(X0, Q(X0)) is the T0-identification space of (X,T) and (Xs0,Q(Xs0)) semi-t0-identification space of (X,T), and that if (X,T) is s-regular and

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1 Irnat. J. M. Math. Si. Vol. 4 No. (1981) SEMI SEARATION AXIOMS AND HYERSACES CHARLES DORSETT Departent of Matheatics, Texas A&M University College Station, Texas (Received April 21, 1980 and in revised for Septeber 4, 1980) ABSTRACT. In this paper exaples are given to show that s-regular and s-noral are independent; that s-noral, and s-regular are not sei topological properties; and that (S(X),E(X)) need not be sei-t I even if (X,T) is copact, s-noral, Also, it is shown that for each space (X,T), s-regular, sei-t2, and T 0. (S(X),E(X)), (S(XO),E(Xo)), and (S(Xs0),E(Xs0)) are hoeoorphic, where (X0, Q(X0)) is the T0-identification space of (X,T) and (Xs0,Q(Xs0)), is the sei-t0-identification space of (X,T), and that if (X,T) is s-regular and then (S(X), E(X)) is sei-t 2. KEF WORDS AND HRASES. Si open sets, si topological propzies, and 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 54AI0, 54B20. i. INTRODUCTION. Sei open sets were first defined and investigated by Levine [i] in DEFINITION I.i. Let (X,T) be a space and let A -X. Then A is sei open, denoted by A E SO(X,T),I there exists U E T such that U GA GU. Since 1963 sei open sets have been used to define and investigate any new topological properties. Maheshwari and rasad [2], [3], and [4] generalized T i, i 0, i, 2, regular, and noral to sei-t i, i 0, 1,2, s-regular, and s-noral,

2 446 C. DORSETT by replacing the word open in the definitions of Ti, i 0, i, 2, regular, and noral by sei open, respectively. Except for s-noral and s-regular, the relationships between these separation axios have been deterined. In this paper, the relationship between s-noral and s-regular is deterined, and sei topological properties and hyperspaces are further investigated. 2. s-regular -NORMAL AND SEMI TOOLOGICAL ROERTIES. Maheshwarl and rasad [4] gave an exaple showing that s-noral does not iply s-regular. That exaple can be cobined with the following exaple to show that s-regular and s-noral are independent. EXAMLE 2.1 Let N denote the natural nubers, let T be the discrete topology on N, let e be the ebedding ap of (N,T) into {If f 6 C*(N,T)}, and let (8N, W) (e(n), e) denote the Stone-ech copactification of (N,T). Fro Willard s book [5], (8N, W) is extreely disconnected, e(n) is open in 8N, and B BN- e(n) is infinite. For each p N let N {n N n < p}. Since for p each p N, there exists a function f N / B x W such that (i) if % {2,...,p}, then fi is an extension of fi-l (2) x i 0 i for all i Np, (3) if i, j Np, then 0-- i n 0--j # iff i ], and (4) B- i 0 i is infinite, then there exists a sequence {(Xn,0 )}n BW such that x 0 for all n N and n ; n n [ # iff n. Let {a } he a sequence such that {a n N}C, SN n n n6 n and an --a iff n, let V {Xn In N} U nn{un 0n e(n)}3 43N, let W 1 be the relative topology on V, and let X V D{an n N}. Since U I is countably infinite, then U 1 {Yn}n N,where Yi YJ iff i j. For each i N, let B i {0 c X- n {{x-i n N} b{an] n # i}) 0 IUn W1 n N, Xn 0 Un except for finitely any n 6N, and a i, Yl for all 0}, and let W2 =IN Bi" Then W I b W2 is a base for a topology S on X, (X,S) is s-regular, sei-t2, and TO, and (X S) is not s-noral since A {a n N} n and C {x n N } are disjoint closed sets and there do not exist disjoint n sei open sets containing A and C, respectively. Seihoeoorphiss and sei topological properties were first introduced and investigated by Crossley and Hildebrand [6].

3 SEMI SEARATION AXIOMS AND HYERSACES 447 DEFINITION 2.1. A i-i function fro one space onto another space is a seihoeoorphis iff iages of sei open sets are sei open. sei open sets are sei open and inverses of A property of topological spaces preserved by seihoeoorphiss is called a sei topological property. Exaple 1.5 in [6], which was used to show that noral and regular are not sei topological properties, also shows that s-noral and s-regular are not sei topological properties. Clearly, sei-ti, i 0, i, 2, are sei topological properties. 3. HYERSACES AND SEMI SEARATION AXIOMS DEFINITION 3.1. Let (X,T) be a topological space, let A cx, and define S(X), S(A), and I(A) as follows: S(X) {F cx F is nonepty and closed}, S(A) {F S(X) F CA}, and I(A) {F S(X) F A }. Denote by E(X) the sallest topology on S(X) satisfying the conditions that if G 6 T, then S(G) 6 E(X) and I(G) E(X). Then (S(X), E(X)) is called a hyperspace [7]. Michael [8] showed that for a space (X,T), B {<G I Gp> p N and Gi T for all i Np {l,j..,p}} is a base for E(X), where N is the {F S(X) F c and i- Gi natural nubers and <G I G > p <Gi>iffiI F G i # for all i Np}, and observed that for each space (X,T), (S(), E(X)) is T O Since T O iplies sei-t0, then for each space (X,T), (S(X), E(X)) is sei-t 0. The following exaple shows that (S(X),E(X)) need not be sei-t 1 even if (X,T) is copact, s-noral, s-regualr, sei-t2, and T 0. EXAMLE 3.1 Let X {a,b,c,d} and T {X,,{b},{d},{b,d},{a,b,d},{b,c,d}}. Then (S(X), E(X)) is not sei-t since 1 {a,b,c}, X S(X) such that {a,b,c} # X and there does not exist a sei open set containing {a,b,c} and not X. In Willard s book [5], T0-identification spaces are discussed. DEFINITION 3.2 Let R be the equivalence relation on a space (X,T) space of (X,T) is defined by xry iff {x-- {-. Then the T0-1dentification (X0, Q(X0)), where X 0 is the set of equivalence classes of R and Q() decoposition topology on X 0, which is T 0. is the

4 448 C. DORSETT This author [9] used T0-identification spaces to show that hyperspaces of spaces, spaces which were first defined and investigated by Davis [i0], are T I. DEFINITION A space (X,T) is R iff for each 0 0 E T and x E 0, {x) c 0. Since T 1 iplies sei-tl, then the hyperspace of each space is sei-t I. Sei open sets were used by Crossley and Hildebran [ii] to define and investigate sei closed sets and sei closure. DEFINITION 3.4. Let (X,T) be a space and let A, B c X. Then A is sei closed iff X-A is sei open and the sei closure of B, denoted by scl B, is the intersection of all sei closed sets containing B. This author [12] used sei closure to define and investigate sei-t 0 identification spaces. DEFINITION 3.5. Let R be the equlvalence relatlon on a space (X,T) defined space of (X,T) is by xry iff scl{x} scl{y}. Then the sei-t0-identification (Xs0 Q(Xs0), where XS0 is the set of equivalence classes of R and Q(Xs0) decoposition topology on XS0 which is senti-t 0, is the This author [13] and [12] showed that the natural ap : (X,T) / (X0,Q(Xo)) -1 is continuous, closed, open, onto, and ((0)) 0 for all 0 T and that the natural ap S: (X,T) / (Xso, Q(Xs0)) is continuous, closed, open, onto, and I(s(0)) 0 for all 0 S0(X,T). These results are used to obtain the following result. THEOREM 3.1. For a space (X,T),(S(X),E(X)), (S(X0),E(X0)), and (S (Xs0), E (Xs0)) are hoeoorphic. ROOF: Let f: (S(X),E(X)). / (S(Xo),E(X0)) and let fs: (S(X),E(X)) / (S(Xs0),E(Xs0)) defined by f(f) (F) and fs(f) S(F). Then f and fs are hoeoorphlss. THEOREM 3.2. If (X,T) is R0,G q T, and F S(X) such that F i then S(G) S(G) and F E l(g). ROOF: Since S(G) c S(G), which is closed, then S(G) c S(G). Let p Then A c G and A S(G). Let <Bi>i=1 B such that A <Bi>i=1.

5 SEMI SEARATION AXIOMS AND HYERSACES 449 # A Bic G B i for all i Np, which iplies G ib i for all i Np. i} cg.rb for all i ie N and e i --N {xi } E S(G)I Thus <Bi>i=l. AE (G) and S(G) c S(G), which iplies For each 16 N let x G i.b Then _{x i. S(G) S(G). Let <Ui>i=1 E 8 such that F <Ui>=I. Then Fc i UIE T and each i 6 N let Yi Bi" en } U Hence i } I(G) C, <Ui>i.1., THEOREM If (X,T) is s-regular and then (S(X),E(X)) is sei-t 2. ROOF: Let A, BE S(X) such that A # B. Then A- B # or B A say B- A # Let x E B- A. Then there exists disjoint sei open sets 0 and W such that x E 0 and A cw. Let U, V ET such that Uc 0 cu and V cw cv. Then I(U) and S(V) are disjoint open sets, B E I(U), and A E S(V) S(V), which iplies S(V) b {A} and I(U) U {B} are disjoint sei open sets. Maheshwari and rasad [4] showed that every s-noral space is s-regular. This result can be cobined with Theore 3.3 to obtain the following corollary. COROLLARY 3.1. If (X,T) is s-noral and R 0, then (S(X),E(X)) is sei-t 2.

6 50 C. DORSETT REFERENCES i. LEVINE, N., Sei Open Sets and Sei Continuity in Topological Spaces, Aer. Math. Monthly, 7_0 (1963), MAHESHWARI, S. and RASAD, R., Soe New Separation Axios, Ann. Soc. Sci. Bruxelles, 89 (1975), MAHESHWARI, S. and RASAD, R., On s-regular Spaces, Glasnik Mat. Ser. III, i0 (30) (1975), MAHESHWARI, S. and RASAD, R., On s-noral Spaces, Bull. Math. de la Soc. Sci. Math. de la R. S. de Rouanie, T 22(70) (1978), WILLARD, S., General Topology, Addlson-Wesley ublishing Copany, CROSSLEY, S. and HILDEBRAND, S., Sei-Topological roperties, Fund. Math., 74 (1972), FRINK, 0., Topology in Lattices, Trans. Aer. Math. Soc., 5_i (1942), MICHAEL, E., Topologies on Spaces of Subsets, Trans Aer. Math. Soc., 7 1 (1951) DORSETT, C., T^-Identification Spaces and Hyperspaces, Ann. Soc. Sci. Bruxelles? 9 1 (1977), i0. DAVIS, A., Indexed Systes of Neighborhoods for General Topological Spaces, Aer. Math. Monthly, 68 (1961), ii. CROSSLEY, S. and HILDEBRAND, S., Sei-closure, Texas J. Science, 22 (1970), DORSETT, C., Sei-T^-Identification Spaces, Sei-Induced Relations, and Sei Separation Axios, Accepted by the Bull Calcutta Math. Soc. 13. DORSETT C., To-Identification Spaces and R 1 18i2) (1978), Spaces, Kyungpook Math. J.,

7 Matheatical robles in Engineering Special Issue on Modeling Experiental Nonlinear Dynaics and Chaotic Scenarios Call for apers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to iprove the operational characteristics of a device or syste. Keying, saturation, hysteretic phenoena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approxiation, around equilibriu points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenoena, engineers started using analytical tools fro Qualitative Theory of Differential Equations, allowing ore precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynaical systes and chaos started to be part of the andatory set of tools for design engineers. This proposed special edition of the Matheatical robles in Engineering ais to provide a picture of the iportance of the bifurcation theory, relating it with nonlinear and chaotic dynaics for natural and engineered systes. Ideas of how this dynaics can be captured through precisely tailored real and nuerical experients and understanding by the cobination of specific tools that associate dynaical syste theory and geoetric tools in a very clever, sophisticated, and at the sae tie siple and unique analytical environent are the subject of this issue, allowing new ethods to design high-precision devices and equipent. Authors should follow the Matheatical robles in Engineering anuscript forat described at rospective authors should subit an electronic copy of their coplete anuscript through the journal Manuscript Tracking Syste at ts.hindawi.co/ according to the following tietable: Guest Editors José Roberto Castilho iqueira, Telecounication and Control Engineering Departent, olytechnic School, The University of São aulo, São aulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Mateática Aplicada e Coputação (LAC), Instituto Nacional de esquisas Espaciais (INE), São Josè dos Capos, São aulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Center for Applied Dynaics Research, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due Deceber 1, 2008 First Round of Reviews March 1, 2009 ublication Date June 1, 2009 Hindawi ublishing Corporation