣ-closed set containing Ħ is called the Ϣ-closure of Ħ and is denoted by cl Ϣ(Ħ)[7]. 2. Ϣ-Pre-Open Set Definition 2.1 [8] In a 𝐺𝑇𝑆 (Z, Ϣ) by an Ϣ-pre-open (in brief, Ϣ − p − o) set, we mean a subset Ħ of Z with Ħ ⊆ intϢclϢ Ħ. An Ϣ-pre-closed (in brief, Ϣ − p − c) set is the complement of an Ϣ-pre-open set. The collection of all Ϣ − p − o (Ϣ − p − c) subsets of Z will be denoted by Ϣ-PO(Z) ( Ϣ-PC(Z), respectively). Proposition 2.2 For a subset Ħ of a (Z, Ϣ) , we have ⋃ intϢα∈ᴧ clϢ Ħα ⊆ intϢclϢ ⋃ Ħαα∈ᴧ . Proof: Ħα ⊆ ⋃ Ħαα∈Λ , for every α ∈ Λ, so clϢĦα ⊆ clϢ ⋃ Ħαα∈Λ for every α ∈ Λ, it follows that intϢclϢĦα ⊆ intϢclϢ ⋃ Ħαα∈Λ ∀𝛼 ∈ Λ. Hence ⋃ intϢα∈ᴧ clϢ Ħα ⊆ intϢclϢ ⋃ Ħαα∈ᴧ . IHJPAS. 36(1)2023 335 Proposition 2.3 The union of any collection of Ϣ − p − o sets is an Ϣ − p − o set. Proof: Let { Ħα: α ∈ ⋀ } be a family of Ϣ − p − o sets, so Ħα ⊆ intϢ clϢ Ħα, ∀𝛼 ∈ Λ. Which means ⋃ Ħαα ∈ ᴧ ⊆ ⋃ intϢα∈ᴧ clϢ Ħα , but ⋃ intϢα∈ᴧ clϢ Ħα ⊆ intϢclϢ ⋃ Ħαα∈ᴧ ( by Proposition 2.2), therefore, we obtain ⋃ Ħα α∈ᴧ ⊆ intϢ clϢ ⋃ Ħαα∈ᴧ , hence ⋃ Ħαα∈ᴧ is an Ϣ − p − o set . Corollary 2.4 The intersection of any collection of Ϣ − p − c sets is an Ϣ − p − c set. Definition 2.5: [6] Let (Z, Ϣ) be a 𝐺𝑇𝑆, and Ħ be a subset of Z 1. The union of all Ϣ − p − o sets contained in Ħ is called the Ϣ-preinterior of Ħ and denoted by pre-intϢĦ. 2. The intersection of all Ϣ − p − c sets containing Ħ is called the Ϣ-preclosuer of Ħ and denoted by pre-clϢĦ. Theorem 2.6 Let Ԋ and Ԏ be subsets of (Z, Ϣ). Then, the following properties are true: 1. Ԋ ⊆ pre-clϢԊ . 2. pre-intϢԊ ⊆ Ԋ . 3. If Ԋ ⊆ Ԏ, then pre-intϢԊ ⊆ pre-intϢԎ . 4. If Ԋ ⊆ Ԏ , then pre-clϢԊ ⊆ pre-clϢԎ . Proof: 1. From Definition of pre-clϢԊ . 2. From Definition of pre-intϢԊ . 3. Let Ԋ ⊆ Ԏ , we have from 2, pre-intϢԊ ⊆ Ԋ , so pre − intϢԊ ⊆ Ԏ , but pre − intϢԎ is the largest Ϣ − p − o set contained in Ԏ . So pre − intϢԊ ⊆ pre − intϢԎ. 4. Let Ԋ ⊆ Ԏ , we have from 1, Ԏ ⊆ pre-clϢԎ , so Ԋ ⊆ pre-clϢԎ , but pre-clϢԊ is the smallest Ϣ − p − c set containing Ԋ . So pre-clϢԊ ⊆ pre-clϢԎ . Proposition 2.7 Let (Z, Ϣ) be a 𝐺𝑇𝑆 let Ӈ be a subset of Z. Then: 1. Ħ is an Ϣ − p − c set, if and only if Ħ = pre-clϢĦ. 2. Ħ is an Ϣ − p − oset, if and only if Ħ = pre-intϢĦ. Proposition 2.8 ⋃ pre −α∈Λ clϢĦα ⊆ pre − clϢ ⋃ Ħαα∈Λ Proof: IHJPAS. 36(1)2023 336 Ħα ⊆ ⋃ Ħαα∈Λ , for every α ∈ Λ, so pre-clϢĦα ⊆ pre-clϢ ⋃ Ħαα∈Λ for every α ∈ Λ , therefore, ⋃ pre −α∈Λ clϢĦα ⊆ pre − clϢ ⋃ Ħαα∈Λ . Remark 2.9 The reverse of Proposition 2.8 is not correct in general, as we show in the following example: For example Z = {a, b, c}, Ϣ = {Z , Ø, {a, b}}, and Ϣ′ = {Z , Ø, {c}}, then: Ϣ-PO(Z) = {Z , Ø, {a}, {b}, {a, b}, {a, c}, {b, c}}. Ϣ-PC(Z) = {Ø , Z, {b, c}, {a, c}, {c}, {b}, {a}}, let Ԋ = {b} and Ԏ = {a}, so pre-clϢ Ԋ = {b} and pre-clϢ Ԏ = {a}, note that Ԋ ∪ Ԏ= {a, b}, and pre-clϢ ( Ԋ ∪ Ԏ) = Z, while pre-clϢԊ ⋃pre-clϢԎ = {a, b}. Hence, pre-clϢ( Ԋ ∪ Ԏ) ⊈ pre-clϢԊ ⋃ pre-clϢԎ. Proposition: 2.10 If Ӈ is any subset of a topological space (Z, Ӽ) , then: 1. [pre − int (Ħ)]c = pre − cl (Ħc). 2. pre − int(Ħc) = [pre − cl (Ħ)]c . 3. Ϣ-Semi-P-Open Set Definition 3.1 A subset G of a 𝐺𝑇𝑆 (Z, Ϣ) is said to be Ϣ-semi-p-open(in brief, Ϣ − sp − o) set if there exists an Ϣ − p − o set Ħ in Z such that Ħ ⊆ G ⊆ pre-clϢĦ. Any subset of Z is called Ϣ-semi-p- closed(in brief, Ϣ − sp − c) set if its complement is Ϣ-semi-p-open set .The collection of all Ϣ − sp − o subsets of Z will be denoted by Ϣ-SPO(Z). The collection of all Ϣ − sp − c subsets of Z will be denoted by Ϣ-SPC(Z). Theorem 3.2 Let (Z, Ϣ) be a 𝐺𝑇𝑆 and G ⊆ Z. Then G is an Ϣ − sp − oset  G ⊆ pre-clϢpre-intϢG . Proof: The "if" part Assume that G is an Ϣ − sp − oset, then there exists a Ϣ − p − o subset Ħ of Z such that Ħ ⊆ G ⊆ pre-clϢĦ, it follows by Theorem 2.6 (4) that pre-intϢĦ ⊆pre-intϢG, but pre-intϢĦ = Ħ, therefore Ħ ⊆ pre-intϢG. It follows by Theorem 2.6 (3) that pre-clϢĦ ⊆ pre-clϢpre- intϢG. Now, we get G ⊆pre-clϢĦ ⊆ pre-clϢpre-intϢG. Thus G ⊆ pre-clϢpre-intϢG. The "only if" part Assume that G ⊆ pre-clϢpre-intϢG, we have to show that G is a Ϣ − sp − oset. Take pre- intϢG = Ħ , then Ħ is a Ϣ − p − o set and Ħ ⊆ G ⊆ pre-clϢĦ. Hence G is an Ϣ − sp − oset. IHJPAS. 36(1)2023 337 Corollary 3.3 Let (Z, Ϣ) be a 𝐺𝑇𝑆 and F ⊆ Z. Then Ӈ is Ϣ − sp − cif and only if pre- intϢ(pre-clϢĦ) ⊆ Ħ. Proof: The "if" part Let F be an Ϣ − sp − c subset of Z, then pre-clϢ Ħ = Ħ (by Proposition 2.9(1)) which implies pre-intϢ(pre-clϢĦ) ⊆ Ħ, since pre-intϢӇ ⊆ Ӈ (by Theorem 2.3(2). The "only if" part Assume that pre- intϢpre-clϢĦ ⊆ Ħ. We have to show Ӈ is an Ϣ − sp − cset. Since pre-intϢpre- clϢĦ ⊆ Ħ, then Ħ c ⊆ [pre- intϢ(pre-clϢ Ħ)] c, so we obtain from Proposition 2.10 Ħc ⊆ pre- clϢ(pre-clϢĦ) c and Ħc ⊆ pre- clϢpre-intϢĦ c. Hence Ħc is an Ϣ − sp − o set by Theorem (2.2.2) which means Ħ is an Ϣ − sp − c. Proposition 3.4 The union of any collection of Ϣ − sp − o sets is an Ϣ − sp − o set. Proof: Let {Gα, α ∈ Λ } be any family of Ϣ − sp − o sets. Then there exists an Ϣ − p − o set Ħα for each Gα , α ∈ Λ such that Ħα ⊆ Gα ⊆ pre-clϢ Ħα, so ⋃ Ħα α∈Λ ⊆ ⋃ Gαα∈Λ ⊆ ⋃ pre −α∈Λ clϢ Ħα, but ⋃ Ħαα∈Λ is an Ϣ − p − o set by Theorem 2.3, and ⋃ pre −α∈Λ clϢĦα ⊆ pre- clϢ ⋃ Ħα α∈Λ by Proposition 2.8. Now we get ⋃ Ħα α∈Λ ⊆ ⋃ Gαα∈Λ ⊆ pre-clϢ ⋃ Ħα α∈Λ . Hence ⋃ Gαα∈Λ is an Ϣ − sp − o set. Corollary 3.5 The intersection of any collection of Ϣ − sp − c sets is an Ϣ − sp − c set. Proof: Let { Fα: α ∈ ⋀ } be any family of Ϣ − sp − c subsets of Z. we have to show that ⋂ Fαα∈ᴧ is an Ϣ − sp − c set, we know that Z − ⋂ Fα = ⋃ (Z − Fα)α∈ᴧα∈ᴧ (De Morgan’s laws). But ⋃ (Z − Fα)α∈ᴧ is an Ϣ − sp − c set, so Z − ⋂ Fαα∈ᴧ is an Ϣ − sp − o set. Hence ⋂ Fαα∈ᴧ is an Ϣ − sp − c . Remark 3.6 The intersection of two Ϣ − sp − o sets need not to be an Ϣ − sp − o set, as we show in the following example: Example Let Z ={a, b, c, d}, Ϣ = {Z, Ø, {a}, {d}, {a, d}}, Ϣ-PO(Z) ={Z, Ø, {a}, {d}, {a, d}, {a, b, d}, {a, c, d}}, and IHJPAS. 36(1)2023 338 ϢSPO(Z)= {Z, Ø, {a}, {d}, {a, b}, {a, c}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}, {a, c, d}}. Let Ԋ = {a, b, c} and Ԏ = {b, d} , Ԋ and Ԏ are Ϣ − sp − o sets, but Ԋ ∩ Ԏ = {b} which is not an Ϣ − sp − o set because there is not Ϣ − p − o set VϢ , therefore V ⊆ {b} ⊆ pre- clϢV. Remark 3.7 If Ԋ and Ԏ are two Ϣ − sp − c sets, then Ԋ ∪ Ԏ need not beϢ − sp − c as we show in the following example: Example From the example of Remark 3.7 Let Ԋ = {d} and Ԏ = {a, c}. Ԋ and Ԏ are Ϣ − sp − c set, but Ԋ ∪ Ԏ ={a, c, d} is not an Ϣ − sp − c set because Z - {a, c, d}={b} is not an Ϣ − sp − o set. The following diagram illustrates the relation among Ϣ-open, Ϣ-pre-open, and Ϣ-semi-p-open set Definition 3.8 1. The union of all Ϣ − sp − o sets contained in Ӈ is called the Ϣ-semi-p-interior of Ħ, denoted by s-p-intϢ(Ħ). 2. The intersection of all Ϣ − sp − c sets containing Ħ is called the Ϣ-semi-p-closure of Ħ, denoted by s-p-clϢ(Ħ). Proposition 3.9 Let Ԋ and Ԏ be two subsets of (Z, Ϣ). Then, the following properties are true: 1. Ԋ ⊆ s − p − clϢ Ԋ. 2. If Ԋ ⊆ Ԏ, then s − p − clϢԊ ⊆ s − p − clϢԎ. 3. s − p − clϢԊ ∪ s − p − clϢԎ ⊆ s − p − clϢ ( Ԋ ∪ Ԏ). 4. s − p − clϢ (Ԋ ∩ Ԏ) ⊆ s − p − clϢ Ԋ ∩ s − p − clϢ Ԏ . Proof: 1. It is clear from Definition 3.14(2). 2. Let Ԋ ⊆ Ԏ , from (1) we have Ԏ ⊆ s − p − clϢ Ԏ, so Ԋ ⊆ s − p − clϢ Ԏ which is Ϣ − sp − c set, but s − p − clϢԊ is the smallest Ϣ − sp − c set containing Ԋ, thus s − p − clϢԊ ⊆ s − p − clϢԎ. Ϣ-open Ϣ-pre-open Ϣ-semi-p-open IHJPAS. 36(1)2023 339 3. Since Ԋ ⊆ Ԋ ∪ Ԏ and Ԏ ⊆ Ԋ ∪ Ԏ, it follows from (1) that 𝑠 − 𝑝 − clϢԊ ⊆ 𝑠 − 𝑝 − clϢ(Ԋ ∪ Ԏ) and 𝑠 − 𝑝 − clϢԎ ⊆ 𝑠 − 𝑝 − clϢ (Ԋ ∪ Ԏ), therefore 𝑠 − 𝑝 − clϢԊ ∪ 𝑠 − 𝑝 − clϢ Ԏ ⊆ 𝑠 − 𝑝 − clϢ (Ԋ ∪ Ԏ). 4. Since ( Ԋ ∩ Ԏ) ⊆ Ӈ and ( Ԋ ∩ Ԏ) ⊆ Ԏ, so semi-p-clϢ( Ԋ ∩ Ԏ) ⊆ semi-clϢԊ and 𝑠 − 𝑝 − clϢ(Ԋ ∩ Ԏ) ⊆ 𝑠 − 𝑝 − clϢԎ, thus 𝑠 − 𝑝 − clϢ( Ԋ ∩ Ԏ) ⊆ 𝑠 − 𝑝 − clϢԊ ∩ 𝑠 − 𝑝 − clϢԎ. Theorem 3.10 Ӈ is Ϣ − sp − c set  Ħ = 𝑠 − 𝑝 − clϢĦ. Proof: Is clear. Corollary 3.11 𝑠 − 𝑝 − clϢZ = Z. Theorem 3.12 Let Ӈ and Ԏ be two subsets of (Z, Ϣ). Then the following properties are true: 1. 𝑠 − 𝑝 − intϢԊ ⊆ Ԋ. 2. If Ԋ ⊆ Ԏ, then 𝑠 − 𝑝 − intϢԊ ⊆ 𝑠 − 𝑝 − intϢ Ԏ. 3. 𝑠 − 𝑝 − intϢ (Ԋ ∩ Ԏ) ⊆ 𝑠 − 𝑝 − intϢ Ԋ ∩ 𝑠 − 𝑝 − intϢ Ԏ 4. 𝑠 − 𝑝 − intϢ Ԋ ∪ 𝑠 − 𝑝 − intϢ Ԏ ⊆ 𝑠 − 𝑝 − intϢ Ԋ ∪ Ԏ). Proof: 1. Clear. 2. Let Ԋ ⊆ Ԏ, from (1) we have 𝑠 − 𝑝 − intϢԊ ⊆ Ԋ, so 𝑠 − 𝑝 − intϢԊ ⊆ Ԏ where 𝑠 − 𝑝 − intϢԊ is Ϣ − sp − o set, but 𝑠 − 𝑝 − intϢԎ is the largest Ϣ − sp − o set contained in Ԏ, hence 𝑠 − 𝑝 − intϢ Ԋ ⊆ 𝑠 − 𝑝 − intϢ Ԏ. 3. Since ( Ԋ ∩ Ԏ) ⊆ Ӈ and ( Ԋ ∩ Ԏ) ⊆ Ԏ, so 𝑠 − 𝑝 − intϢ( Ԋ ∩ Ԏ) ⊆ 𝑠 − 𝑝 − intϢԊ and 𝑠 − 𝑝 − intϢ(Ԋ ∩ Ԏ) ⊆ 𝑠 − 𝑝 − intϢԎ, so 𝑠 − 𝑝 − intϢ( Ԋ ∩ Ԏ) ⊆ 𝑠 − 𝑝 − intϢԊ ∩ 𝑠 − 𝑝 − intϢԎ. 4. Since Ԋ ⊆ Ԋ ∪ Ԏ and Ԏ ⊆ Ԋ ∪ Ԏ, then 𝑠 − 𝑝 − intϢԊ ⊆ 𝑠 − 𝑝 − intϢ (Ԋ ∪ Ԏ) and 𝑠 − 𝑝 − intϢԎ ⊆ 𝑠 − 𝑝 − intϢ (Ԋ ∪ Ԏ). Thus 𝑠 − 𝑝 − intϢԊ ∪ 𝑠 − 𝑝 − intϢ Ԏ ⊆ 𝑠 − 𝑝 − intϢ (Ԋ ∪ Ԏ). Theorem 3.13 Ħ is an Ϣ − sp − o setĦ = 𝑠 − 𝑝 − intϢĦ . Proof: Is Clear. Corollary 3.14 𝑠 − 𝑝 − intϢØ = Ø IHJPAS. 36(1)2023 340 4. (Ϣ𝟏, Ϣ𝟐)-semi-p-continuous function Definition 4.1:[8] Let (Z, Ϣ1) and (Y, Ϣ2) be two GTS ,s. A function f: Z → Y is said to be (Ϣ𝟏, Ϣ𝟐)-continuous function if the inverse image of any Ϣ2-open subset of Y is an Ϣ1-open set in Z. Definition 4.2:[9] A function𝑓: (Z, Ϣ1) → (Y, Ϣ2) is called (Ϣ1, Ϣ2)-M- pre-open function if the direct image of any Ϣ1- pre-open set in Z is an Ϣ2- pre-open set in Y. Definition 4.3: A function 𝑓: (Z, Ϣ1) → (Y, Ϣ2) is called (Ϣ1, Ϣ2)-M-semi-p-open ((Ϣ1, Ϣ2)-M-semi-p- closed) function if the direct image of any Ϣ1-semi-p-open (Ϣ1-semi-p-closed) set in Z is an Ϣ2-semi-p-open (Ϣ2-semi-p-closed ) set in Y. Definition 4.4 A function 𝑓: (Z, Ϣ1) → (Y, Ϣ2) is said to be (Ϣ1, Ϣ2)-semi-p-continuous function if the inverse image of any Ϣ2-open set in Y is an Ϣ1-semi-p-open set in Z. Theorem 4.5 A function 𝑓: (Z, Ϣ1) → (Y, Ϣ2) is an (Ϣ1, Ϣ2)-semi-p-continuous function  the inverse image of any Ϣ2-closed set in Y is an Ϣ1-semi-p-closed set in Z . Proof: The "if" part. Let F be any Ϣ2-closed set in Y, thus (Y– F) is an Ϣ2-open set in Y, then f −1(Y − F) is an Ϣ1-semi-p-open set in Z ( since f is an (Ϣ1, Ϣ2)-semi-p-continuous function), but f −1(Y − F) = Z − f −1(F), then f −1(F) is an Ϣ1-semi-p-closed set. The "only if" part. Let Ħ be any Ϣ2-open set in Y, thus (Y– Ħ) is an Ϣ2-closed set in Y, then f −1(Y − Ħ) is an Ϣ1-semi-p-closed set in Z (by hypothesis) but f −1(Y − Ħ) = Z − f −1(Ħ), then f −1(Ħ) is an Ϣ1-semi-p-open set in Z , therefore f is an (Ϣ1, Ϣ2)-semi-p-continuous function. Definition 4.6 A function 𝑓: (Z, Ϣ1) → (Y, Ϣ2) is said to be (Ϣ1, Ϣ2)-semi-p-irresolute function if the inverse image of any Ϣ2-semi-p-open set in Y is an Ϣ1-semi-p-open set in Z Theorem 4.7 A function 𝑓: (Z, Ϣ1) → (Y, Ϣ2) is an (Ϣ1, Ϣ2)-semi-p-irresolute function the inverse image of each Ϣ2-semi-p-closed set in Y is an Ϣ1-semi-p-closed set in Z . IHJPAS. 36(1)2023 341 Proof: The "if" part. Let F be any Ϣ2-semi-p-closed set in Y, thus (Y– F) is an Ϣ2-semi-p-open set in Y, then f −1(Y − F) is an Ϣ1-semi-p-open set in Z (since f is an (Ϣ1, Ϣ2)-semi-p-irresolute function) , but f −1(Y − F) = Z − f −1(F), therefore f −1(F) is an Ϣ1-semi-p-closed set. The "only if" part . Let Ħ be any Ϣ2-semi-p-open set in Y, thus (Y– Ħ) is an Ϣ1-semi-p-closed set in Y then f −1(Y − Ħ) is an Ϣ1-semi-p-closed set in Z (by hypothesis) , but f −1(Y − Ħ) = Z − f −1(Ħ), then f −1(Ħ) is an Ϣ1-semi-p-open set in Z, therefore f is an (Ϣ1, Ϣ2)-semi-p-irresolute function. Proposition 4.8 Every (Ϣ1, Ϣ2)-semi-p-irresolute function is an (Ϣ1, Ϣ2)-semi-p-continuous function. Proof: Let 𝑓 be any (Ϣ1, Ϣ2)-semi-p-irresolute function from (Z, Ϣ1) into (Y, Ϣ2) . Let Ħ by any Ϣ2- open in Y , thus Ħ is an Ϣ2-semi-p- open set (Corollary 3.11), then f −1(Ħ) is an Ϣ1-semi-p-open set in Z(since 𝑓 is (Ϣ1, Ϣ2)-semi-p-irresolute function), therefore 𝑓 is an (Ϣ1, Ϣ2)-semi-p- continuous function. Remark 4.9 The reverse of Proposition 4.7 is not correct in general as we show in the following example: Example Let Z = {1,2,3,4}, Ϣ1 = {Z, Ø, {1}, {4}, {1,4}}, Ϣ1 − PO(Z) = {Z, Ø, {1}, {4}, {1,4}, {1,2,4}, {1,3,4}}, and Ϣ1 − SPO(Z) = Ϣ1 − PO(Z) ∪ {{1,2}, {1,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {2,3,4}, }. Let Y = {a, b, c, d}, Ϣ2 = {∅, {b, d}} , Ϣ2 − PO(Y) = {∅, {b, d}, {b}, {d}}, Ϣ2 − SPO(Y) = ℙ(Y) (The power set of Y). Define 𝑓 ∶ (Z, Ϣ1 ) → (Y, Ϣ2) such that 𝑓(1) = 𝑓(2) = {𝑑}, 𝑓(3) = {𝑏} 𝑓 is an (Ϣ1, Ϣ2)-semi-p-continuous function. But not (Ϣ1, Ϣ2)-semi-p-irresolute function, since {b} is an Ϣ2-semi-p-open set in Y, but f −1({ b}) = {3} is not an Ϣ1-semi-p-open set in Z. Proposition 4.10 Every (Ϣ1, Ϣ2)-continuous function is an (Ϣ1, Ϣ2)-semi-p-continuous function. Proof: Let 𝑓 be any (Ϣ1, Ϣ2)- continuous function from (Z, Ϣ1) into (Y, Ϣ2) . Let Ħ by any Ϣ2-open in Y , it follows from Definition 4.1 that f −1(Ħ) is an Ϣ1- open set in Z, but every Ϣ1- open set is an Ϣ1-semi-p –open. Therefore 𝑓 is an (Ϣ1, Ϣ2)-semi-p-continuous function. IHJPAS. 36(1)2023 342 Remark 4.11 The reverse of Remark 4.9 is not correct in general as we show in the following example: Example Let Z = {1,2,3}, Ϣ1 = {∅, {1}, {2}, {1,2}}, and Ϣ1 − PO(Z) = {∅, {1}, {2}, {1,2}}, Ϣ1 − SPO(Z) = ℙ(𝑍) (The power set of Z). Let Y = {a, b, c, d}, Ϣ2 = {∅, {b, d}} , Ϣ2 − PO(Y) = {∅, {b, d}, {b}, {d}}, Ϣ2 − SPO(Y) = ℙ(Y) (The power set of Y). Define 𝑓 ∶ (Z, Ϣ1 ) → (Y, Ϣ2) such that 𝑓(1) = 𝑓(2) = {𝑎}, 𝑓(3) = {𝑏}, 𝑓is an (Ϣ1, Ϣ2)-semi-p-continuous function, but it is not an (Ϣ1, Ϣ2)-continuous function, since {b, d} is an Ϣ2-open set in Y, but f −1({b, d}) = {3} is not an Ϣ1-open set in Z . Proposition 4.12 The composition of (Ϣ1, Ϣ2)-semi-p-irresolute function and (Ϣ2, Ϣ3)-semi-p-irresolute function is an (Ϣ1, Ϣ3)-semi-p- irresolute function. Proof Let f: (Z, Ϣ1 ) → (Y, Ϣ2) be (Ϣ1, Ϣ2)-semi-p-irresolute function and g ∶ (Y, Ϣ2 ) → (W, Ϣ3) be (Ϣ2, Ϣ3)-semi-p-irresolute functions, we have to show that g ∘ f ∶ (Z, Ϣ1 ) → (W, Ϣ3) is an (Ϣ1, Ϣ3)-semi-p-irresolute function. Let Ӈ be any Ϣ3-semi-p-open set in W, then (g ∘ f) −1(Ħ) = f −1 ∘ g−1(Ħ) = f −1(g−1(Ħ)), but g−1(Ħ) is an Ϣ2-semi-p-open set in Y ( since g is an (Ϣ2, Ϣ3)-semi-p-irresolute function ), and f −1(g−1(Ħ)) is an Ϣ1-semi-p-open set in Z ( since f is an (Ϣ1, Ϣ2)-semi-p-irresolute functions ) , therefore g ∘ f is an (Ϣ1, Ϣ3)-semi-p-irresolute functions . Remark 4.13 The composition of (Ϣ1, Ϣ2)-semi-p-continuous function and (Ϣ2, Ϣ3)-semi-p-continuous function need not to be (Ϣ1, Ϣ3)-semi-p-continuous function as we show in the following example: Example Let Z = {1, 2, 3}, Ϣ1 = {Z , Ø, {1, 2}} , Y = {a, b , c}, Ϣ2 = {Y , Ø, {a, b}} , W = {i, j, k}, Ϣ3 = {W , Ø, {i, k}}, Ϣ1-PO(Z)= {Z, Ø, {1}, {2}, {1, 2}, {1, 3}, {2, 3}} = Ϣ1-SPO(Z) Ϣ2-PO(Y)= {Y, Ø, {a}, {b}, {a, b}, {a, c}, {b, c} } = Ϣ2-SPO(Y), and Ϣ3-PO(W)= {W, Ø, {i}, {k}, {i, j}, {i, k}, {j, k} } = Ϣ3-SPO(W) Define 𝑓 ∶ (Z, Ϣ1) → (Y, Ϣ2) by f(1) = f(3) = {b}, f(2) = {c}. IHJPAS. 36(1)2023 343 And 𝑔: (Y, Ϣ2) → (W, Ϣ3) by g(a) = g(c) = {j}, g(b) = {k}. Then 𝑔 ∘ 𝑓: (Z, Ϣ1) → (W, Ϣ3) is defined by: g ∘ f (1) = g(f(1)) = g(b) = {k}, g ∘ f (2) = g(f(2)) = g(c) = {j}, g ∘ f (3) = g(f(3)) = g(b) = {k}, f is an (Ϣ1, Ϣ2)-semi-p-continuous function and g is an (Ϣ2, Ϣ3)-semi-p-continuous function. But g ∘ f is not an (Ϣ1, Ϣ3)-semi-p-continuous function, since {i, k} is an Ϣ3-semi-p-open set in W, but f −1({i, k} ) = {3} is not Ϣ1-semi-p-open set in Z. Proposition 4.14 The composition of an (Ϣ1, Ϣ2)-semi-p-continuous function and (Ϣ2, Ϣ3)- continuous function is an (Ϣ1, Ϣ3)-semi-p-continuous function. Proof: Let f ∶ (Z, Ϣ1 ) → (Y, Ϣ2) be any (Ϣ1, Ϣ2)-semi-p-continuous function and g: (Y, Ϣ2 ) → (W, Ϣ3) be any (Ϣ2, Ϣ3)- continuous function. We have to show that g ∘ f ∶ (Z, Ϣ1 ) → (W, Ϣ3) is an (Ϣ1, Ϣ3)-semi-p- continuous function. Let Ħ be any Ϣ3- open set in W. Then, g−1(Ħ) is an Ϣ2-open set in Y (since g is an (Ϣ2, Ϣ3)-continuous function), so f −1(g−1(Ħ) is an Ϣ1-semi-p-open set in Z ( since f is an (Ϣ1, Ϣ2)-semi-continuous function ), but (g ∘ f) −1(Ħ) = f −1 ∘ g−1(Ħ) = f −1(g−1(Ħ). Hence g ∘ f is an (Ϣ1, Ϣ3)-semi-p-continuous function. Theorem 4.15 Let 𝑓 ∶ (Z, Ϣ1 ) → (Y, Ϣ2) be an onto function, then 𝑓 is an (Ϣ1, Ϣ2)-M-semi-p-open function if and only if it is an (Ϣ1, Ϣ2)-M-semi-p-closed function. Proof: The "if" part. Let F be any Ϣ1-semi-p-closed set, so (Z − F) is an Ϣ1-semi-p-open set, then 𝑓(Z − F) is an Ϣ2-semi-p-open set ( since 𝑓 is an (Ϣ1, Ϣ2)-M-semi-p-open function), but 𝑓(Z − F) = Y − 𝑓(F), therefore 𝑓(F) is an Ϣ2-semi-p-closed. Hence 𝑓 an (Ϣ1, Ϣ2)-M-semi- p-closed function. The "only if" part. Let Ħ be any Ϣ1-semi-p-open set , so (Z − Ħ) is an Ϣ1-semi-p-closed set, then 𝑓(Z − Ħ) is an Ϣ2-semi-p-closed set ( since 𝑓 is an (Ϣ1, Ϣ2)-M-semi-p-closed function), but f(Z − Ħ) = Y − 𝑓(Ħ), therefore 𝑓(Ħ) is an Ϣ2-semi-p-open. Hence 𝑓 an (Ϣ1, Ϣ2)-M-semi- p-closed function. Theorem 4.16 Let 𝑓 ∶ (Z, Ϣ1 ) → (Y, Ϣ2) be a bijective function, then 𝑓 is an (Ϣ1, Ϣ2)-M-semi-p-open function,  𝑓 −1 ∶ (Y, Ϣ2 ) → (Z, Ϣ1) is an (Ϣ1, Ϣ2)-semi-p-irresolute function. Proof The "if" part. Suppose that f is an (Ϣ1, Ϣ2)-M-semi-p-open function, to show that f −1 is an (Ϣ1, Ϣ2)-semi-p-irresolute function. Let Ħ be any Ϣ1-semi-p-open set in Z, then (f −1)−1(Ħ) = IHJPAS. 36(1)2023 344 f(Ħ) is an Ϣ2-semi-p-open set in Y (since f is an (Ϣ1, Ϣ2)-M-semi-p-open function), so f −1is an (Ϣ1, Ϣ2)-semi-p- irresolute function. The "only if" part. Suppose that f −1 is an (Ϣ1, Ϣ2)-semi-p-irresolute function, to show that f is an (Ϣ1, Ϣ2)-M-semi-p-open function. Let Ħ be any Ϣ1-semi-p-open set in Z, then (f −1)−1(Ħ) = f(Ħ) is an Ϣ2-semi-p-open set in Y(since f −1is an (Ϣ1, Ϣ2)-semi-p-irresolute function), so f is an (Ϣ1, Ϣ2)-M-semi-p-open function. Definition 4.17 A bijection function f ∶ (Z, Ϣ1) → (Y, Ϣ2) is called (Ϣ1, Ϣ2)-semi-p-homeomorphism function if f is both (Ϣ1, Ϣ2)-semi-p-irresolute function and (Ϣ1, Ϣ2)-M-semi-p-open function. References 1.Engelking, R., General Topology, Sigma Ser. 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