2009) 3( 22مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد --Cالتراص من نوع رنا بهجت اسماعیل جامعة بغداد -ابن الهیثم - كلیة التربیة -قسم الریاضیات الخالصة ـمیناه ة بعــض " -cالتـراص مـن نــوع "قمنـا فـي هــذا البحـث بتعریــف نـوع جدیـد مــن التـراص اسـ كـذلك قمنــا بدراسـ .-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 Department of Mathematics,College of Education Ibn-Al-Haitham , University of Baghdad Abstract In this paper, we introduce a new type of compactness which is called "-c- compactness". Also, we study some properties of this type of compactness and the relationships among it and compactness, -compactness and c-compactness. 1. Introduction and Preliminaries A topological space (X,) is said to be c-compact space 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.Njasted [2] introduced "-open set" in topology [A subset A of a topological space X is said to be "-open set if A  int (cl(int(A)))], and he proved that the family of all "-open sets in a space (X,) is a topology on X, which is finer than  and denoted by . -open sets are discussed in [3], [4], [5], some concepts were studied as follows: i. The complement of an -open 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 topological space X the -derived of A is the set of all elements x satisfies the condition, that for every -open set V contains x, implies V\{x}A  . In 1985, the term of "-compactness" was used for the first time by S.N.M aheshwari and Thakur [6]. A space X is called -compact space if every -open cover for X has a finite subcover. In this paper we shall introduce a new concept of compactness, which is called an "-c- compactness" where [A topological space X is said to be -c-compact space if for every - closed set A  X, each family of -open sets in X which covers A, there is a finite subfamily W such that {-cl U :U  W} covers A]. We discuss some properties of this kind of compactness and give some propositions, corollaries and examples After investigating the relationships among compact spaces,c- compact spaces, -compact spaces and -c-compact spaces are considered. 1.1 Definition [1] A topological space (X,) is said to be c-compact 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. 1.2 Proposition [1] Every compact space is c-compact. 1.3 Remark The implication in proposition (1.2) is not reversible, for example: A space (N,) where,  = {Un = {1,2,…,n}n  N}  {N,} is c-compact which is not compact. 1.4 Proposition [1] A T3-c-compact space is compact. 1.5 Definition [6] IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 A space X is said to be -compact space if every -open cover of X has a finite subcover. 1.6 Proposition [6] Every -compact space is compact. 1.7 Remark The opposite direction of proposition (1.6) may be false, for example: Let X = {0}  N and  = { ,{0},X} be a topology on X. Evidently, X is a compact space. However, it is not -compact space. 1.8 Proposition [6], [7] If all nowhere dense subsets of a topological space X are finite, then the concepts of compactness and -compactness are concident. In propositions (1.9) and (1.11) we shall discuss the relationships between -compatness and c-compactness. 1.9 Proposition Every -compact space is c-compact. Proof: Follows directly from propositions (1.6) and (1.2). 1.10 Remark The opposite direction of proposition (1.9) may be false, see the example in remark (1.3), (N,) is c-compact space which is not -compact, since {{1,n} n  N} is -open cover for N which has no finite subcover. 1.11 Proposition If all nowhere dense subsets of a T3- space X are finite, then X is -compact space, whenever it is c-compact.. Proof: Follows from propositions (1.4) and (1.8). 2. -c-compactness 2.0 Introduction In this section we shall introduce a new type of compactness which is termed "-c- compactness", we shall study further properties of this type of compactness. Examples were constructed to show the relationships among "compact, c-compact, -compact and -c- compact space". Several propositions of these spaces are given also 2.1 Definition A topological space (X,) is said to be -c-compact space if for each -closed set A  X, each family of -open subset of X which covers A has a finite subfamily whose -closures in X covers A. 2.2 Proposition An -compact space is -c-compact. Proof: Let A be an -closed subset of an -compact space X and {U: } be a family of -open sets in X which covers A, implies, {U: }  {X – A} is an -open cover of X which is - compact space, then there is a finite family { U i :i = 1,2,…,n}  {X – A} covers X. But (X – A) covers no part 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 space. 2.3 Corollary If every nowhere dense subset of a topological space (X,) is finite, then X is -c- compact space whenever it is compact. IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Proof: Follows from propositions (1.8) and (2.2). 2.4 Corollary If every nowhere dense set is finite in a T3-c-compact space (X,), then it is -c-compact space. Proof: Follows from propositions (1.4) and corollary (2.3). 2.5 Remark The opposite direction of proposition (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} is an -open cover for N which has no finite subcover. So N is not - compact space. But N is -c-compact, since N is the unique -closed set contains 1. In the following proposition we put some condition to make the -c-compact space an - compact space. 2.6 Proposition A T3--c-compact space is -compact. Proof: Let X be a T3--c-compact space, if it is not -compact, then there is an -open cover for X say {U: } which has no finite subcover. Since X is -c-compact space, 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 T1-space, then {x} is closed set and since x  U i , then y  {x} for each y  U i and X is regular space, implies for each y  U i , there are two open 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 compact 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}, then V is an open set contains x. On the other side, let V =  {Vy :y  U i } implies V is an open set contains U i . So V  V =  . In view of, every open set is -open, hence, x  -derived U i which is a contradiction. thereupon, X is -compact space. 2.7 Corollary A T3--c-compact space is compact. Proof: In view of, every -compact space is compact, then proposition (2.6) is applicable.  2.8 Remark In general, -c-compact space need not be compact as the following example 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 space, since N is the unique -closed set contains 1. But N is not compact space. In corollary (2.4), we discussed the relationship between, c-compact and -c-compact space, in one side, the other side of this relation we shall descry in the following proposition. 2.9 Proposition An -c-compact space is c-compact. Proof: IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Let X be an -c-compact space. If it is not c-compact space, then there is a closed set A  X, and a family of open sets in X say {U:} covers A. But for each n  N, implies A   {cl U i ,i = 1,2,…,n}. On the other side, clearly A is -closed subset of an -c-compact space X and {U:} is an -open cover for A in X, then there exists n  N such that A  {-cl U i : i = 1,2,…,n}. This 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 , implies 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 open set is -open then V is -open set implies x  -derived U i which is a contradiction. Therefore, X is c-compact space whenever it is -c-compact. The following diagram shows the relationships among the different types of compactness that we studied in this paper. 3. Certain Fundamental Properties of  -c-compact Spaces In this section, we shall discuss some properties of the new kind of compactness which we introduced in this paper. + 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 proposition (3.3) we shall discuss the heredity property in -c-compact spaces. 3.1 Remark -c-compactness is not a hereditary property . For example: Let X = N {-1,0} and  = P(N)  {HX-1,0HX – H is finite}. Clearly: (X,) is -c-compact space, since the complement of each -closed set which contains (-1) or (0) is finite set. Now, take N as a subspace of (X,). It is clear that the induced topology on N is the discrete topology on N Hence, N is not -c-compact space. The above example shows that if Y is an open subspace of an -c-compact space (X,), then Y need not be -c-compact. 3.2 Remark [4], [6] i. If Y is an open subset of a topological space X, then every -open set in Y is an -open set in X. ii. If Y is an open, -closed subspace of an -compact space X, then Y is -compact. 3.3 Proposition If Y is an open and -closed subspace of an -c-compact space X, then Y is -c- compact. The proof of this proposition will take effect in virtue of remark (3.2).  3.4 Definition [8], [9] A function f :(X,)  (Y,) is said to be "*-continuous", if and only if the inverse image of every -open subset of Y is an -open 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), implies, -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, suppose -cl(f -1(B))  f -1(-cl B) for each B  Y. To prove f is *- continuous function. We must prove if A ia an -closed set in Y, then f - 1(A) is an -closed set in X. It is enough to prove that -cl(f -1(A))  f -1(A). Since A is -closed set in Y, then -cl(A) = A and by hypothesis, -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 Prposition The *-continuous image of an -c-compact space is -c-compact. Proof: Let (X,) be an -c-compact space, and f :(X,)  (Y,) be an *-continuous onto function. To prove (Y,) is -c-compact space. Let A be an -closed subset of Y, and {U: } be an -open cover in Y for A. Since f is *-continuous, then f -1(A) is an -closed set in X and { f -1(U):} is a family of -open sets in X covering f -1(A) and X is -c- compact space, then there is 1, 2,…,n such that {-cl(f -1( U i )):i = 1,2,…,n} covers f -1(A), implies { 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 space. 3.8 Proposition [9], [10] Every continuous, onto, open function is *-continuous. 3.9 Corollary An -c-compactness is a topological property . Proof: Follows from propositions (3.8) and (3.7). 4. Conclusion and Recommandation We introduced a new type of compactness which is called -c-compact and discussed the relationships among this type and some types of compactness like, compact, -compact and c-compact. Also, we some examples to explain the direction that not hold and we put some condition to make that false direction valid. In future, we shall study strongly c-compact, semi--c-compact, semi-p-compact and semi-p-c-compact. References 1. Viglion, G. (1969), "C-Compact Spaces", Duke Math. J.,36:761-764. 2. Njastad, O. (1965), "On Some Classes of Nearly Open Sets", Pacific J.M ath.,15:961-970. 3. Caldas, M. and Jafari, S. (2001), "Some Properties of Contra--Continuous Functions", Mem. Fac.Sci. Kochi Univ. (Math.), 22:19-28. 4. Maheshwari, S.N. and Thakur, S.S. (1980),"On -Sets", Joffnabha J.Math.,11:209-214. 5. Kumar, M.Veera, (2002),"Pre-Semi-Clsed Sets', Indian Journal of Mathematics, 4492:165- 181. 6. Maheshwari, S.N. and Thakur, S.S. (1985), "On -Compact Spaces", Bulletin of the Institute of Mathematics , Academic Sinica, 13(4 ):341-347. 7. Noiri, T. and Maio, G.Di. (1988), "Properties of -c-compact Spaces', Rendiconti Circ. Math. Palermo. Ser II, 18:359-69. 8. Navalagi, G.B. (1991), "Definition Bank in General Topology", 45 G. 9. Rielly, I.L. and M.K., (1985), "On -Continuity in Topological Saces", Acta Mathematics Hungarica,45. 10. Ali, N.M . (2004),"On New Types of Weakly Open Sets", M.Sc.Thesis, University of Baghdad.