Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers Science and Technology, 8 (2003) 61-65 © 2003 Sultan Qaboos University Characterizations of K- Semimetric 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. Kة من الصنف لفضاءات شبه المترياتمييز عبدالعظيم مؤات محمد . بطرق مختلفة ومتكافئةKنقوم في هذا البحث بكتابة الفضاءات شبه المترية من الصنف : خالصة ABSTRACT: In this paper, we prove, for a space X, the following are equivalent: 1. X is a ω ∆1 space with a regular-Gδ-diagonal, 2. X is a ω ∆2 space with a regular-Gδ-diagonal, 3. X is a semi-developable space with Gδ (3) -diagonal, 4. X is a ω ∆1-space with a Gδ(3)-diagonal, 5. X is a ω ∆2 -space with a Gδ(3)-diagonal, 6. X is a q, β -space with a G*δ (2)-diagonal, 7. X is a semi-developable space with G*δ (2)-diagonal, 8. X is a semimetrizable, c-stratifiable space, 9. X is a c-Nagata β -space, 10. X is a K-semimetrizable. KEYWORDS: ω ∆ - space,sSemi- developable space, -semimetrizable space, K β -space, G *δ (2)-diagonal, Gδ (3)- diagonal, regular-Gδ -diagonal, semi-stratifiable, c -semi-stratifiable. 1. Introduction A space X is semimetrizable if there exists a real valued function d on X X× such that 1. d x ( ) ( ), = , 0.y d y x ≥ 0 2. d x if and only if ( ), y = .x y= . 3. for ,M X x M⊂ ∈ if and only if ( ) ( ){ }, inf , : M d x y y M= ∈ 0.= 0 d x If in addition, d satisfies. 4. d H whenever and K are disjoint compact subsets of ( ), K > H X , then X is said to be semimetrizable (Arhangel'skii, 1966) . Let { K − } Nn nG ∈ be a sequence of covers of a space X . 1. Suppose { } Nn nG ∈ satisfies the following property: if, ( ), ,nx st x G∈ n then the sequence nx has a cluster point. 61 ADBUL M. MOHAMAD (a) If, for each , G is an open cover of Nn ∈ n X , then X is called a ω ∆ -space (Borges, 1968). (b) If, for each , is an open subset of Nn ∈ (st , )nx G X , then X is called a 1ω ∆ -space (Gittings, 1975). (c) If, for each , Nn ∈ x ∈ Int , then X is called a ( , nx G )st ω ∆2–space (Gittings, 1975). . 2. If for each x X∈ , ( ){ } Nn∈ , nx Gst is a local base at x , then X is called a semi-developable space. If in addition, for each , Nn ∈ ( )n,Gst is an open subset of x X , then X is called a semi- developable space. 3. If, for each , GNn ∈ n is an open cover of X and for each x X∈ , ( ) { }3 , ,n nst x G x=∩ then X has a G diagonal. ( )3 −δ 4. If, for each , Nn ∈ nG is an open cover of X and for each x X∈ , ( ) { }2 ,n nst x G x=∩ , then X has a G diagonal. * (2δ )- 5. If, for each , st is an open subset of Nn ∈ ( , ) nx G X and for each x X∈ , ( ) { },n nst x G x=∩ , then X has a S diagonal. 2 − 6. If, for each , Nn ∈ x ∈ Int st and for each ( , nx G ) x X∈ , { }( , ) ,n nst x G x=∩ then X has a 2α -diagonal. 7. If, for each , GNn ∈ n is an open cover of X and for any pair of distinct points , , x y X∈ there exist neighborhoods U and of V x and , respectively, and , such that y Nn ∈ (st , )nU G V = , φ∩ equivalently, ( , nst V G ) = , U φ∩ then X has a regular- Gδ -diagonal. A COC-map (= countable open covering map) for a topological space X is a function from N X× into the topology of X such that for every x X∈ , and n N∈ , x∈g(n, x) and g(n + 1, x) g(n,x). A space ⊆ X is called β -space if X has a COC-map g such that if ( ), nx g n x∈ for every , then the sequence N∈n nx has a cluster point. A space X is called q space if − X has a COC-map g such that if ( ,n )x g n x∈ for every then the sequence N,n ∈ nx has a cluster point. A space X is called c-semi-stratifiable (Martin, 1973) (c-stratifiable) if there is a sequence ( ),g n x of open neighborhoods of x such that for each compact set ( ){ }, if ( , )= , : ,g n K g n x x K⊂ ∈∪K X then ( ){ } ( ){ }( ), : 1 , : 1g n K n K g n K n K≥ = ≥ =∩ ∩ . The COC-map : Ng X τ× → is called a c-semi-stratification (c-stratification) of X . A space X is called c-Nagata if it is first countable, c- stratifiable space. Throughout this paper, all spaces are assumed to be T 2 − spaces unless otherwise stated explicitly. The letter always denotes the set of all positive integers. N 2. Main results Lemma 1 : Every space with a G diagonal has a G diagonal. (3) -δ * (2) -δ Proof. Let { } Nn nG ∈ be a G diagonal sequence for (3) -δ X . We want to prove that ( ) { }2Nn st x∈∩ , nG x= for every x X∈ . Suppose we have ( )2 ,n t x G∈ nq s . For every open set such that q and for each ∩ U U∈ Nn ∈ 2 ( , ) .nst x G U φ≠∩ 62 CHARACTERIZATIONS OF K-SEMIMETRIC SPACES In particular, if G nG∈ is such that q G∈ then 2 ( , ) .nst x G G φ≠∩ So, q s As this holds for all , it follows that 3 ( , ). nt x G∈ n x q= . Lemma 2: Any space with a Gδ* (2)-diagonal is a c-stratifiable space. Proof. Let nG be a sequence of open covers of a space X such that ( )2N , {n nst x G x∈ =∩ }. Define a COC-map by g ( ) ( ), , ng n x st x G= . We must prove that ( ),g n K K=∩ for any compact subset of X . Let p K∉ . Then, for each , there exists an integer k K∈ ( )n k such that ( )( )2 , .n kp st k G∉ Therefore there is an open set U(k) containing p such that ( ) ( )( ), .2 n kst k GU k φ= K K ∩ ,k k Since is compact, we can find a finite number of points of such that 1 2 ,..., rk ( )( ){ },i n kk G : 1, 2,...,st i r= covers . Let K ( ){ }maxn n= : 1,k i 2,..., ,i r= and ( )1 .i iU U k== ∩ Then ( ), .nU st k G φ∩ = That is,U g ( ),n K φ=∩ . This implies ( ), .p g n K∉ Theorem 1: Every 1ω ∆ -space with S2-diagonal is an o-semidevelopable space. Proof. Let { } Nn nG ∈ be a countable family of covers of a space X illustrating that X is a ω ∆ 1- space. Since X has an -diagonal, there exists a sequence 2S :n Nν n ∈ of covers of X such that, for each x X∈ and (N, , nn st x )ν∈ is an open subset of X and ( ) {n x }N ,n st x ν∈ =∩ . For each , let Nn ∈ ( ) ( ){ }1 1: , , , 1, 2,..., .n nn i i i i i i i iu U U G V G G V i nν= == = ∩ ∈ ∈ =∩ ∩ It is easy to see that u refines u for all 1n + n Nn ∈ and that, for each x X∈ , ( ) { }N ,n nst x u x∈ =∩ . Furthermore, for each x X∈ and Nn ∈ ( ) ( )( ) ( )( )1 1, ,n nn i i i ist x u st x G st x ,ν= == ∩ ∩ ∩ and thus st is an open subset of ( , nx u ) X . Also it is easy to check that : Nnu n ∈ is a ω ∆ 1- sequence for X . It remains to show that : Nnu n ∈ is a semi-development for X . Suppose instead that : Nnu n ∈ is not a semi-development for X . Then there is a point x , an open neighborhood W of x , and a sequence nx such that for all , n ( ),n nx st x u∈ and nx W∉ . Since :nu n ∈ N is a 1ω ∆ -sequence for X , the sequence nx has a cluster point p . Clearly p W∉ so p x≠ . By choice of :n nν ∈ N , there are in and a neighborhood V of k N p such that ( ), kx .V st ν φ=∩ Now for n , k≥ ( ) ( ) ( ), k, ,n nt x u st x k sx us t x ν∈ ⊂ ⊂ so .nx V∉ This contradicts the fact that p is a cluster point of nx . Thus : ∈ Nnu n is a semi-development for X . Theorem 2: The following are equivalent for a regular 2ω ∆ -space X : (1) X is semimetrizable; (2) X is semi-stratifiable; (3) X is θ -refinable and has a Gδ-diagonal; (4) X has a G*δ-diagonal; 63 ADBUL M. MOHAMAD (5) X has 2α -diagonal. (6) X is semidevelopable. Proof. The only implications requiring comment are (5) ⇒ (6) and (6) (1). To prove (5) ⇒ (6), let { ⇒ }nG be a countable family of covers of X illustrating that X is a 2ω ∆ -space. Let : n Nnν ∈ be an 2α -sequence for X . Let the sequence :nu n N∈ be defined as in the proof of Theorem 2.3. Since for each x X∈ and Nn ∈ , ( ) )( ) ( )x∩1 1n n= = , ,i∩ ∩x u I ( ν x x u∈ : ∈ N X ω ∆ X ω ∆ X X ω ∆ X ω ∆ X X X X β X 2∆ β ⇒ ω ∆ n x 0n 0n n> ) U⊂ ( 2 , nGy V { }0 1 2,nxN n , n x ( ,y G )st (3 , . ⇔ sphere ered at X ( ,x ), y < / .n { }i= 1/ spher center . ( nG,t x ( < ( ( ), ,n i i iIntst nst x G Inst= ), .n we have Inst It follows, exactly as before, that nu n is a semi-development for X . The implication (6) (1) follows from (Alexander, 1971), Theorem 1.3. ⇒ Theorem 3: For a space X, the following are equivalent: 1. is a 1-space with a regular -Gδ-diagonal, 2. is a 2-space with a regular-Gδ-diagonal, 3. is a semi-developable space with Gδ (3)-diagonal, 4. is a 1-space with a G δ (3)-diagonal, 5. is a 2-space with a G δ (3)-diagonal, 6. is a q, β -space with a G *δ (2)-diagonal, 7. is a semi-developable space with G *δ (2)-diagonal, 8. is a semimetrizable, c-stratifiable space, 9. is a c-Nagata -space, 10. is a K-semimetrizable. Proof. It is clear that 1 ⇒ 2, 3⇒ 4, 4⇒ 5, 8⇒ 9. The implication 5⇒ 6 follows by Lemma 2.5 and since every ω -space is a q, β -space. The implication 6⇒ 7 follows by facts every β -space with a G *δ -diagonal is a semi-stratifiable space, every q-space with a G *δ -diagonal is first countable and every first countable, semi-stratifiable space is a semimetrizable. The implication 7 8 follows by Lemma 2.2 and since every T⇒ 0 semi-developable space is a semimetrizable. The implication 9⇒ 8 follows by facts every c-stratifiable, -space is semi-stratifiable and every first countable, semi-stratifiable space is a semimetrizable. 1 8 follows by Lemma 2.2, Theorem 2.3. For 2⇒ 3. Suppose that X is a 2 -space with a regular-Gδ-diagonal. Every space with a regular-Gδ-diagonal has a G *δ -diagonal. By Theorem 2.4, X is a semi-developable space. Let { }nG be a semi-development and regular- Gδ-diagonal-sequence. To see that G satisfies the Gδ(3)- diagonal-sequence, let x y≠ points in X , U and V open sets containing and respectively, and an integer such that if , then no member of G meets both U and V . Let and be integers such that st and y n 1n 2n ( 1 , nx G ) ⊂st . ma= n . Then no member of G meets both st and . Thus ( ), nG n )nGy s∉ t y For 10 3. Let { }1/ cent .n =G n It is clear that x nG is a sequence of covers of and if and only if )nGy st∈ ( 1d x Therefore nG is a semidevelopment for X . Now let nterior of e ed atnG n It is clear that nG is a sequence of open covers of X and if then )y s∈ ), y 1/ n.d x If there exist distinct points x and x 64 CHARACTERIZATIONS OF K-SEMIMETRIC SPACES y such that ( )3 , ny st x G∈ for all n N∈ , then there are sequences nx and ny such that ( ) ( )n n, , ,n nx st x G∈ y st y G∈ and ( ),nt x Gny s∈ n . Let { } { }1 :nK x x n ω= ∈∪ and { }2K y= { }:ny n ω∈∪ 1 2K K φ=∩ ( )1 2, 0K ,= n X d X ( ) ( ){ }, 1y / inf j= ∈ : , ist y G∉ ( , n )x st y G∈ ( , H ( ), .y 1/ n< H ) 0.= nx ny K H ( ),n nd x y < 1/ n X nx ny in x x y ( ) 1/ i< ( ), kx G, . We may assume with both sets compact. But d K a contradiction. Conversely, let G be a semi-development and Gδ(3)-diagonal-sequence for . Define a semimetric on by N xd x . From the definition if and only if d x Assume there exist disjoint compacta K and such that d K We can find two sequences and in and respectively, such that . Note that is sequential and T so that 2 and have convergent subsequences. Let and in y be subsequences of nx and ny converging to and , respectively. Without loss of generality, we may assume ( ), in x < k 1/ id x and for each Since d x it follows that there is no such that . This contradiction completes the proof. , in y 3t d y y sN.i ∈ ( , 1/ i in n y i) < ∉ References ALEXANDER, C.1971. Semi-developable space and quotient images of metric spaces, Pacific J. Math 37: 277-293. ARHANGEL'SKII, A. 1966. Mappings and spaces, Russian Math. Surveys, 21:115-162. BORGES, C. 1968. On metrizability of topological spaces, Canad. J. Math. 20:795-804. GITTINGS, R. 1975. On o-semimetrizable space, Studies in topology, Academic Press, New York, 179-188. GRUENHAGE, G. 1984.Generalized metric spaces, Handbook of Set-theoretic Topology 423- 501. MARTIN, H.W.1973. Metrizability of M-spaces, Canad. J. Math., 25:840-841. Received 15 May 2001 Accepted 20 February 2003 65 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 : äÞæã Ýí åÐÇ ÇáÈÍË ÈßÊÇÈÉ � References