Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers Science and Technology, 7 (2002) 279-281 © Sultan Qaboos University Metrization of Weakly Developable Spaces Abdul M. Mohamad Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al Khod 123, Muscat, Sultanate of Oman, Email: mohamad@squ.edu.om. عيفةمترية الفضاءات النامية الض عبدالعظيم مؤآت محمد . نقدم في هذا البحث النظرية المترية للفضاءات النامية الضعيفة : خالصة ABSTRACT: In this note, we present metrization of weak developability. Keywords: Weakly Developable, Metrizable, Weak Base. 1. Introduction M artin (1976) introduced the concept of weak developability in order to study the problem of the metrization of spaces with weak bases. A space X is weakly developable if and only if there is a sequence { }n n NG ∈ of covers of X such that for each x X∈ ,{ ( ) }, :nst x G n N∈ is a weak base at x . The sequence { }n n NG ∈ is said to be a weak development for the space X . If each consists of open sets, then {nG }n n NG ∈ is a development for the space X and X is a developable space (Gruenhage, 1984). The idea of weak base was introduced by Arhangel'skii (1966) in the study of symmetrizable spaces. It is more convenient to use the form of Siwiec, (1974) and Franklin, (1965). A collection w of subsets of a space X is called a weak base for X provided that to each x X∈ , there exists w such that x w⊂ 1. Each member of contains xw x . 2. For any two members W and W1 2 of , there is a Wxw 3 in xw , such that W W3 1 W⊂ ∩ 2 . 3. A subset of F X is closed if and only if for every point x F∉ , there exists a W in such that F W xw φ∩ = . If to each x X∈ we assign a collection xw of supersets of { }x such that { }:x x X= ∪ ∈W w is a weak base by virtue of the collections , i.e., the collectionsxw xw x satisfy conditions (1), (2) and (3) of the preceding paragraph, then we say that the collection is a local weak base at w x for each x X∈ . It is easy to show that a subset O of a space X with local weak bases { }:xw x X∈ is open if and only if for each x O∈ , there is a member W of the local weak base of xw x X∈ with . W O⊂ In this study, we prove a metrization theorem for weakly developable spaces. We assume throughout this note that all spaces are T . A topological space is a T -space if, and only if, for each pair 0 0 x and of distint points, there is a nighborhood of one point to which the other does not y 279 ADBUL M. MOHAMAD belong. Also, we let denote the set of all positive integers. For a collection G of subsets of a space N X , we define ( ) { }, :x G g x g G= ∪ ∈ ∈ andst ( ) ( ) ( ){ }, , :2st x G st y G st x G= ∪ ∈ , .y }nG X :ny n N K∈ y X ∈ y :y n N∈ K X N∈ : N ∈ }nG ( Gy ∈ ∈ y s }:i y∪ X N i t∈ }n n ∈ :n− F x 2. Main Results 2.1 lemma 1 Let { n N∈ be a weak development of a space X . Then for every compact subset K of and any sequence of points in , there is a point in and a subsequence :ny i N of the sequences :ny n N∈ which converges to . Proof. Let be a compact subset of K X and let be any sequence of points in . Suppose there is no subsequence of :ny n N∈ which converges to a point of { }:ny n− ∈ N . Then, we note that { }:nF y n= N∈ is a closed subset of X . For if it is not true then for some point we shall have y X F∈ − ( ) 1 ,t y G∈ny s i F∩ for each i N∈ . This will imply that the subsequence :ny i will converge to , which will contradict our assumption. Therefore, is closed. y F n Define { }n iF y i n= ≥ for each . Similarly, one can show that F is closed for each . Consider the open cover { n N∈ n n ∈ }:nX F n− N∈ of in K X . It is easy to see that it does not contain a finite subcover of which contradicts the fact that is compact. Hence, the sequences K K :ny n N have a convergent subsequence. 2.2 lemma 2 Let { n N∈ be a weak development of a space X , which satisfies the following condition: For any closed subset F of X and any point ,y X F∈ − there is an n such that st N∈ ( )G ),, n nst F .φ∩ = Then every compact subset of X is closed. Proof. Let K be a compact subset of X . Suppose is not closed. Then there is a point such that K y X K∈ − ( )i K,y Gst φ∩ ≠ for all i . Thus for each i , let . Hence, the sequence N N ( ,i t y G∈ )i K :iy i N∈ converges to . Put y { } {F y= ∈i N . We claim that is a closed subset of F . For if not, then there is a point such that z X∈ − F ( ), iGst z F φ∩ ≠ for all i ∈ . Without loss of generality, let for all ( ), iz Gy s F∩ i N∈ . This is not possible since a weakly developable space is T and hence by the hypothesis there is an 1 n N∈ such that ( ) ( ), ,yn nGst z G st φ∩ = . Define { { }:n iF y i= ≥ y∪ for each n N∈ . Clearly, is closed for each . Therefore, nF N { }X F n N∈ is an open cover of with no finite subcover of giving a contradiction. K K ∩ 2.3 Theorem The following are equivalent for a space X . 1. The space X is metrizable. 2. The space X has a weak development { }n n NG ∈ such that for any closed subset of X and any point ,X F∈ − there is an i such thatN∈ ( ) ( ), ,i iG st F Gst y φ∩ = 280 METRIZATION OF WEAKLY DEVELOPABLE SPACES 3. The space X has a weak development { }n n NG ∈ such that if where is compact and V is open, then st for some . ,A V⊂ A ( ), nA G V⊂ n 4. The space X has a weak development { }n n NG ∈ such that if ,x V∈ where V is open, then there exists a neighborhood U of x and ,n V∈ for which ( ), .nG V⊂st U 5. The space X has a weak development { }n n NG ∈ such that if x V∈ is open, then there exists an n N for which ∈ ( ), .2 nst x G V⊂ Proof. The implication 1 is clear. The implications 3 4 , and 5 are proved in Martin (1976, Theorem 2.5 and Theorem 2.6). To prove . Let A be any compact subset of 2 5 1 3 ⇒ ⇒ 2 ⇒ 4 ⇒ ⇒ X and let V be an open subset of X containing . Suppose that A ( ) (X ), .iG Ust A φ∩ − ≠ Let ( )i,x st A G∈ ( )X V∩ − . Hence, by Lemmas 2.1 and 2.2 there is a subsequence :kix k ∈ N of the sequence :ix i ∈ y G N ( ) st , which converges to a point in . Now, by hypothesis there is a such that st y A .Nj ∈ (( V ), i X G ), i φ∩ − = This leads to a contradiction. 3. Acknowledgement The author is grateful to Prof. David Gauld for his kind help and suggestions on this paper. References ARHANGEL’SKII, A. 1966. Mappings and spaces, Russian Math. Surveys, 21:115--162. FRANKLIN, S. 1965. Spaces in which sequences suffice. II, Fund. Math. 57:107-115. GRUENHAGE, G. 1984. Generalized metric spaces, in Handbook of Set-theoretic Topology, pp. 423-501. MARTIN, H. 1976. Weak bases and metrization, Trans. Amer. Math. Soc. 222: 338-344. SIWIEC, F. 1974. On defining a space by a weak base, Pacific J. Math. 52:233-245. Received 14 May 2001 Accepted 6 June 2002 281 Abdul M. Mohamad Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al Khod 123, Muscat, Sultanate of Oman, Email: mohamad@squ.edu.om. ÎáÇÕÉ : äÞÏã Ýí åÐÇ ÇáÈÍË ÇáäÙÑí� References