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UDC 513.831 Results on Simply-Continuous Functions Al Bayati Jalal Hatem Hussein Department of Mathematics Peoplesβ Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, 117198, Russia In this paper, we introduce new classes of simply continuous functions as generalization of continuous function. We obtain their characterizations, their basic properties and their relationships with other forms of generalized continuous functions between topological spaces. Key words and phrases: simply-opened set, simply-closed set, πΏ-set, semi-g-regular space, almost continuous function. 1. Introduction Let (π, π ) be a topological space. For a subset π of π, the closure, the interior, and the complement of π with respect to (π, π ) is denoted by ππ π, πππ‘ π, and π π , respectively. Any function is assumed to be mapping between two topological spaces. Recently there has been some interest in the notion of a simply-opened subset of topological space (π, π ). According to Neubrunnovia [1] a subset π of a space (π, π ) is called simply-opened if it is π = π βͺ π , where π is open and π is nowhere dense (= ππ€π, πππ‘ (ππ π ) = Ξ¦) subset of a space (π, π ). In [2] Ganster, Reilly and Vamanmurthy showed that a subset π of a space (π, π ) is simply-opened if and only if it is intersection of semi-opened and semi-closed subsets of a space (π, π ). A set π΅ of a topological space (π, π ) is simply-closed if and only if π΅ π is simplyopened. The class of all simply-opened subsets of the space (π, π ) will be denoted by ππ π(π). Semi-opened subsets of a topological space were defined by Levine [3]. Recall that π is called semi-opened [3] if π β ππ(πππ‘ π). A semi-closed set is a set, whose complement is semi-opened. A subset π of a topological space (π, π ) is called πΏ β π ππ‘(see [4]) if πππ‘(ππ π) β ππ(πππ‘ π). Intersection of all semi-closed sets containing π΄ is called the semi-closed of π΄ (see [5]) and is denoted by π ππ(π΄). By a semi-clopen set we mean the set, which is semi-opened and semi-closed at the same time. Recall that a subset A of a space (π, π ) is called regular open if π΄ = πππ‘ ππ π΄. The concept of semi-reopening was introduced by D. Andrijevic (see [6] and [7]). These sets were called π½ β open sets. A subset π of a space π is called semi-preopen if π β ππ(πππ‘(ππ(π))). A subset π΄ of a space π is said to be regular open (respectively regular closed)if π΄ = πππ‘(ππ(π΄)) (respectively π΄ = ππ(πππ‘(π΄))) (see [7]). In [8] the definition of semi-g-regular topological space was introduced. A topological space (π, π ) is said to be semi-g-regular if for each sg-closed set π΄ and each point π₯β / π΄, there exist disjoint semi-opened sets π, π β π such that π΄ β π and π₯ β π . A subset π΄ of a topological space (π, π ) is called semi-generalized closed (see [9]), briefly sg-closed, if π ππ(π΄) β π whenever π΄ β π and π is a semi-opened. A function π : (π, π ) ββ (π, π) is said to be semi-continuous [3] (resp. irresolute [10], perfectly contra-irresolute [11], semi-precontinuous [12], almost continuous [13] (written as a.c. π), contra-irresolute [11]) if the inverse image of every open subset (resp. semi-opened, semi-opened, open, regular open, semi-opened) subset 10 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 1, 2012. Pp. 9β13 of (π, π) is semi-opened (resp. semi-opened, semi-clopen, semi-preopen, open, semiclosed) in (π, π ). A function π : (π, π ) ββ (π, π) between two topological spaces is said to be pre-semi-opened [10] if and only if, the image of every semi-opened set in (π, π ) is semi-opened in (π, π). A function π : (π, π ) ββ (π, π) between two topological spaces is said to be semi-homeomorphism [10] if and only if π is one-to-one, onto, irresolute, and presemi-opened. Def 1. Let π : (π, π ) ββ (π, π) be a function. We say that f is a simplycontinuous(resp. ws-continuous, ss-continuous if the inverse image of every open(resp. simply-opened, simply-opened) set in π is simply-opened(resp. open, simply-opened) set in π. We give now an example to show that the simply-continuous function appears to be not necessary continuous function. Example 1. Let π : β ββ β where β is the real line with the usual topology and π (π₯) = 1 when (0 6 π < 1) and π (π₯) = 0 otherwise. It is very easy to see that π is simply-continuous but not continuous function. 2. Properties of simply-continuous, ss- continuous, wscontinuous functions Lemma 1 (see [14]). Every semi-opened subset of a space (π, π ) is a simplyopened set. Theorem 1. Every perfectly contra-irresolute function is a simply-continuous function. Proof. Let π : (π, π ) ββ (π, π) be a perfectly contra-irresolute function and let π be an open subset of π . Then by lemma 1 we have that π is simply-opened and since f is perfectly contra -irresolute function, π β1 (π) is semi-clopen subset of π (π β1 (π) = π β1 (π) β© π β1 (π)) as intersection of semi-opened and semi-closed and so, π β1 (π) is simply-opened by the definition of simply-opened in [2]. Theorem 2. Every ws-continuous is a semi-precontinuous function. Proof. Let π : (π, π ) ββ (π, π) be a ws-continuous function and letπ΄ be an open and so semi-opened subset of π . By lemma 1 π΄ is simply-opened subset of π , so π β1 (π΄) is open subset of π and so semi-opened, but it is very easy to see that, every semi-opened set is semi-preopen, so π β1 (π΄) is semi-preopen. The converse of above theorem need not be true by the following example: Example 2. Let π = {π, π, π}, π1 = {Ξ¦, π, π, π, π, π}, π = {1, 2, 3, 4}, π2 = {Ξ¦, 1, 1, 2, 1, 2, 3, π }. A function π : (π, π1 ) ββ (π, π2 ) is defined by π (π) = 1, π (π) = 3, π (π) = 2. Here ππ π(π) = {Ξ¦, {π}, {π}, {π, π}, {π, π}, {π, π}, π}. Then π is semi-precontinuous. But π is not ws-continuous since π β 1({1, 2}) = {π, π}, which is not open in π1 . Theorem 3. Every ws-continuous function is an almost continuous function. Proof. Let π : (π, π ) ββ (π, π) be a ws-continuous function and let π΄ be a regular open subset of π . Since every regular open set is open set, and so semi-opened subset of π , by lemma 1, π΄ is simply-opened subset of π , so π β1 (π΄) is an open subset of π. Jalal Hatem Hussein Al Bayati Results on Simply-Continuous Functions 11 Proposition 1 (see [15]). If π is a topological space and π is a dense subspace of π, then for any π΄ β π , π΄ is nowhere dense in π if and only if π΄ is nowhere dense in π. Theorem 4. Let (π, π ) be a topological space, (π0 , π0 ) be a subspace of (π, π ), π΄ β π0 β π and π0 β ππ π(π). If π΄ β ππ π(π) then π΄ β ππ π(π0 ). Proof. Since π΄ is simply-opened in (π, π ) then π΄ = π βͺ π , where π is open and π is nowhere dense subsets of π, so π΄ = π0 β©π΄ = π0 β©(π βͺπ ) = (π0 β©π)βͺ(π0 β©π ). But (π0 β© π) and (π0 β© π ) are open and nowhere dense subsets of the space (π0 , π0 ) respectively. Thus π΄ is simply-opened in (π0 , π0 ). Lemma 2 (see [14]). Let π and π be subsets of a space (π, π ) such that π is a simply-opened set and π is an open set. Then π β© π is a simply-opened set. Theorem 5. If π : (π, π ) ββ (π, π) is a simply-continuous function and π0 is an open set in π, then the restriction π /π0 : π0 ββ π is simply-continuous. Proof. Since π is simply-continuous, for any open set π in π , π β1 (π ) is a simplyopened set in π. Hence by lemma 2, we have that π β1 (π ) β© π0 is a simply-opened set in π. Since π0 is an open set, by theorem 8, (π /π0 )β1 (π ) = π β1 (π ) β© π0 is a simply-opened set in π0 , which implies that π /π0 is simply-continuous. Theorem 6. Let π : (π, π ) ββ (π, π) be a function between two topological spaces. Then the following statements are equivalent: (1) π is strongly simply-continuous; (2) if π₯ β π and V is simply-opened in π containing π (π₯), then there exists π simplyopened in π containing π₯ such that π (π ) β π . Proof ((1) =β (2)). Let π be simply-opened in π and π (π₯) β π . Since π is ss-continuous, π β1 (π ) is simply-opened in π and π₯ β π β1 (π ). Put π = π β1 (π ). Then π₯ β π and π (π ) = π (π β1 (π )) β π . ((2) =β (1)). Let π be simply-opened in π and π₯ β π β1 (π ). Then π (π₯) β π therefore, there exists a ππ₯ simply-opened in π such that π₯ β ππ₯ and π (ππ₯ ) β π , therefore π₯ β ππ₯ β π β1 (π ). This implies that π β1 (π ) is union of simply-opened sets, hence π β1 (π ) is simply-opened in π so π is strongly simply-continuous. Theorem 7. Let (π, π ) and (π, π) be two topological spaces. The function π : (π, π ) ββ (π, π) is simply-continuous function if and only if, for every closed subset π΅ of π , π β1 (π΅) is simply-closed in π. Proof. Necessity. If π : (π, π ) ββ (π, π) is simply-continuous function, then for every open subset π of π , π β1 (π) is simply-opened in π. If π΅ is any closed subset of π , then π΅ π is open. Thus π β1 (π΅ π ) is simply-opened, but π β1 (π΅ π ) = (π β1 (π΅))π so that π β1 (π΅) is simply-closed. Sufficiency. If for all closed subset π΅ of π , π β1 (π΅) is simply-closed in π, and if π is any open subset of π , then ππ is closed. Also, π β1 (ππ ) = (π β1 (π))π is simply-closed. Thus π β1 (π) is simply-opened. Theorem 8. Let π : (π, π ) ββ (π, π) be two topological spaces. The one-to-one and onto function π : (π, π ) ββ (π, π) is ws-continuous (resp. ss-continuous) if and only if, for every simply-closed subset π΅ of π , π β1 (π΅) is closed(resp. simply-closed) in π. Proof. In the same way of proving theorem 7 we can proof this theorem. Lemma 3 (see [8]). For a topological space (π, π ) the following statements are equivalent: 12 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 1, 2012. Pp. 9β13 (1) (π, π ) is semi-g-regular; (2) every sg-closed set π΄ is an intersection of semi-clopen sets. By the upper Lemma we have the following proposition: Proposition 2. A subset π΅ of a semi-g-regular topological space is simply-opened if and only if it is sg-closed set. Theorem 9. Let π : (π, π ) ββ (π, π) be a simply-continuous (resp. ws-simply continuous, ss-continuous) function between two semi-g-regular topological spaces. Then the inverse image of every sg-closed set is sg-closed set. Proof. Let π : (π, π ) ββ (π, π) be a simply-continuous (resp. ws-simply continuous, ss-continuous) function between two semi-g-regular topological spaces and let π΄ be a sg-closed subset of π and π a semi-g-regular topological space, then, by Proposition 2, π΄ is simply-opened subset of π , so π β1 (π΄) is simply-opened subset of π whatever π be. But π is semi-g-regular topological space, so π β1 (π΄) is sg-closed set. Lemma 4 (see [11]). Let π : (π, π ) ββ (π, π) be a map. Then the following conditions are equivalent: (i) π is contra-irresolute; (ii) the inverse image of each semi-closed set in π is semi-open in π. Theorem 10. Every contra-irresolute function is simply-continuous. Proof. Let π΄ be open in (π, π) where π : (π, π ) ββ (π, π) is contra-irresolute function, so π΄π is closed and so semi-closed and by the contra-irresoluteness, lemma 4, of the function we have, π β1 (π΄π ) to be semi-open and so, by lemma 1 is simply-closed in (π, π ), so we have that (π β1 (π΄π ))π = π β1 (π΄) is simply-opened. References 1. Neubrunnovia. On Transfinite Sequences of Certain Types of Functions // Acta Fac. Rer. Natur. Univ. Com. Math. β 1975. β Vol. 30. β Pp. 121β126. 2. Ganster M., Reilly I. L., Vamanmurthy M. K. Remarks on Locally Closed Sets // Mathematical Pannonica. β 1992. β Vol. 3(2). β Pp. 107β113. 3. Levine N. Semi-Open Sets and Semi-Continuity in Topological Space // Amer. Math. Monthly. β 1963. β Vol. 70. β Pp. 36β41. 4. Chattopadhyay C., Bandyopadhyay C. On Structure of πΏ β π ππ‘π . β 1991. 5. Crossly S. G., Hildebrand S. K. Semi-Closure // Texas J. Sci. β 1971. β Vol. 22. β Pp. 99β112. 6. Andrijevic D. Semi-Preopen Sets // Ibid. β 1986. β Pp. 24β32. 7. Stone M. H. Applications of the theory of Boolean Rings to General Topology // TAMS. β 1937. β Vol. 41. β Pp. 375β381. 8. Ganster M., Jafari S., Navalagi G. B. On Semi-g-Regular and Semi-g-Normal Spaces // Demonstration Math. β 2002. β Vol. 35. β Pp. 414β421. 9. Bhattacharyya P., Lahiri B. K. Semi-Generalized Closed Sets in Topology // Indian J. of Math. β 1987. β Vol. 29. β Pp. 375β382. 10. Crossly S. G., Hildebrand S. K. Semi-Topological Properties // Fund. Math. β 1974. β Vol. 74. β Pp. 233β254. 11. Cueva M. C. Weak and Strong Forms of Irresolute Maps // Internat. J. Math. Math. Sci.Vol. β 2000. β Vol. 23. β Pp. 253β259. 12. Navalagi G. B. Semi-Precontinuous Functions and Properties of Generalized SemiPreclosed Sets in Topological Spaces // IJMMS Internat. J. Math. Math. Sci. β 2002. β Vol. 29. β Pp. 85β98. 13. Singal M. K., Singal A. R. Almost-Continuous Mappings // Yokohama Math. J. β 1968. β Vol. 15. β Pp. 63β73. Jalal Hatem Hussein Al Bayati Results on Simply-Continuous Functions 13 14. Kljushin V. L., Jalal Hatem Hussein A. B. On Simply-Open Sets // Bulletin of PFUR. Series βMathematics. Informatics. Physicsβ. β 2011. β Vol. 3. β Pp. 34β 38. 15. Maxim R. Burke A. W. M. Models in which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set // Canadian Journal of Mathematics. β 2005. β Vol. 57. β Pp. 1129β1144. Π£ΠΠ 513.831 ΠΠ΅ΠΊΠΎΡΠΎΡΡΠ΅ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΡ ΠΎ ΠΏΡΠΎΡΡΠΎ-Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΡΡ ΡΡΠ½ΠΊΡΠΈΡΡ ΠΠ»Ρ ΠΠ°ΡΡΠΈ ΠΠΆΠ΅Π»Π°Π» Π₯Π°ΡΠ΅ΠΌ Π₯ΡΡΡΠ΅ΠΉΠ½ ΠΠ°ΡΠ΅Π΄ΡΡ Π²ΡΡΡΠ΅ΠΉ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΠΊΠΈ Π ΠΎΡΡΠΈΠΉΡΠΊΠΈΠΉ ΡΠ½ΠΈΠ²Π΅ΡΡΠΈΡΠ΅Ρ Π΄ΡΡΠΆΠ±Ρ Π½Π°ΡΠΎΠ΄ΠΎΠ² 117198, ΡΠ». ΠΠΈΠΊΠ»ΡΡ ΠΎ-ΠΠ°ΠΊΠ»Π°Ρ, Π΄.6, ΠΠΎΡΠΊΠ²Π°, Π ΠΎΡΡΠΈΡ Π ΡΡΠΎΠΉ ΡΡΠ°ΡΡΠ΅ Π²Π²Π΅Π΄Π΅Π½Ρ ΡΡΠΈ ΠΊΠ»Π°ΡΡΠ° ΡΡΠ½ΠΊΡΠΈΠΉ, Π½Π°Π·ΡΠ²Π°Π΅ΠΌΡΡ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²Π΅Π½Π½ΠΎ ΠΏΡΠΎΡΡΠΎ-Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΡΠΌΠΈ, ΡΠΈΠ»ΡΠ½ΠΎ ΠΏΡΠΎΡΡΠΎ-Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΡΠΌΠΈ ΠΈ ΡΠ»Π°Π±ΠΎ ΠΏΡΠΎΡΡΠΎ-Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΡΠΌΠΈ ΠΊΠ°ΠΊ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΡ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΡΡ ΡΡΠ½ΠΊΡΠΈΠΉ. ΠΠΎΠ»ΡΡΠ΅Π½Ρ ΠΈΡ Ρ Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ, ΠΎΡΠ½ΠΎΠ²Π½ΡΠ΅ ΡΠ²ΠΎΠΉΡΡΠ²Π° ΠΈ ΠΎΠΏΠΈΡΠ°Π½Ρ ΠΈΡ ΡΠ²ΡΠ·ΠΈ Ρ Π΄ΡΡΠ³ΠΈΠΌΠΈ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΡΠΌΠΈ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½ΡΡ ΠΎΡΠΎΠ±ΡΠ°ΠΆΠ΅Π½ΠΈΠΉ ΡΠΎΠΏΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ². ΠΠ»ΡΡΠ΅Π²ΡΠ΅ ΡΠ»ΠΎΠ²Π°: ΠΏΡΠΎΡΡΠΎΠΎΡΠΊΡΡΡΡΠ΅ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°, ΠΏΡΠΎΡΡΠΎΠ·Π°ΠΌΠΊΠ½ΡΡΡΠ΅ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π°, ΠΏΠΎΠ»Ρg-ΡΠ΅Π³ΡΠ»ΡΡΠ½ΠΎΠ΅ ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²ΠΎ, ΠΏΠΎΡΡΠΈ Π½Π΅ΠΏΡΠ΅ΡΡΠ²Π½Π°Ρ ΡΡΠ½ΠΊΡΠΈΡ.