Yarmouk University, Department of Mathematics, P.O Box 566 zip code: 21163, Irbid, Jordan.

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1 Acta Scientiarum ε-open sets SSN printed: SSN on-line: Doi: /actascitechnolv35i Talal Ali Al-Hawary Yarmouk University, Department of Mathematics, PO Box 566 zip code: 21163, rbid, Jordan ABSTRACT n th paper, we introduce the relatively new notion of ζ -open subset which strictly weaker than open We prove that the collection of all ζ -open subsets of a space forms a topology that finer than the original one Several characterizations properties of th class are also given as well as connections to other well-known "generalized open" subsets Keywords: ζ -open, Countable set, Anti locally countable space Conjuntos aberto ζ RESUMO Este artigo apresenta um novo conceito de subconjunto aberto ζ que é muito ma fraco que o aberto Foi comprovada que a coleta de todos os subconjuntos abertos ζ de um espaço forma uma topologia ma fina do que a original Várias caracterizações e propriedades desta classe são também apresentadas, bem como conexões com outros bem conhecidos subconjuntos abertos generalizados Palavras-chave: aberto ζ, conjunto contável, espaço contável anti-localmente ntroduction Let ( X, ) be a topological space (or simply, a space) f A X, then the closure of A, the interior of A the derived set of A will be denoted by Cl (, nt( d (, respectively f no ambiguity appears, we use o A, A ' A instead A subset A X called semi-open (simply, SO) (LEVNE, 1963) if there exts an open set O such that O A O Clearly A a semi-open set if only if A Cl( nt( ) A complement of a semi-open set called semi-closed (simply, SC) The semiinterior of A the union of all semi-open subsets contained in A denoted by snt ( A called preopen( simply, PO) (MASHHOUR et al, 1982a b) A nt( Cl( ) A called α open (NJASTAD, 1965) if A nt( Cl( nt( )) β open (ABD EL- MONSEF et al, 1983) if A Cl( nt( Cl( )) Finally, A called regular-open (simply, RO) (WL- DEEP et al, 1983) if A = nt( Cl( ) Complements of regular-open sets are called regular-closed (simply, RC) The collection of all SO (resp, PO, RO, RC, α open β open) subsets of X denoted SO ( X, ) (resp, PO( X, ), RO( X, ), RC( X, ), α ( X, ) β ( X, ) We remark that α ( X, ) a topological space α ( X, ) = SO( X, ) PO( X, ) A space ( X, ) called locally countable (P-space, anti locally countable, respectively) if each x X has a countable neighborhood (countable intersections of open subsets are open, non-empty open subsets are uncountable, respectively) For the preceding notions, the reader referred to Crossley Hildebr (1971), Ganster Reilly (1990), Mashhour et al (1982), Munkres (1975) Tong (1986) n th paper, we introduce the relatively new notions of ζ -open, which weaker than the class of open subset n section 2, we also show that the collection of all ζ -open subsets of a space ( X, ) forms a topology that finer than we investigate the connection of ζ -open notion to other classes of "generalized open" subsets as well as several characterizations of ζ -open ζ -closed notions via the operations of interior closure n section 3, several interesting properties constructions ofζ -open subsets are dcussed in the case of anti locally countable spaces Acta Scientiarum Technology Maringá, v 35, n 1, p , Jan-Mar, 2013

2 112 Al-Hawary ζ -open set We begin th section by introducing the notion of ζ -open ζ -closed subsets Definition 1 A subset A of a space ( X, ) called ζ -open if for every x A, there exts an open subset U X containing x such that U \ snt( countable The complement of an ζ - open subset called ζ -closed Clearly every open set ζ -open, but the converse needs not be true Example 1 Let X = { b} = { φ, X,{ a}} Set A = {b} Then A ζ -open but not open Another interesting example for the infinite case giving next Example 2 Consider the real line R with the topology = { U R : R \ U finite or 0 R \ U} set A = R \ Q {0} Then A not open while it ζ -open Next, we show that the collection of all ζ -open subsets of a space ( X, ) forms a topology ζ that contains Theorem 1 f ( X, ) a space, then ( X, ζ ) a space such that ζ Proof We only need to show ( X, ζ ) a space Clearly since φ X are open, they are ζ -open f A, B ζ x A B, then there ext open sets U,V in X both containing x such that U \ snt( V \ snt( are countable Now x U V for every y ( U V ) \ snt( A = ( U V ) \ ( snt( snt( ) either y U \ snt( or y V \ snt( Thus ( U V ) \ snt( A U \ snt( or ( U V ) \ snt( A U \ snt( thus ( U V ) \ snt( A countable Therefore, A B ζ f { A α : α Δ} a collection of ζ -open subsets of X, then for every x A α, x Aβ for some β Δ Hence there exts an open subset U of X containing x such that U \ snt( countable Now as U \ snt( A ) U \ snt( U \ snt(, α U \ snt( Aα ) countable hence Aα ζ Corollary 1 f ( X, ) a p-space, then = ζ Next we show that ζ -open notion independent of both PO SO notions Example 3 Consider R with the stard topology Then Q PO but not ζ -open Also [0,1] SO but not ζ -open Example 4 n Example 1, {b} ζ -open but neither PO nor open Next we characterize ζ when X a locally countable space Theorem 2 f ( X, ) a locally countable space, then ζ the dcrete topology Proof Let A X x A Then there exts a countable neighborhood U of x hence there exts an open set V containing x such that V U Clearly V \ snt( U \ snt( U thus V \ snt( countable Therefore A ζ - open so ζ the dcrete topology Corollary 2 f ( X, ) a countable space, then ζ the dcrete topology The following result, in which a new characterization of ζ -open subsets given, will be a basic tool throughout the rest of the paper Lemma 1 A subset A of a space X ζ -open if only if for every x A, there exts an open subset U containing x a countable subset C such that U C snt( Proof Let A ζ x A, then there exts an open subset U containing x such that U \ snt( ) countable Let C = U \ snt( = U ( X \ snt( ) Then U C snt( Conversely, let x A Then there exts an open subset U containing x a countable subset C such that U C snt( Thus U \ snt( = C countable The next result follows easily from the definition the fact that the intersection of ζ -closed sets again ζ -closed Lemma 2 A subset A of a space X ζ -closed if only if Cl ζ ( = A We next study restriction deletion operations Theorem 3 f A ζ -open subset of X, then ζ A ( ζ Acta Scientiarum Technology Maringá, v 35, n 1, p , Jan-Mar, 2013

3 ζ -Open sets 113 Proof Let G ζ A Then G = H for some ζ -open subset H For every x G, there ext V H, V A containing x countable sets D H D A such that VH DH snt(h ) VA DA snt( Therefore, x A ( VH V A, DH DA countable A ( V H V ) ( D A H D ( VH V ( X DH ) ( X D = ( VH DH ) ( VA D snt( H ) snt( = snt( H = snt( G) snt ( G) Therefore, G ( A ) ζ Corollary 3 f A open subset of X, then ζ A ( ζ n the next example, we show that if A in the preceding Theorem not ζ -open, then the result needs not be true Example 5 Consider R with the stard topology let A = R \ Q Then A not ζ -open so not open As (0,1) ζ -open, then D = ( 0,1) ζ A while if D ( ζ then for every x D, here exts U A a countable C A such that U C snt(d) = φ Thus U C hence U countable which a contradiction n the next example, we show that ( A ) ζ needs not be a subset of ζ A Example 6 Consider R with the stard topology, A = Q B=(0,2) f B ζ A, then B = D for some D ζ which impossible as 2 B A On the other h to show B ( ζ, let x B f x A, pick q 1, q 2 A such that < q < x < q 2 let < U = ( q1, q2) Then x U φ B = snt( f x A, then q q \ A such that A 1, 2 R 0 < q 1 < x < q2 < 2 let U = ( q1, q2) Then x U φ B = snt( Thus in both cases B ( ζ Lemma 3 f X a Lindelof space, then A- snt( countable for every closed subset A ζ Poof Let A be a closed set such that A ζ For every x A, there exts an open set V x containing x such that V x snt( countable Thus { V x : x A} an open cover for A as A lindelof, it has a countable subcover { V n : n N} Now A snt( = ( Vn snt( ) which n N countable Corallary 4 f X a second countable space, then A-snt( countable for every closed subset A ζ Theorem 4 Let ( X, ) be a space C X ζ -closed Then Cl ( C) K B for some closed subset K a countable subset B Proof Let C be ζ -closed Then X-C ζ -open hence for every x X C, there exts an open set U containing x a countable set B such that U B snt( X C) X Cl ( C) Thus Cl ( C) X ( U X ( U ( X ) X (( X U ) ( X U ) B Letting K=X-U Then K closed such Cl ( C K B ) Anti-locally countable spaces n th section, several interesting properties constructions of ζ -open subsets are dcussed in case of anti locally countable spaces Theorem 5 A space ( X, ) anti locally countable if only if X, ) anti locally ( ζ countable Proof Let A ζ x A Then by Lemma 1, there exts an open subset U X containing x a countable C such that U C snt( Hence snt( uncountable so A The converse follows from the fact that every open subset ζ -open Corollary 5 f ( X, ) anti locally countable space A ζ -open, then Cl ( = Cl ζ ( Proof Clearly Cl( Cl ζ ( On the other h, let x Cl( B be an ζ -open subset containing x Then by Lemma 1, there exts an open subset V containing x a countable set C such that V C snt( Thus ( V C) snt( ( V C snt( As x V x Cl(, V = φ then as V A are ζ - open, V ζ -open as X ani locally countable, V uncountable so Acta Scientiarum Technology Maringá, v 35, n 1, p , Jan-Mar, 2013

4 114 Al-Hawary V C Thus B uncountable as it contains the uncountable set snt( Therefore, B φ which means x Cl ζ ( By a similar argument, we can easily prove the following result: Corollary 6 ( X, ) anti locally countable A ζ -closed, nt ( = nt ζ ( Theorem 6 Let ( X, ) countable space Then α( X, ) α( X, ζ ) Proof f A α ( X, ), then A nt ( Cl ( nt ( )) by Corollary 5, A nt( Cl ζ ( nt( )) Now by Corollary 6 as Cl ( nt( ) ζ -closed, A nt ( Cl ( nt ( )) which means ζ ζ ζ A α( X, ζ ) The converse of the preceding result needs not be true as shown next Example 7 Consider R with the stard topology let A = R \ Q Then A α( X, ζ ), but A α ( X, ) Similarly, one can show that in an anti locally countable space, β ( X, ζ ) β ( X, ) Theorem 7 Let ( X, ) countable space Then d ( = d ζ ( for every subset A X Proof f x d( V any ζ -open subset containing x, then there exts an open subset U containing x a countable C such that U C snt( V ) V Thus ( U C) ( A { snt( V ) ( A { V ( A { o as x d( V open containing x, we have V o ( A { φ so V ( A { φ Therefore x d ζ ( The converse obvious as every open subset ζ -open Theorem 8 Let ( X, ) countable space Then RO( X, ζ ) = RO( X, ) Proof f A RO( X, ), then A = nt( Cl( ) by Corollary 5, A = nt( Cl ζ ( ) Now by Corollary 6 as Cl ζ ( ζ -closed, A = ntζ ( Cl ζ ( ) which means A RO( X, ζ ) Conversely, if A RO X, ), then ( ζ A = nt ( Cl ( ) Then as A ζ -open, by ζ ζ Corollary 5, A = nt ( Cl ( ) ζ as Cl ( ζ -closed being a closed set, then A = nt( Cl( ) which means A RO( X, ) The converse of the preceding result needs not be true as shown next Example 8 Let X={b,c,d,e} = { φ, X,{ a},{ b},{ b, c},{ b, c, d}} Then ( X, ) not an anti locally countable space such that RO( X, ) = { φ, X} while RO ( X, ζ ) = We end th section by showing that ( X, ζ ) Urysohn when ( X, ) an anti locally countable space Definition 2 [8] A space ( X, ) Urysohn if for every two dtinct points x y in X, there exts two open subsets U V such that x U, y V Cl ( U ) Cl ( V ) = φ Corollary 7 Let ( X, ) countable space that Urysohn Then ( X, ζ ) Urysohn Proof f x y in X, then there exts U, V such that x U, y V Cl ( U ) Cl ( V ) = φ By Corollary 5, Cl ( U ) Cl ( V ) = Cl ( U ) Cl ( V ) = φ ζ ζ Conclusion The relatively new notions of ζ -open, which weaker than the class of open subset was introduced t showon that the collection of all ζ -open subsets of a space ( X, ) forms a topology that finer than connections of ζ -open notion to other classes of "generalized open" subsets as well as several characterizations of ζ -open ζ -closed notions via the operations of interior closure were investigated Several interesting properties constructions ofζ -open subsets were dcussed in the case of anti locally countable spaces Acknowledgements The authors would like to thank the referees for useful comments suggestions References CROSSLEY, S; HLDEBRAND, S Semi-closure Texas Journal of Science, v 22, n 2, p , 1971 CROSSLEY, S; HLDEBRAND, S Semi-topological properties Fundamenta Mathematicae, v 71, n 1, p , 1972 Acta Scientiarum Technology Maringá, v 35, n 1, p , Jan-Mar, 2013

5 ζ -Open sets 115 ABD EL-MONSEF, M E; EL-DEEB, S N; MAHMOUD, R A β -open sets β -continuous mappings Bulletin of the Faculity of Science, v 12, n 1, p 77-90, 1983 GANSTER, M; RELLY, A decomposition of continuity Acta Mathematica Hungaric v 56, n 3-4, p , 1990 LEVNE, N Semi-open sets semi-continuity in topological spaces American Mathematical Monthly, v 70, n 1, p 36-41, 1963 MASHHOUR, A; HASANEN, ; EL-DEEB, S A note on semi-continuity precontinuity ndian Journal of Pure Applied Mathematics, v 13, n 10, p , 1982a MASHHOUR, A S; ABD EL MONSEF, M E; EL DEEP, S N On precontinuous weak precontinuous mappings Proceedings Mathematical Physical Engineering Sciences, v 53, n 53, p 47-53, 1982b MUNKRES, J Topology a first course New Jersey: Prentice Hall nc, 1975 NJASTAD, O On some classes of nearly open sets Pacific Journal of Mathematics, v 15, n 3, p , 1965 TONG, J A Decomposition of continuity Acta Mathematica Hungaric v 48, n 1-2, p 11-15, 1986 TONG, J On decomposition of continuity in topological spaces Acta Mathematica Hungaric v 54, n 1-2, p 51-55, 1989 WL-DEEP, S N; HUSANEN, A; MASHHOUR, A S; NOR, T On p-regular spaces Bulletin Math'ematique de la Soci'et'e des Sciences Math'ematiques de la R'epublique Socialte de Roumanie, v 27, n 75, p 51-55, 1983 Received on December 21, 2011 Accepted on April 25, 2012 License information: Th an open-access article dtributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, dtribution, reproduction in any medium, provided the original work properly cited Acta Scientiarum Technology Maringá, v 35, n 1, p , Jan-Mar, 2013