CUBO A Mathematical Journal Vol.20, No¯ 01, (31–39). March 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000100031 Pre-regular sp-Open Sets in Topological Spaces P. Jeyanthi and P. Nalayini Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur-628 215, Tamil Nadu, India. jeyajeyanthi@rediffmail.com,nalayini4@gmail.com T. Noiri Shiokita - cho, Hinagu, Yatsushiro - shi, Kuvempu University Kumamoto - ken, 869-5142 Japan t.noiri@nifty.com ABSTRACT In this paper, a new class of generalized open sets in a topological space, called pre- regular sp-open sets, is introduced and studied. This class is contained in the class of semi-preclopen sets and cotains all pre-clopen sets. We obtain decompositions of regular open sets by using pre-regular sp-open sets. RESUMEN En este art́ıculo se introduce y estudia una nueva clase de conjuntos abiertos gener- alizados en un espacio topológico, llamados conjuntos pre-regulares sp-abiertos. Esa clase está contenida en la clase de conjuntos semi-preclopen y contiene todos los conjun- tos pre-clopen. Obtenemos descomposiciones de conjuntos abiertos regulares usando conjuntos pre-regulares sp-abiertos. Keywords and Phrases: Generalized open sets, preopen, regular open, pre-regular sp-open, decompositions of complete continuity. 2010 AMS Mathematics Subject Classification: 54A05. http://dx.doi.org/10.4067/S0719-06462018000100031 Ignacio Castillo Ignacio Castillo Ignacio Castillo Ignacio Castillo 32 P. Jeyanthi, P. Nalayini and T. Noiri CUBO 20, 1 (2018) 1 Introduction In general topology, by repeated applications of interior (int) and closure (cl) operators several different new classes of sets are defined in the following way. Definition 1. A subset A of a space X is said to be i) semi-open [10] if A ⊆ cl(intA). ii) preopen [11] if A ⊆ int(clA). iii) semi-preopen [2] or β-open [1] if A ⊆ cl(int(clA)). iv) α-open [12] if A ⊆ int(cl(intA)). v) regular open [13] if A = int(clA). vi) b-open [3] if A ⊆ cl(intA) ∪ int(clA). vii) pre-regular p-open [9] if A = pint(pclA). The complements of the above open sets are called their respective closed sets. Definition 2. A subset A of a space X is called a q-set [14] or δ-set [5] if int(clA) ⊆ cl(intA). In this paper, we introduce and study a new class of sets, called pre-regular sp-open sets using pre-interior and semi-preclosure operators. This class is contained in the class of semi-preclopen sets and cotains all pre-clopen sets. Moreover, we investigate the relationship between this class of sets and other class of open sets. By using pre-regular sp-open sets, we obtain decompositions of regular open sets. In the last section, we obtain decompositions of complete continuity. Throughout this paper (X, τ) (briefly X) denotes a topological space on which no separation axioms are assumed, unless explicity stated. We recollect some of the relations that, together with their duals, we shall use in the sequel. Proposition 1. [2] Let A be a subset of a space X. Then i) pclA = A ∪ cl(intA) and pintA = A ∩ int(clA). ii) spclA = A ∪ int(cl(intA)) and spintA = A ∩ cl(int(clA)). iii) pint(spclA) = (A ∩ int(clA)) ∪ int(cl(intA)). iv) pcl(spintA) = (A ∪ cl(intA)) ∩ cl(int(clA)). Definition 3. A function f : X → Y is called completely continuous [4] (resp. α-continuous [8],semi-continuous [10], q-continuous [14] ) if the inverse image of every open subset of Y is a regular open (resp. α-open, semi-open, a q-set) subset of X. CUBO 20, 1 (2018) Pre-regular sp-Open Sets in Topological Spaces 33 2 pre-regular sp-open sets In this section, we define and characterize pre-regular sp-open sets and study some of their prop- erties. Definition 4. A subset A of a topological space (X, τ) is said to be pre-regular sp-open if A = pint(spclA). The complement of a pre-regular sp-open set is said to be pre-regular sp-closed. We denote the collection of all pre-regular sp-open (resp. preopen, preclosed, pre-semiopen, pre-semiclosed, pre-clopen, pre-semiclopen) sets of (X, τ) by PRSPO(X) (resp. PO(X), PC(X), PSO(X), PSC(X), PCO(X), PSCO(X)). Theorem 2.1. Let (X, τ) be a topological space and A, B subsets of X. Then the following hold: i) If A ⊆ B, then pint(spclA) ⊆ pint(spclB). ii) If A ∈ PO(X, τ), then A ⊆ pint(spclA). iii) If A ∈ SPC(X, τ), then pint(spclA) ⊆ A. iv) We have pint(spcl(pint(spclA))) = pint(spclA). v) If A ∈ SPC(X, τ), then pintA is a pre-regular sp-open set. Proof. i) Suppose that A ⊆ B. Then pint(spclA) ⊆ pint(spclB). ii) Suppose that A ∈ PO(X, τ). Since A ⊆ spclA, we have A ⊆ pint(spclA). iii) Suppose that A ∈ SPC(X, τ). Since pintA ⊆ A, we have pint(spclA) ⊆ A. iv) We have pint(spcl(pint(spclA))) ⊂ pint(spcl(spclA)) = pint(spclA) and pint(spcl(pint(spclA))) ⊃ pint(pint(spclA)) = pint(spclA). Hence pint(spcl(pint(spclA))) = pint(spclA). v) Suppose that A ∈ SPC(X, τ). By (i), we have pint(spcl(pintA)) ⊆ pint(spclA) = pintA. On the other hand, we have pintA ⊆ spcl(pintA). Therefore pintA ⊆ pint(spcl(pintA)) and hence pint(spcl(pintA)) = pintA. Remark 2.2. The family of pre-regular sp-open sets is not closed under finite union as well as finite intersection. It will be shown in the following example. Example 2.3. Let X = {a, b, c, d} and τ = {∅, {a, b}, {a, b, c}, {a, b, d}, X}. Then {a} and {b} are pre-regular sp-open sets but their union {a, b} is not a pre-regular sp-open set. Moreover, {a, c, d} and {b, c, d} are pre-regular sp-open but their intersection {c, d} is not a pre-regular sp-open set. Theorem 2.5 and 2.6 give the characterizations of pre-regular sp-open sets. 34 P. Jeyanthi, P. Nalayini and T. Noiri CUBO 20, 1 (2018) Theorem 2.4. Let (X, τ) be a topological space. For a subset A of X, the following are equivalent: i) A is pre-regular sp-open. ii) A = spclA ∩ int(clA). iii) A = pintA ∪ int(cl(intA)). Proof. It follows form Proposition 1.3. Theorem 2.5. Let (X, τ) be a topological space. A subset A of X is pre-regular sp-open if and only if it is preopen and semi-preclosed . Proof. Let A be pre-regular sp-open. Then A = pint(spclA). Hence pintA = pint(pint(spclA)) = pint(spclA) = A. Thus A is preopen. By Theorem 2.5, A = pintA ∪ int(cl(intA)) and int(cl(intA)) ⊆ A. Therefore, A is semi-preclosed. Conversely assume that A is both preopen and semi-preclosed. Then A = pintA and A = spclA. Now pint(spclA) = pintA = A. Hence A is pre-regular sp-open. Corolary 1. For a topological space (X, τ), we have PO(X) ∩ PC(X) ⊆ PRSPO(X) ⊆ SPO(X) ∩ SPC(X). Proof. This is obvious. Remark 2.6. The converse inclusions in Corollary 2.7 need not be true as the following examples show. Example 2.7. Let X = {a, b, c, d} and τ = {∅, {a}, {b}, {a, b}, {b, c}, {a, b, c}, X}. Then {a, d} is semi-preclopen but not pre-regular sp-open. Example 2.8. Let X = {a, b, c, d} and τ = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {c, d}, {a, c, d}, {a, b, c}, {b, c, d}, X}. Then {c} is pre-regular sp-open but it is not pre-clopen. Theorem 2.9. In any space (X, τ), the empty set is the only subset which is nowhere dense and pre-regular sp-open. Proof. Suppose A is nowhere dense and pre-regular sp-open. Then by Theorem 2.5, A = pint(spclA) = spclA ∩ int(clA) = spclA ∩ ∅ = ∅. Remark 2.10. The notions of pre-regular sp-open sets and open sets (hence α-open sets, semi- open sets, q-sets) are independent of each other. It is shown in [5] and [14] that every semi-open set is a q-set, that is, a δ-set. CUBO 20, 1 (2018) Pre-regular sp-Open Sets in Topological Spaces 35 Example 2.11. Let X = {a, b, c} and τ = {∅, {a, b}, X}. Then {a, b} is open hence α-open, semi- open, a q-set but it is not pre-regular sp-open. Also, {a} is pre-regular sp-open but it is not a q-set. Theorem 2.12. Every regular open set is pre-regular sp-open. Proof. Let A be regular open. Then A = int(clA). By Proposition 1.3, pint(spclA) = (spclA) ∩ int(cl(spclA)) = spclA ∩ int(cl[A ∪ int(cl(intA))]) = spclA ∩ int(clA) = spclA ∩ A = A. This shows that A is pre-regular sp-open. The above disscusion can be summarized in the following diagram: DIAGRAM regular open ⇒ open ⇒ α-open ⇒ semi-open ⇒ q-set ⇓ ⇓ ⇓ pre-regular sp-open ⇒ preopen ⇒ b-open ⇒ semi-preopen Remark 2.13. A q-set and a semi-preopen set are independent by Example 2.13 and the following example. Example 2.14. Let R be the real numbers with the usual topology. Then for each x ∈ R, cl(int(cl{x})) = ∅ and it does not contain {x}. Hence {x} is not semi-preopen. But int(cl{x}) = cl(int{x}) = ∅ and {x} is a q-set. Theorem 2.15. Every pre-regular p-open set is pre-regular sp-open. Proof. Let A be pre-regular p-open. Then A = pint(pclA) and A is preopen. Since spclA ⊆ pclA, we have pint(spclA) ⊆ pint(pclA) = A. On the other hand, we have A ⊆ spclA. Since A is preopen, A = pintA ⊆ pint(spclA). Hence A = pint(spclA). Theorem 2.16. For a subset A of a space X, the following are equivalent: i) A is regular open. ii) A is pre-regular sp-open and a q-set. iii) A is α-open and semi-preclosed. Proof. i) ⇒ ii). Let A be regular open. Then, by Theorem 2.14 A is pre-regular sp-open and also by Diagram, A is a q-set. ii) ⇒ i). Since A is a q-set, int(clA) ⊂ cl(intA) and int(clA) ⊂ int(cl(intA)) ⊂ int(clA). Therefore, we have int(clA) = int(cl(intA)). By using Theorem 2.5, we obtain int(clA) = [A ∪ int(clA)] ∩ int(clA) = [A ∪ int(cl(intA))] ∩ int(clA) = spclA ∩ int(clA) = A. i) ⇒ iii). Let A be regular open. Then A is open and A = int(clA) = int(cl(intA)). Therefore, every regular open set is α-open and semi-preclosed. 36 P. Jeyanthi, P. Nalayini and T. Noiri CUBO 20, 1 (2018) iii) ⇒ i). Let A be α-open and semi-preclosed. Then int(cl(intA)) ⊂ A ⊂ int(cl(intA)). Therefore, A = int(cl(intA)) and hence int(clA) = int(cl(int(cl(intA)))) = int(cl(intA)) = A. Hence A is regular open. Corolary 2. Suppose A is pre-regular sp-open. Then the following are hold: i) If A is open, then A is regular open. ii) If A is closed, then A is clopen. iii) If A is semi-open, then A is regular open. iv) If A is semi-closed, then A is α-open and semi-preclosed. Proof. Since A is pre-regular sp-open, by Theorem 2.5 A = spclA∩int(clA) = pintA∪int(cl(intA)). i) Suppose A is open. Then by Diagram, A is a q-set and by Theorem 2.18, we have A is regular open. ii) Suppose A is closed. Now A = spclA ∩ int(clA) = spclA ∩ intA = intA. Hence A is open and hence clopen. iii) Since every semi-open set is a q-set, by Theorem 2.18 A is regular open. iv) Suppose A is semi- closed. Then int(clA) ⊆ A. This implies int(clA) ⊂ intA ⊂ cl(intA). Hence A is a q-set and by Theorem 2.18, A is α-open and semi-preclosed. Remark 2.17. In a partition space (X, τ), a subset A of X is preopen if and only if A is pre-regular sp-open. Theorem 2.18. If a space (X, τ) is submaximal, then any finite intersection of pre-regular sp-open sets is pre-regular sp-open. Proof. Let {Ai|i ∈ I} be a finite family of pre-regular sp-open sets. Then {Ai|i ∈ I} is a finite family of preopen sets. Since X is submaximal, ⋂ i∈I Ai is pre open. Therefore by Theorem 2.2 (ii), ⋂ i∈I Ai ⊆ pint(spcl( ⋂ i∈I Ai). On the other hand, for each i ∈ I, we have ⋂ i∈I Ai ⊆ Ai and by Theorem 2.2 (i) pint(spcl( ⋂ i∈I Ai))) ⊆ pint(spclAi). Since pint(spclAi) = Ai, we have pint(spcl( ⋂ i∈I Ai))) ⊆ ⋂ i∈I Ai. Hence pint(spcl( ⋂ i∈I Ai))) = ⋂ i∈I Ai. Theorem 2.19. If A is pre-regular sp-closed and a rare set of a space (X, τ), then A is semi- preopen. CUBO 20, 1 (2018) Pre-regular sp-Open Sets in Topological Spaces 37 Proof. Since A is pre-regular sp-closed, by Theorem 2.5 A = pcl(spintA) = spintA ∪ cl(intA). Since A is a rare set, intA = ∅. Thus A = spintA. Hence A is semi-preopen. Recall that a space (X, τ) is said to be an extremally disconnected if the closure of every open subset of X is open. Moreover, it is shown in [7] (X, τ) is extremally disconnected if and only if SPO(X) = PO(X). Theorem 2.20. For an extremally disconnected space (X, τ), the following are equivalent: i) A is pre-regular sp-open. ii) A is pre-regular sp-closed. iii) A is pre-clopen. iv) A is semi-preclopen. Proof. (i) ⇔ (iii). Suppose that A is pre-regular sp-open. Then by Theorem 2.6, A is preopen and semi-preclosed. Since X is extremally disconnected, A is pre-clopen. Hence A is pre-closed. The converse is obvious by Theorem 2.6. (ii) ⇔ (iv). Let A be pre-regular sp-closed. Then X\A is pre-regular sp-open and by (i) ⇔ (iii) X\A is pre-clopen. Therefore, A is semi-preclopen. The converse is obvious. (iii) ⇔ (iv). This is obvious. Recall that a space (X, τ) has the property Q [10] if int(clA) = cl(intA) for all subset A of X. Theorem 2.21. Let (X, τ) be a space with the property Q. For a subset A ⊆ X, the following properties are equivalent: i) A is pre-regular sp-open. ii) A is pre-regular sp-closed. iii) A is regular open. iv) A is regular closed. Proof. (i) ⇔ (iii). By Proposition 1.3, pint(spclA) = [A ∩ int(clA)] ∪ int(cl(intA)) = [A ∩ int(clA)] ∪ int(int(clA)) = int(clA). This completes the proof. (ii) ⇔ (iv). By Proposition 1.3, pcl(spintA) = [A ∪ cl(intA)] ∩ cl(int(clA)) = [A ∪ cl(intA)] ∩ cl(cl(intA)) = cl(intA). This completes the proof. (iii) ⇔ (iv). This is obvious. 38 P. Jeyanthi, P. Nalayini and T. Noiri CUBO 20, 1 (2018) 3 Decompositions of complete continuity In this section, the notion of pre-regular sp-continuous functions is introduced and the decompo- sitions of complete continuity are discussed. Definition 5. A function f : X → Y is said to be pre-regular sp-continuous (briefly, prsp- continuous) if f−1(V) is pre-regular sp-open in X for each open subset V of Y. By Theorems 2.18 and Daigram, we have the following main theorem Theorem 3.1. For a function f : X → Y, the following properties are equivalent: i) f is completely continuous. ii) f is prsp-continuous and continuous. iii) f is prsp-continuous and α-continuous. iv) f is prsp-continuous and semi-continuous. v) f is prsp-continuous and q-continuous. Remark 3.2. As shown by the following examples, prsp-continuity and continuity (hence α- continuity, semi-continuity, q-continuity) are independent of each other. Example 3.3. Let X = {a, b, c}, τ = {∅, {a}, X} and σ = {∅, {a, b}, X}. 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Rao, q-sets in topological spaces , Prog of Maths., 36 (1-2) (2002), 159-165. Introduction pre-regular sp-open sets Decompositions of complete continuity