2009) 3( 22مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد --Cالتراص من نوع رنا بهجت اسماعیل جامعة بغداد -ابن الهیثم - كلیة التربیة -قسم الریاضیات الخالصة كـذلك قمنــا بدراسـة بعــض " -cالتـراص مـن نــوع "قمنـا فـي هــذا البحـث بتعریــف نـوع جدیـد مــن التـراص اســمیناه .-cوالتراص من نوع -خواصه والعالقة بینه وبین التراص والتراص من نوع IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009  - C-Compactness R. B. Esmaeel Departme nt of Mathematics,College of Education I bn-Al-Haitham , Unive rsity of Baghdad Abstract In this p aper, we introduce a new ty p e of comp actness which is called "-c- comp actness". Also, we st udy some p rop erties of this ty p e of comp actness and the relationship s among it and comp actness, -comp actness and c-compactness. 1. Introduction and Preliminaries A top ological sp ace (X,) is said to be c-compact sp ace if for each closed set A  X, each open cover of A contains a finite subfamily W such that {cl v: v  W} covers A, [1]. In 1965, O.Njast ed [2] introduced "-op en set" in top ology [A subset A of a top ological sp ace X is said to be "-op en set if A  int (cl(int(A)))], and he p roved that the family of all "-op en sets in a sp ace (X,) is a top ology on X, which is finer than  and denoted by . -op en sets are discussed in [3], [4], [5], some concepts were st udied as follows: i. The comp lement of an -op en set is called -closed set and the intersection of all -closed sets contains a set A which is called the -closure of A and denoted by -clA. So, -clA is an -closed set and proved (-clA = A iff A is -closed set). ii. If A be a subset of a top ological sp ace X the -derived of A is the set of all elements x satisfies t he condition, that for every -op en set V contains x, imp lies V\{x}A . In 1985, the term of "-comp actness" was used for the first time by S.N.M aheshwari and Thakur [6]. A sp ace X is called -comp act sp ace if every -op en cover for X has a finite subcover. In this p aper we shall introduce a new concept of comp actness, which is called an "-c- comp actness" where [A top ological sp ace X is said to be -c-compact sp ace if for every - closed set A  X, each family of -op en sets in X which covers A, t here is a finite subfamily W such that {-cl U :U  W} covers A]. We discuss some p rop erties of this kind of comp actness and give some p rop ositions, corollaries and examp les Aft er invest igating the relationship s among comp act sp aces,c- comp act sp aces, -comp act sp aces and -c-compact sp aces are considered. 1.1 De fini tion [1] A t op ological sp ace (X,) is said to be c-comp act if for each closed set A  X, each op en cover of A contains a finite subfamily W such that {cl v: v  W} covers A. 1.2 Proposi tion [1] Every comp act sp ace is c-comp act. 1.3 Remark The implication in p rop osition (1.2) is not reversible, for examp le: A sp ace (N,) where,  = {Un = {1,2,…,n}n  N}  {N,} is c-compact which is not comp act. 1.4 Proposi tion [1] A T 3-c-compact sp ace is compact. 1.5 De fini tion [6] IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 A sp ace X is said to be -comp act sp ace if every -op en cover of X has a finite subcover. 1.6 Proposi tion [6] Every -comp act sp ace is compact. 1.7 Remark The opp osite direction of prop osition (1.6) may be false, for examp le: Let X = {0}  N and  = {,{0},X} be a top ology on X. Evidently , X is a comp act sp ace. However, it is not -comp act sp ace. 1.8 Proposi tion [6], [7] If all nowhere dense subsets of a top ological sp ace X are finite, then the concepts of comp actness and -comp actness are concident. In p rop ositions (1.9) and (1.11) we shall discuss the relationships between -comp atness and c-comp actness. 1.9 Proposi tion Every -comp act sp ace is c-comp act. Proof: Follows directly from prop ositions (1.6) and (1.2). 1.10 Remark The opp osite direction of prop osition (1.9) may be false, see the examp le in remark (1.3), (N,) is c-compact sp ace which is not -comp act, since {{1,n} n  N} is -op en cover for N which has no finite subcover. 1.11 Proposi tion If all nowhere dense subsets of a T3- sp ace X are finite, then X is -comp act sp ace, whenever it is c-comp act.. Proof: Follows from p rop ositions (1.4) and (1.8). 2. -c-compactness 2.0 Introduction In this section we shall introduce a new ty p e of comp actness which is termed "-c- comp actness", we shall st udy further prop erties of this ty p e of comp actness. Examp les were constructed to show the relationship s among "comp act, c-compact, -comp act and -c- comp act sp ace". Several p rop ositions of these sp aces are given also 2.1 De fini tion A t op ological sp ace (X,) is said to be -c-compact sp ace if for each -closed set A  X, each family of -op en subset of X which covers A has a finite subfamily whose -closures in X covers A. 2.2 Proposi tion An -comp act sp ace is -c-compact. Proof: Let A be an -closed subset of an -comp act sp ace X and {U: } be a family of -op en sets in X which covers A, imp lies, {U: }  {X – A} is an -op en cover of X which is - comp act sp ace, then there is a finite family { U i :i = 1,2,…,n}  {X – A} covers X. But (X – A) covers no p art from A, implies, { U i :i = 1,2,…,n} covers A. So {-closur U i : i = 1,2,…,n} covers A. Hence, X is -c-compact sp ace. 2.3 Corollary If every nowhere dense subset of a top ological sp ace (X,) is finite, then X is -c- comp act sp ace whenever it is comp act. IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Proof: Follows from p rop ositions (1.8) and (2.2). 2.4 Corollary If every nowhere dense set is finite in a T3-c-compact sp ace (X,), then it is -c-compact sp ace. Proof: Follows from p rop ositions (1.4) and corollary (2.3). 2.5 Remark The opp osite direction of prop osition (2.2) may be untrue. For examp le: Let N be the set of all natural numbers, and let  = {,{1},N} be a top ology on N. Then {{1,n} n  N} is an -op en cover for N which has no finite subcover. So N is not - comp act sp ace. But N is -c-compact, since N is t he unique -closed set contains 1. In the following p rop osition we put some condition to make the -c-compact sp ace an - comp act sp ace. 2.6 Proposi tion A T 3--c-compact sp ace is -comp act. Proof: Let X be a T3--c-compact sp ace, if it is not -comp act, then there is an -op en cover for X say {U: } which has no finite subcover. Since X is -c-compact sp ace, then there is a finite subfamily { U i : i = 1,2,…,n} such that {-closure U i : i = 1,2,…,n} covers X. This means, there existS x  X such that x  -cl U i and x  U i for some i = 1,2,…,n. Implies x  -derived U i for some i = 1,2,…,n. Now, since X is T 1-sp ace, then {x} is closed set and since x  U i , then y  {x} for each y  U i and X is regular sp ace, implies for each y  U i , there are two op en sets Vy and Vy such that y  Vy and {x}  Vy and Vy ,  Vy = . Implies, {x}   {Vy: y  U i } and U i  {Vy :y  U i }. But {x} is comp act set, then there is a finite subset of U i say {y1, y2, …, yn} such that {x}   { V jy  : j = 1,2,…,n}. Now, let V =  { V jy  : j = 1,2,…,n}, t hen V is an open set contains x. On t he other side, let V =  {Vy :y  U i } imp lies V is an open set contains U i . So V  V = . In view of, every op en set is -op en, hence, x  -derived U i which is a contradiction. thereup on, X is -comp act sp ace. 2.7 Corollary A T 3--c-compact sp ace is compact. Proof: In view of, every -comp act sp ace is compact, then prop osition (2.6) is ap p licable.  2.8 Remark In general, -c-compact sp ace need not be comp act as the following examp le shows: Let N be the set of all natural numbers and let  = {Un un = {1,2,…,n}; n  N}  {,N}. Then (N,) is -c-compact sp ace, since N is the unique -closed set contains 1. But N is not comp act sp ace. In corollary (2.4), we discussed the relationship between, c-compact and -c-compact sp ace, in one side, the other side of this relation we shall descry in the following p rop osition. 2.9 Proposi tion An -c-compact sp ace is c-comp act. Proof: IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Let X be an -c-compact sp ace. If it is not c-compact sp ace, then there is a closed set A  X, and a family of op en sets in X say {U:} covers A. But for each n  N, imp lies A   {cl U i ,i = 1,2,…,n}. On t he other side, clearly A is -closed subset of an -c-compact sp ace X and {U:} is an -op en cover for A in X, then there exists n  N such that A  {-cl U i : i = 1,2,…,n}. T his means, there exists x  A such that x  -cl U i and x  cl U i for some i = 1,2,…,n. Since x  cl U i , imp lies xU i and x  derived U i . But x  -cl U i , then x  -derived U i . Since x  derived U i then there exists an open set say V such that x V and V \{x} U i =. In view of, every op en set is -op en then V is -op en set imp lies x  -derived U i which is a contradiction. Therefore, X is c-comp act sp ace whenever it is -c-compact. The following diagram shows t he relationships among the different ty p es of comp actness that we studied in this p aper. 3. Ce rtain Fundamental Properties of -c-compact S paces In this section, we shall discuss some p rop erties of the new kind of comp actness which we introduced in this p aper. + c-compact -c- compact + + T3 - compact Every nowhere dense set is finite T3 compact IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 In remark (3.1) and p rop osition (3.3) we shall discuss the heredity p rop erty in -c-compact sp aces. 3.1 Remark -c-compactness is not a hereditary p rop erty . For examp le: Let X = N {-1,0} and  = P(N)  {HX-1,0HX – H is finite}. Clearly : (X,) is -c-compact sp ace, since the comp lement of each -closed set which contains (-1) or (0) is finite set. Now, take N as a subsp ace of (X,). It is clear that the induced top ology on N is the discrete top ology on N Hence, N is not -c-compact sp ace. The above examp le shows that if Y is an op en subsp ace of an -c-compact sp ace (X,), then Y need not be -c-compact. 3.2 Remark [4], [6] i. If Y is an op en subset of a top ological sp ace X, t hen every -op en set in Y is an -op en set in X. ii. If Y is an open, -closed subsp ace of an -comp act sp ace X, then Y is -comp act. 3.3 Proposi tion If Y is an op en and -closed subsp ace of an -c-compact sp ace X, then Y is -c- comp act. The proof of this p rop osition will take effect in virtue of remark (3.2).  3.4 De fini tion [8], [9] A function f :(X,)  (Y,) is said to be "*-continuous", if and only if the inverse image of every -op en subset of Y is an -op en subset of X. 3.5 Remark [10] A function f :(X,)  (Y,) is said to be "*-continuous", if and only if the inverse image of every -closed subset of Y is an -closed subset of X. 3.6 Lemma A function f :(X,)  (Y,) is *-continuous if and only if -closure (f -1 (B))  f -1 (- closure((B)) for each B  Y. Proof: Necessity , let f :(X,)  (Y,) be an *-continuous function, let B  Y. Now, since, B  -cl B, then (f -1 (B))  f -1 (-cl B), imp lies, -cl(f -1 (B))  -cl( f -1 (-cl B)). In virtue of remark (3.5), f -1 (-cl B) is an -closed set in X. So -cl( f -1 (-cl B)) = f -1 (-cl B). Therefore -cl(f -1 (B))  f -1 (-cl B). Sufficiency, sup p ose -cl(f -1 (B))  f -1 (-cl B) for each B  Y. To p rove f is *- continuous function. We must p rove if A ia an -closed set in Y, then f - 1 (A) is an -closed set in X. It is enough to p rove that -cl(f -1 (A))  f -1 (A). Since A is -closed set in Y, then -cl(A) = A and by hy p othesis, -cl(f -1 (A))  f -1 (-cl (A)) implies,-cl(f -1 (A))  f -1 (A). So f -1 (A) is an -closed set in X and f is *-continuous function. 3.7 Prposi tion The *-continuous image of an -c-compact sp ace is -c-compact. Proof: Let (X,) be an -c-compact sp ace, and f :(X,)  (Y,) be an *-continuous onto function. To p rove (Y,) is -c-compact sp ace. Let A be an -closed subset of Y, and {U: } be an -op en cover in Y for A. Since f is *-continuous, t hen f -1 (A) is an -closed set in X and { f -1 (U):} is a family of -op en sets in X covering f -1 (A) and X is -c- comp act sp ace, then there is 1, 2,…,n such that {-cl(f -1 ( U i )):i = 1,2,…,n} covers f -1 (A), imp lies { f (-cl(f -1 ( U i ))): i = 1,2,…,n} covers A. In virtue of lemma (3.6), { f (f - 1 (- IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 cl( U i ))):i = 1,2,…,n} covers A. Since f is onto function,{-cl( U i ):i = 1,2,…,n} covers A. Hence, Y is -c-compact sp ace. 3.8 Proposi tion [9], [10] Every continuous, ont o, op en function is *-continuous. 3.9 Corollary An -c-compactness is a top ological p rop erty . Proof: Follows from p rop ositions (3.8) and (3.7). 4. Conclusi on and Recommandation We introduced a new ty p e of comp actness which is called -c-compact and discussed the relationships among this ty p e and some ty p es of comp actness like, comp act, -comp act and c-comp act. Also, we some examp les to exp lain the direction that not hold and we p ut some condition to make that false direction valid. In future, we shall st udy st rongly c-compact, semi--c-compact, semi-p -comp act and semi-p -c-compact. Re ferences 1. Viglion, G. (1969), "C-Comp act Sp aces", Duke M ath. J.,36:761-764. 2. Njastad, O. (1965), "On Some Classes of Nearly Op en Sets", Pacific J.M ath.,15:961-970. 3. Caldas, M . and Jafari, S. (2001), "Some Prop erties of Contra--Continuous Functions", M em. Fac.Sci. Kochi Univ. (M ath.), 22:19-28. 4. M aheshwari, S.N. and Thakur, S.S. (1980),"On -Sets", Joffnabha J.M ath.,11:209-214. 5. Kumar, M .Veera, (2002),"Pre-Semi-Clsed Sets', Indian Journal of M athematics, 4492:165- 181. 6. M aheshwari, S.N. and Thakur, S.S. (1985), "On -Compact Sp aces", Bulletin of the Inst itut e of M athematics , Academic Sinica, 13(4 ):341-347. 7. Noiri, T. and M aio, G.Di. (1988), "Prop erties of -c-compact Sp aces', Rendiconti Circ. M ath. Palermo. Ser II, 18:359-69. 8. Navalagi, G.B. (1991), "Definition Bank in General Top ology ", 45 G. 9. Rielly , I.L. and M .K., (1985), "On -Continuity in Top ological Saces", Acta M athematics Hungarica,45. 10. Ali, N.M . (2004),"On New Ty p es of Weakly Op en Sets", M .Sc.Thesis, University of Baghdad.