اولت ارص من نوع -c. الخلاصة رنا بهجت اسماعیل مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد 22 (3) 2009

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1 مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد 22 (3) 2009 الت ارص من نوع -C- - جامعة بغداد رنا بهجت اسماعیل قسم الریاضیات - كلیة التربیة- ابن الهیثم الخلاصة قمنا في هذا البحث بتعریف نوع جدید من الت ارص اسمیناه "الت ارص من نوع "-c كذلك قمنا بد ارسة بعض خ اوصه والعلاقة بینه وبین الت ارص اولت ارص من نوع - اولت ارص من نوع -c.

2 - C-Compactness R. B. Esmaeel Department of Mathematcs,College of Educaton Ibn-Al-Hatham, Unversty of Baghdad Abstract In ths paper, we ntroduce a new type of compactness whch s called "-ccompactness". Also, we study some propertes of ths type of compactness and the relatonshps among t and compactness, -compactness and c-compactness. 1. Introducton and Prelmnares A topologcal space (X,) s sad to be c-compact space f for each closed set A X, each open cover of A contans a fnte subfamly W such that {cl v: v W} covers A, [1]. In 1965, O.Njasted [2] ntroduced "-open set" n topology [A subset A of a topologcal space X s sad to be "-open set f A nt (cl(nt(a)))], and he proved that the famly of all "-open sets n a space (X,) s a topology on X, whch s fner than and denoted by. -open sets are dscussed n [3], [4], [5], some concepts were studed as follows:. The complement of an -open set s called -closed set and the ntersecton of all -closed sets contans a set A whch s called the -closure of A and denoted by -cla. So, -cla s an -closed set and proved (-cla = A ff A s -closed set).. If A be a subset of a topologcal space X the -derved of A s the set of all elements x satsfes the condton, that for every -open set V contans x, mples V\{x}A. In 1985, the term of "-compactness" was used for the frst tme by S.N.M aheshwar and Thakur [6]. A space X s called -compact space f every -open cover for X has a fnte subcover. In ths paper we shall ntroduce a new concept of compactness, whch s called an "-ccompactness" where [A topologcal space X s sad to be -c-compact space f for every - closed set A X, each famly of -open sets n X whch covers A, there s a fnte subfamly W such that {-cl U :U W} covers A]. We dscuss some propertes of ths knd of compactness and gve some propostons, corollares and examples After nvestgatng the relatonshps among compact sp aces,ccompact spaces, -compact spaces and -c-compact spaces are consdered. 1.1 Defnton [1] A topologcal space (X,) s sad to be c-compact f for each closed set A X, each open cover of A contans a fnte subfamly W such that {cl v: v W} covers A. 1.2 Proposton [1] Every compact space s c-compact. 1.3 Remark The mplcaton n proposton (1.2) s not reversble, for example: A space (N,) where, = {U n = {1,2,,n}n N} {N,} s c-compact whch s not compact. 1.4 Proposton [1] A T 3 -c-compact space s compact. 1.5 Defnton [6]

3 A space X s sad to be -compact space f every -open cover of X has a fnte subcover. 1.6 Proposton [6] Every -compact space s compact. 1.7 Remark The opposte drecton of proposton (1.6) may be false, for example: Let X = {0} N and = {,{0},X} be a topology on X. Evdently, X s a compact space. However, t s not -compact space. 1.8 Proposton [6], [7] If all nowhere dense subsets of a topologcal space X are fnte, then the concepts of compactness and -compactness are concdent. In propostons (1.9) and (1.11) we shall dscuss the relatonshps between -compatness and c-compactness. 1.9 Proposton Every -compact space s c-compact. Follows drectly from propostons (1.6) and (1.2) Remark The opposte drecton of proposton (1.9) may be false, see the example n remark (1.3), (N,) s c-compact space whch s not -compact, snce {{1,n} n N} s -open cover for N whch has no fnte subcover Proposton If all nowhere dense subsets of a T 3 - space X are fnte, then X s -compact space, whenever t s c-compact.. Follows from propostons (1.4) and (1.8). 2. -c-compactness 2.0 Introducton In ths secton we shall ntroduce a new type of compactness whch s termed "-ccompactness", we shall study further propertes of ths type of compactness. Examples were constructed to show the relatonshps among "compact, c-compact, -compact and -ccompact space". Several propostons of these spaces are gven also 2.1 Defnton A topologcal space (X,) s sad to be -c-compact space f for each -closed set A X, each famly of -open subset of X whch covers A has a fnte subfamly whose -closures n X covers A. 2.2 Proposton An -compact space s -c-compact. Let A be an -closed subset of an -compact space X and {U : } be a famly of -open sets n X whch covers A, mples, {U : } {X A} s an -open cover of X whch s - compact space, then there s a fnte famly { U : = 1,2,,n} {X A} covers X. But (X A) covers no part from A, mples, { U : = 1,2,,n} covers A. So {-closur U : = 1,2,,n} covers A. Hence, X s -c-compact space. 2.3 Corollary If every nowhere dense subset of a topologcal space (X,) s fnte, then X s -ccompact space whenever t s compact.

4 Follows from propostons (1.8) and (2.2). 2.4 Corollary If every nowhere dense set s fnte n a T 3 -c-compact space (X,), then t s -c-compact space. Follows from propostons (1.4) and corollary (2.3). 2.5 Remark The opposte drecton of proposton (2.2) may be untrue. For example: Let N be the set of all natural numbers, and let = {,{1},N} be a topology on N. Then {{1,n} n N} s an -open cover for N whch has no fnte subcover. So N s not - compact space. But N s -c-compact, snce N s the unque -closed set contans 1. In the followng proposton we put some condton to make the -c-compact space an - compact space. 2.6 Proposton A T 3 --c-compact space s -compact. Let X be a T 3 --c-compact space, f t s not -compact, then there s an -open cover for X say {U : } whch has no fnte subcover. Snce X s -c-compact space, then there s a fnte subfamly { U : = 1,2,,n} such that {-closure U : = 1,2,,n} covers X. Ths means, there exsts x X such that x -cl U and x U for some = 1,2,,n. Imples x -derved U for some = 1,2,,n. Now, snce X s T 1 -space, then {x} s closed set and snce x U, then y {x} for each y U and X s regular space, mples for each y U, there are two open sets V y and V y such that y V y and {x} V y and V y, V y =. Imples, {x} {V y : y U } and U {V y :y U }. But {x} s compact set, then there s a fnte subset of U say {y 1, y 2,, y n } such that {x} { V : j = 1,2,,n}. y j Now, let V = { V y j : j = 1,2,,n}, then V s an open set contans x. On the other sde, let V = {V y :y U } mples V s an open set contans U. So V V =. In vew of, every open set s -open, hence, x -derved U whch s a contradcton. thereupon, X s -compact space. 2.7 Corollary A T 3 --c-compact space s compact. In vew of, every -compact space s compact, then proposton (2.6) s applcable. 2.8 Remark In general, -c-compact space need not be compact as the followng example shows: Let N be the set of all natural numbers and let = {U n u n = {1,2,,n}; n N} {,N}. Then (N,) s -c-compact space, snce N s the unque -closed set contans 1. But N s not compact space. In corollary (2.4), we dscussed the relatonshp between, c-compact and -c-compact space, n one sde, the other sde of ths relaton we shall descry n the followng proposton. 2.9 Proposton An -c-compact space s c-compact.

5 Let X be an -c-compact space. If t s not c-compact space, then there s a closed set A X, and a famly of open sets n X say {U :} covers A. But for each n N, mples A {cl U, = 1,2,,n}. On the other sde, clearly A s -closed subset of an -c-compact space X and {U :} s an -open cover for A n X, then there exsts n N such that A {-cl U : = 1,2,,n}. Ths means, there exsts x A such that x -cl U and x cl U for some = 1,2,,n. Snce x cl U, mples xu and x derved U. But x -cl U, then x -derved U. Snce x derved U then there exsts an open set say V such that x V and V \{x} U =. In vew of, every open set s -open then V s -open set mples x -derved U whch s a contradcton. Therefore, X s c-compact space whenever t s -c-compact. The followng dagram shows the relatonshps among the dfferent types of compactness that we studed n ths paper. compact + Every nowhere dense set s fnte T3 + - compact c-compact T3 + -ccompact 3. Certan Fundamental Propertes of -c-compact Spaces In ths secton, we shall dscuss some propertes of the new knd of compactness whch we ntroduced n ths paper.

6 In remark (3.1) and proposton (3.3) we shall dscuss the heredty property n -c-compact spaces. 3.1 Remark -c-compactness s not a heredtary property. For example: Let X = N {-1,0} and = P(N) {HX-1,0HX H s fnte}. Clearly: (X,) s -c-compact space, snce the complement of each -closed set whch contans (-1) or (0) s fnte set. Now, take N as a subspace of (X,). It s clear that the nduced topology on N s the dscrete topology on N Hence, N s not -c-compact space. The above example shows that f Y s an open subspace of an -c-compact space (X,), then Y need not be -c-compact. 3.2 Remark [4], [6]. If Y s an open subset of a topologcal space X, then every -open set n Y s an -open set n X.. If Y s an open, -closed subspace of an -compact space X, then Y s -compact. 3.3 Proposton If Y s an open and -closed subspace of an -c-compact space X, then Y s -ccompact. The proof of ths proposton wll take effect n vrtue of remark (3.2). 3.4 Defnton [8], [9] A functon f :(X,) (Y,) s sad to be "*-contnuous", f and only f the nverse mage of every -open subset of Y s an -open subset of X. 3.5 Remark [10] A functon f :(X,) (Y,) s sad to be "*-contnuous", f and only f the nverse mage of every -closed subset of Y s an -closed subset of X. 3.6 Lemma A functon f :(X,) (Y,) s *-contnuous f and only f -closure (f -1 (B)) f -1 (closure((b)) for each B Y. Necessty, let f :(X,) (Y,) be an *-contnuous functon, let B Y. Now, snce, B -cl B, then (f -1 (B)) f -1 (-cl B), mples, -cl(f -1 (B)) -cl( f -1 (-cl B)). In vrtue of remark (3.5), f -1 (-cl B) s an -closed set n X. So -cl( f -1 (-cl B)) = f -1 (-cl B). Therefore -cl(f -1 (B)) f -1 (-cl B). Suffcency, suppose -cl(f -1 (B)) f -1 (-cl B) for each B Y. To prove f s *- contnuous functon. We must prove f A a an -closed set n Y, then f - 1 (A) s an -closed set n X. It s enough to prove that -cl(f -1 (A)) f -1 (A). Snce A s -closed set n Y, then -cl(a) = A and by hypothess, -cl(f -1 (A)) f -1 (-cl (A)) mples,-cl(f -1 (A)) f -1 (A). So f -1 (A) s an -closed set n X and f s *-contnuous functon. 3.7 Prposton The *-contnuous mage of an -c-compact space s -c-compact. Let (X,) be an -c-compact space, and f :(X,) (Y,) be an *-contnuous onto functon. To prove (Y,) s -c-compact space. Let A be an -closed subset of Y, and {U : } be an -open cover n Y for A. Snce f s *-contnuous, then f -1 (A) s an -closed set n X and { f -1 (U ):} s a famly of -open sets n X coverng f -1 (A) and X s -ccompact space, then there s 1, 2,, n such that {-cl(f -1 ( U )): = 1,2,,n} covers f -1 (A), mples { f (-cl(f -1 ( U ))): = 1,2,,n} covers A. In vrtue of lemma (3.6), { f (f - 1 (-

7 cl( U ))): = 1,2,,n} covers A. Snce f s onto functon,{-cl( U ): = 1,2,,n} covers A. Hence, Y s -c-compact space. 3.8 Proposton [9], [10] Every contnuous, onto, open functon s *-contnuous. 3.9 Corollary An -c-compactness s a topologcal property. Follows from propostons (3.8) and (3.7). 4. Concluson and Recommandaton We ntroduced a new type of compactness whch s called -c-compact and dscussed the relatonshps among ths type and some types of compactness lke, compact, -compact and c-compact. Also, we some examples to explan the drecton that not hold and we put some condton to make that false drecton vald. In future, we shall study strongly c-compact, sem--c-compact, sem-p-compact and sem-p-c-compact. References 1. Vglon, G. (1969), "C-Compact Spaces", Duke Math. J.,36: Njastad, O. (1965), "On Some Classes of Nearly Open Sets", Pacfc J.M ath.,15: Caldas, M. and Jafar, S. (2001), "Some Propertes of Contra--Contnuous Functons", Mem. Fac.Sc. Koch Unv. (Math.), 22: Maheshwar, S.N. and Thakur, S.S. (1980),"On -Sets", Joffnabha J.Math.,11: Kumar, M.Veera, (2002),"Pre-Sem-Clsed Sets', Indan Journal of M athematcs, 4492: Maheshwar, S.N. and Thakur, S.S. (1985), "On -Compact Spaces", Bulletn of the Insttute of Mathematcs, Academc Snca, 13(4 ): Nor, T. and Mao, G.D. (1988), "Propertes of -c-compact Spaces', Rendcont Crc. Math. Palermo. Ser II, 18: Navalag, G.B. (1991), "Defnton Bank n General Topology", 45 G. 9. Relly, I.L. and M.K., (1985), "On -Contnuty n Topologcal Saces", Acta Mathematcs Hungarca, Al, N.M. (2004),"On New Types of Weakly Open Sets", M.Sc.Thess, Unversty of Baghdad.