@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 On Semi-p-Proper Mappings Narjis. A.U. Dawood Dept. of Mathematics/ College of Education for Pure Science (Ibn Al-Haitham)/ University of Baghdad Received in:3 July 2012 Accepted in:21 January 2013 Abstract The aim of this paper is to introduce a new type of proper mappings called semi-p-proper mapping by using semi-p-open sets, which is weaker than the proper mapping. Some properties and characterizations of this type of mappings are given. Keywords : Proper mapping, semi-p-open sets. 290 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Introduction One of the important concepts in topology and in mathematics in general is the concept of mapping, in this paper we give a new class of mappings. The class of proper mappings was first introduced by Valnsteln in 1947 and studied by Lary in 1950 and Bourbaki in 1951, who defined the proper mapping as (if f : X → Y is continuous and f × IZ is closed for each space Z then f is called a proper mapping). For more details see [1], [2], [3], [4], [5] and [6]. In this paper we introduced and studied semi-p-proper mappings. 2- Semi-p-Open Sets Throughout this paper X, Y and Z are topological spaces or space only with no separation axioms assumed unless otherwise stated. The interior and closure of a subset A of a topological space will be denoted by intA and clA respectively. In this section we shall recall some needed definitions, propositions, and properties of semi-p-open sets and pre-open sets which we shall use to define semi-p-proper mapping, some of these propositions or definition are given for the first time by the author. Definition 2.1: A subset A of a space X is called (1) pre-open if A ⊆ int(clA), [7]. (2) semi-p-open if U ⊆ A ⊆ pre-cl U for some pre-open set U in X and pre-cl U is the smallest pre-closed set which contains U, [8]. The complement of pre-open (respectively semi-p-open) set is called a pre-closed (respectively semi-p-closed) set. Remarks 2.2: [9] (1) X and φ are pre-open (semi-p-open)sets. (2) The union of any family of pre-open (semi-p-open) set is a pre-open (semi-p-open) set. (3) The intersection of two pre-open (semi-p-open) sets need not to be pre-open (respectively semi-p-open). Examples 2.3: [9], [4] and [10] (1) If X is any infinite set and τco is the cofinite topology on X, then the family of pre-open sets PO(X) and the family of semi-p-open sets S-PO(X) are equal and PO(X) = S-PO(X) = {U ⊆ X:U is an infinite set or U = φ}. (2) In the Discrete space (X,D) we have PO(X) = SPO)X) = D = ℙ(X) = the family of all subsets of X. (3) In the indiscrete space (X,I) where I = {X, φ}, PO(X) = SPO(X) = ℙ(X). (4) In the space (ℝ,τu) where ℝ is the set of real numbers and τu is the usual topology on ℝ, PO(X) = SPO(X) = τu ∪ {Q, Irr} where Q is the rationals and Irr is the irrationals. Proposition 2.4: [11] Let A be any subset of the space X, then (1) (intA)c = clAc. (2) (clA)c = intAc. Proposition 2.5: [4] Let A be any subset of a space X, then A is semi-p-open set iff A ⊆ pre-cl(pre-intA) where pre-clA is the smallest pre-closed set which contains A, and pre-intA is the largest pre- open set contained in A. 291 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Corollary 2.6: Let A be a subset of a space X. A is semi-p-closed set iff pre-int(pre-clA) ⊆ A. Proof: (⇒) Let A be a semi-p-closed subset of X, then pre-clA = A, which implies pre-int(pre-clA) ⊆ A since pre-intA ⊆ A. (⇐) Suppose pre-int(pre-clA) ⊆ A. To prove A is semi-p-closed set. Since pre-int(pre-clA) ⊆ A, so we get Ac ⊆ [pre-int(pre-clA)]c = pre-cl(pre-clA)c = pre-cl(pre-intAc) Hence Ac is a pre-open set by proposition 2.5 which means A is pre-closed. ■ Corollary 2.7: Let A be a subset of a space X. A is semi-p-open set iff A ⊆ cl(pre-intA). Proof: Since pre-clA ⊆ clA and by proposition 2.5. ■ Proposition 2.8: [9] Let A and B be subsets of a space X, then pre-int(A ∩ B) = pre-intA ∩ pre-intB. Proposition 2.9: Let A be an open set in a space X and B is a semi-p-open set, then A ∩ B is a semi-p- open set. Proof: we shall use corollary 2.7 to prove this proposition; that is we must show that A ∩ B ⊆ cl(pre-int(A ∩ B)). Let x ∉ cl(pre-int(A ∩ B)) which implies ∃ U open in X s.t. x ∈ U and U ∩ pre-int(A ∩ B) = φ. Then U ∩ (pre-intA ∩ pre-intB) = φ (by proposition 2.8) So we get (U ∩ pre-intA) ∩ pre-intB = φ Hence (U ∩ A) ∩ pre-intB = φ since A is open Which means U ∩ A is open in X Now if x ∈ A then x ∈ U ∩ A and (U ∩ A) ∩ pre-intB = φ which means x ∉ cl(pre-intB) but B ⊆ cl(pre-intB), since B is semi-p-open So x ∉ B ⇒ x ∉ A ∩ B. ■ Proposition 2.10: If A is closed in X and B is semi-p-closed then A ∩ B is semi-p-closed set in X. Proof: (A ∩ B)c = Ac ∪ Bc where Ac is open and Bc is semi-p-open so Ac ∪ Bc is semi-p-open. ■ Proposition 2.11: Let A be a pre-open subset of the space X, and let B be a pre-open subset of the space Y then A×B is pre-open in the product space X×Y. Proof: A ⊆ int(clA) and B ⊆ int(clB), A×B ⊆ int(clA) × int(clB) = int(cl(A×B)). So A×B is pre-open in X×Y. ■ Proposition 2.12: Let A be a subset of a space X, and B a subset of a space Y then 1) pre-cl(A×B) = pre-clA × pre-clB. 2) Pre-int(A×B) = pre-intA × pre-intB. Proof: Obvious. ■ 292 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Corollary 2.13: The product of any two semi-p-open sets in X and Y respectively is semi-p-open in X×Y. Proof: Let A be a semi-p-open subset of X, and let B be a semi-p-open subset of Y, then A ⊆ pre-cl(pre-intA) and B ⊆ pre-cl(pre-intB) by proposition 2.5 which implies A×B ⊆ pre-cl(pre-intA) × pre-cl(pre-intB) = pre-cl(pre-intA×B) by proposition 2.12. ■ Corollary 2.14: If A is a nonempty subset of X, and B a non empty subset of Y then: (1) If A is pre-closed subset of X and B is pre-closed subset of Y then A×B is pre-closed subset of X×Y. (2) If A is a semi-p-closed subset of X and B is a semi-p-closed subset of Y then A×B is semi-p-closed subset of X×Y. Proof: The proofs of parts (1) and (2) are similar, so we prove part (2) because our work about semi-p-closed sets. (2) A is semi-p-closed subset of X means pre-int(pre-clA) ⊆ A. By corollary 2.14 On the other hand B is semi-p-closed subset of Y implies pre-int(pre-clB) ⊆ B. So we have pre-int(pre-clA) × pre-int(pre-clB) ⊆ A×B. Hence pre-int(pre-cl A×B) ⊆ A×B. ■ It known, that a mapping f : X → Y is continuous if the inverse image of each closed set in Y is closed in X. On the other hand f is an open (closed) mapping if the image of each, open (closed) subset of X is open (closed) in Y, [12]. Now, we shall recall two definitions of a mapping using semi-p-closed sets, these are known by, semi-p-closed and semi-irresolute mappings. Definition 2.15: [9] Let f : X → Y be a mapping, f is called: (1) semi-p-closed if f (A) is semi-p-closed in Y whenever A is closed in X. (2) semi-p-irresolute if f – 1(B) is semi-p-closed in X whenever B is semi-p-closed in Y. Remarks 2.16: [9] (1) The composition of two semi-p-closed mappings need not be semi-p-closed in general. (2) The composition of a semi-p-closed and continuous mappings is semi-p-closed. (3) Every homeomorphism is a semi-p-closed mapping. In the following definition we give a new type of homeomorphisms named semi-p- homeomorphism which is weaker than the concept homeomorphism. Definition 2.17: A mapping f : X → Y is called semi-p-homeomorphism if it is bijective, continuous and semi-p-closed. Remark 2.18: It is clear that every homeomorphism is semi-p-homeomorphism, but not converse. For example: Let f: (ℝ,D) → (ℝ,I ) be a mapping defined by f (x) = x ∀ x ∈ ℝ. It is clear that f is not closed, which means that f is not a homeomorphism. 293 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 On the other hand f is semi-p-homeomorphism, since it is bijective, continuous and semi-p- closed mapping, since S-PO(ℝ,I) =ℙ(ℝ) by examples 2.3 (3). Proposition 2.19: Let X, X′, X″ be three topological spaces, and let f : X → X′, g: X′ → X″ be two mappings, then: (1) If g∘f is semi-p-closed and if f is continuous and surjective, then g is semi-p-closed. (2) If g∘f is semi-p-closed and if g is semi-p-irresolute and injective, then f is semi-p-closed. Proof: We have (1) X X Xf g g f→ ′ ′′→ →  Let B be a closed subset of X′ to prove g(B) is semi-p-closed in X″. Now B closed in X′ and f is continuous, then f – 1(B) is closed in X, but g∘f is semi-p-closed, so (g∘f )( f – 1(B)) is semi-p- closed in X″, but (g∘f )( f – 1(B)) = g(B) since f is surjective, hence g is semi-p-closed. (2) Let A be a closed subset of X to prove f (A) is semi-p-closed in X′. Since A is closed in X and g∘f is semi-p-closed, we get (g∘f )(A) is semi-p-closed in X″, but g is semi-p-irresolute, which implies g– 1((g∘f )(A)) is semi-p-closed subset of X′, but g– 1((g∘f )(A)) = f (A) since g is injective, which means that f is semi-p-closed mapping. ■ 3- Semi-p-Proper Mapping In this section we shall define a new type of proper mappings named semi-p-proper mapping by using the concept of semi-p-open set which depends on the, pre-open set. Recall that a mapping f from a topological space X into a topological space Y is called proper if f is continuous and f × IZ : X×Z → Y×Z is closed for every topological space Z, [3]. Definition 3.1: Let f be a mapping from a topological space X into a topological space Y, then f is called a semi-p-proper mapping if f is continuous and f×IZ:X×Z→Y×Z is semi-p-closed for any topological space Z. Remarks 3.2: (1) Every proper mapping is semi-p-proper since every closed mapping is semi-p-closed, but the converse is not true in general. For example: Let ℝ be the set of real numbers with the usual topology then i: Q → ℝ is semi-p-proper mapping which is not proper because it is not closed, because Q is semi-p-closed subset of ℝ which is not closed. (2) Let X be any topological space, and F be a closed subset of X, then the inclusion mapping i: F → X is semi-p-proper since it is a proper mapping, i is proper because it is a closed mapping since F is closed in X and, so every closed subset of F is closed in X. (3) The identity Ix: X → X is proper and so it is semi-p-proper, IX is proper since it is 1-1 and a closed mapping.. In the following theorem, we give a characterization for semi-p-proper mappings. Theorem 3.3: Let f : X → Y be a continuous injection mapping, then the following statements are equivalant: (1) f is semi-p-proper. (2) f is semi-p-closed. 294 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 (3) f is semi-p-homeomorphism from X onto a semi-p-closed subset of Y. Proof: (1) ⇒ (2) Let f be semi-p-proper. To prove f is semi-p-closed mapping. Take Z = {p} then by hypothesis the mapping f×IZ:X×{p}→Y×{p} is semi-p-closed, but X×{p} and Y×{p} are homeomorphic to X and Y respectively, then f is semi-p-closed mapping. (2) ⇒ (3) f is continuous, injection and semi-p-closed by hypothesis, so it is semi-p- homeomorphism from X onto f (X) which is semi-p-closed subset of Y. (3) ⇒ (1) Let f be semi-p-homeomorphism mapping from X into a semi-p-closed subset of Y, i.e. f : X → f(X) is semi-p-homeomorphism mapping. Now, let Z be any topological space, so IZ: Z → Z the identity mapping on Z is homeomorphism, which implies f×IZ:X×Z→Y×Z is semi-p-homeomorphism from X×Z onto the closed subspace f(X)×Z of Y×Z and therefore is a semi-p-closed mapping. ■ Proposition 3.4: The composition of two semi-p-proper mapping is semi-p-proper. Proof: Let X, X′ and X″ be topological spaces, and let f : X → X′ and g: X′ → X″ be two semi-p-proper mappings. To prove g∘f is semi-p-proper. g∘f is a continuous mapping, since f and g are continuous. Now let Z be any topological space, then f×IZ is semi-p-closed mapping since f is semi-p-proper by hypothesis, on the other hand g×IZ is continuous since g and IZ are continuous by [3,p.44], so (g×IZ)∘(f×IZ) is semi-p-closed mapping by remarks 2.16 (2) which implies (g∘f )×IZ is semi-p-closed. ■ Proposition 3.5: Let X, X′ and X″ be three topological spaces, and let f : X → X′ and g: X′ → X″ be mappings, then: (1) If f is continuous surjective mapping, g is continuous and g∘f is semi-p-proper, then g is semi-p-proper. (2) If f is continuous, g is semi-p-irresolute injective mapping and g∘f is semi-p-proper then f is semi-p-proper. Proof: (1) Let Z be any topological space, then f×IZ is continuous and onto since f and IZ are so, on the other hand (g∘f )×IZ is semi-p-closed since g∘f is semi-p-proper which implies that g×IZ is semi-p-closed by proposition 2.19(1). (2) Le Z be any topological space, then the mapping g×IZ: X′×Z → X″×Z is semi-p- irresolute injective mapping since g and IZ are so. On the other hand (g∘f )×IZ: X×Z → X″×Z is semi-p-closed since g∘f is semi-p-proper, which implies that f×IZ: X×Z → X′×Z is semi-p-closed by proposition 2.19(2). ■ Proposition 3.6: Let X1, Y1, X2 and Y2 be topological spaces, and let f1: X1 →Y1, f2: X2 →Y2 be mappings, then f1× f2 is semi-p-proper if f1 and f2 are semi-p-proper mappings. Proof: Suppose f1 and f2 are semi-p-proper mappings to show that f1× f2 is semi-p-proper. Let Z be any topological space, then: f1× f2×IZ = (f1× 2Y I ×IZ)∘( 1x I ×f2×IZ) as shown in the following diagram X 2 Z 1 Y Z1 2 1 2 Z I I I I 1 2 1 2 1 2 I X X Z X Y Z Y Y Zf f f f × × × × → × × × × → × × → × × 295 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 but f1× 2Y I ×IZ is continuous since f1, 2Y I and IZ are continuous. On the other hand 1x I ×f2×IZ is semi-p-closed since 1x I is semi-p-closed and f2×IZ is semi-p-closed since f2 is semi-p-proper mapping, hence by remarks 2.16(2) f1× f2×IZ is semi-p-closed which means that f1× f2 is semi- p-proper. ■ 4- Future Work (1) We can use the concept of semi-p-compactness to characterize semi-p-proper mappings. (2) We can use the concept of semi-p-proper mapping to define a new kind of spaces called semi-p-proper G-spaces. References 1. Al-Aubaidy, A.I., (2003), On Semi-Proper G-Spaces, M.Sc. Thesis, University of Al- Mustansiriyah. 2. AL-Saidy, Z.A.D., (2005), A Weak Form of Proper Actions, M.Sc. Thesis, University of Al-Mustansiriyah. 3. Bourbaki, N., (1989), Elements of Mathematica, General Topology, Springer-Verlage Berlin Heidelberg, NewYork, London, Paris, Tokyo, 2nd Edition. 4. Gasim, S.G., (2006), On Semi-p- Compact Spaces, M.Sc. Thesis, University of Baghdad. 5. Mustafa, H.J. and Ali, H.J., (2002), On Proper Mapping, J.Babylon, 6, No.3. 6. Wahid, A.H.A., (2005), On L-Proper Actions, M.Sc. Thesis, University of Al-Mustansiriyah. 7. Mashhour, A., Abd-Monsef, M. and Hasanein, I.A., (1984), On Pretopological Spaces, Bull. Math.Soc.Sci.R.S.R. 28, No.76:39-45. 8. Navalagi, G.B., (1991), Definition Bank in General Topology, 54G. 9. Al-Khazraji, R.B., (2004), On Semi-p-Open Sets, M.Sc. Thesis, University of Baghdad. 10. Steen, L.A. and Seebach, J.A., (1970), Counter Examples in Topology, Holt Rinehart Wilson, New York. 11. Sharma, L.J.N., (2000), Topology, Krishma Prakashan Medis (p)ltd. India, Twenty Fifth Edition. 12. د.سمیر بشیر حریر، مقدمة في التبولوجیا العامة، جامعة الموصل 296 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 p– حول الدوال شبھ السدیدة من النمط داود نرجس عبد الجبار جامعة بغداد / )ابن الھیثم( للعلوم الصرفة كلیة التربیة / قسم الریاضیات 2013كانون الثاني 21قبل البحث في: ، 2012تموز 3: البحث فيستلم أ ةصالخال ، باستخدام p –الھدف من ھذا البحث ھو اضافة نمط جدید من الدوال السدیدة و تسمى الدالة شبھ السدیدة من نوع سة بعض خواص ھذه الدوال وھي أضعف من الدوال السدیدة. تم برھنة ودرا p-المجموعات شبھ المفتوحة من نمط ومكافئاتھا. . p –الدوال السدیدة، المجموعات شبھ المفتوحة من النمط : الكلمات المفتاحیة 297 | Mathematics