Quotient Spaces Determined by Algebras of Continuous Functions

QUOTIENT SP A CES DETERMINED BY ALGEBRAS OF CONTINUOUS FUNCTIONS ALDO J. LAZAR Abstract. W e pro v e that if X is a lo cally compact σ -compact space then on its quot ient, γ ( X ) say , determined b y the algebra of all real v alued bounded con tinuo us functions on X , the quotient topology and the completely r egular topology defined b y this algebra are equal. It follows from th is that if X is second count able lo cally compact then γ ( X ) is s econ d coun table lo cally compact Hausdorff if and only if it is fir st coun table. The interest in these results originated in [1] and [7] where the pr imitive ideal space of a C ∗ -algebra wa s considered. 1. Introduction The primitive ideal space, Pr im ( A ), o f a C ∗ -algebra A with its hull-kernel top ol- ogy has some pleasant prop erties: it is a lo cally compact Bair e space, see [5, Corol- lary 3 .3.8 a nd Coro llary 3.4.1 3]. How ever, no better separ ation pro perty than T 0 can b e exp ected in g eneral. The abs e nc e of the Hausdorff sepa r ation pr oper t y in Prim( A ) justified a s tudy in [1 1] of the collection o f all the closed limit sets of a topolo g ical space. It should be added that the closed limit subsets of Prim( A ) corres p ond, b y one of the bijections detaile d in [5, Prop osition 3.2.1], to some dis- tinguished idea ls of A . An inv estigation of the topo logies on the class o f these ideals that mirro r top ologies on the collection of closed limit s ets of P rim( A ) was per formed in [1]. W e intend here to pursue the study beg un in [2] of the top olo- gies on the q uotien t of Prim( A ) deter mined by the a lgebra of the bounded scalar functions. W e adopted a pure ly top ological se tting so no knowledge of the theo r y of C ∗ -algebra s is needed for r eading this pap er. The method w e c hose is to substi- tute for the quotient of a p ossibly non-Hausdo rff space X , via a homeomor phism, a quotien t of a certain Ha usdorff h yp erspac e of X . In the following X denotes a lo cally compact space that is, X is a s pa ce in which every po in t has a neighbour hoo d base of compact sets. The a lgebra of all complex 1991 Mathematics Subje ct Classific ation. Prim ary: 54B15; Secondary: 54B20, 54D45, 46L05. Key wor ds and phr ases. lo cally compact space, quotien t s pace , F ell’s topology . 1 2 ALDO J. LAZAR v alued b ounded co n tinu ous functions on X , denoted C b ( X ), induces an equiv alence relation on X : x 1 ∼ x 2 if f ( x 1 ) = f ( x 2 ) for every f ∈ C b ( X ). W e let γ ( X ) be the quotient space of this equiv a lence rela tion; the quotient map q : X − → γ ( X ) was called in [3 , section II I.3] the complete regulariza tion o f X a nd this constructio n was discussed in [2 ] for the sp ecial case of the primitiv e ideal space of a C ∗ -algebra . As in [2], w e endow γ ( X ) with the topolo gy τ cr , the w eak top ology defined b y the bo unded con tinuous functions on X viewed as functions on the quotient. Another natural topo logy on γ ( X ) is the quotient top ology τ q . Alwa ys τ cr ⊂ τ q ; it w as shown in [2] that if X is compact or if q is o p en for τ q or τ cr then τ q = τ cr . The question whether these topologies a re equal for an y lo cally compact space was left op en there. Of co urse, C b ( γ ( X )) is alwa ys the same for b oth top ologies. W e shall prov e in Section 2 that if X is σ -compact then τ cr = τ q . Recen tly D. W. B. Somerse t found a n ex ample of a lo cally co mpact space X for which these t wo top ologies are differ en t. The example a ppears in an app endix to this pap er and only its la st pa ragraph, where it is s hown that the topolo gical s pace cons tructed there is homeomor phic to the primitive ideal s pace of a C ∗ -algebra , needs a minimum knowledge of op erator algebra theory . In Sectio n 3 we discuss the cla ss of seco nd countable loca lly compact spa ces X for whic h γ ( X ) is a s econd countable lo cally compact Hausdorff space. Suc h s pa ces when ser ving as primitiv e ideal spa ces of C ∗ -algebra s were consider ed in [7 ] and [8]. W e g iv e a characterization of these space s by using the to ols develop ed in Section 2. The family of all closed subsets of X will b e denoted by F ( X ) and its subfamily that co nsists of all the no nempt y clo sed subsets of X will b e denoted F ′ ( X ). W e shall equip F ( X ) with tw o top ologies: the F ell top olog y , denoted here τ s , that w as defined in [9], and the lo wer s emifinite top ology of Michael, whic h w e denote τ w , see [12]. A base for τ s consists of the family of a ll the sets U ( C, Φ) := { S ∈ F ( X ) | S ∩ C = ∅ , S ∩ O 6 = ∅ , O ∈ Φ } where C is a compac t subset o f X a nd Φ is a finite family o f o pen subsets of X . The hyperspace ( F ( X ) , τ s ) is alwa ys co mpact Hausdorff ([9, Lemma 1 and Theorem 1]). If X is second co untable then this hyper space is metrizable ([4, Le mme 2]). The family of all the sets U ( ∅ , Φ) is a base for τ w . The map η X given b y η X ( x ) := { x } QUOTIENT SP ACES DETERMINED BY ALGEBRAS OF CONTINUOUS FUNCTIONS 3 from X to F ( X ) is τ w -contin uo us; it is not τ s -contin uo us in general. It is one-to- o ne exactly when X is a T 0 space. A subset S of X is called a limit subset if there is a net in X that conv er ges to all the p oint s of S . The family of all the clos e d limit subsets of X will b e denoted L ( X ); it is a compact Hausdo r ff spa c e with its relative τ s -top ology , metrizable if X is second countable, see [4, Th ´ eor` eme 12 and Lemme 2]. W e put L ′ ( X ) := L ( X ) \ {∅} . Then ( L ′ ( X ) , τ s ) is a lo cally compact Hausdorff space; if X is compact then one easily sees that ∅ is an isolated point of ( L ( X ) , τ s ) hence ( L ′ ( X ) , τ s ) is compact. The family of all (closed) maximal limit subsets is denoted ML ( X ) and ML s ( X ) stands for its τ s -closure in L ′ ( X ). 2. equiv alence rela tio ns o n L ′ ( X ) W e define on L ′ ( X ) a n equiv alence relation: we say that A ∼ 1 B if there is a finite sequence { F i | 0 ≤ i ≤ n } in L ′ ( X ) with F 0 = A and F n = B such that F i ∩ F i +1 6 = ∅ , 0 ≤ i ≤ n − 1. Let now C b ∼ ( L ′ ( X )) b e the alge bra of all C -v alued b ounded τ s -contin uo us functions on L ′ ( X ) that are constant on the equiv alence classes with resp ect to ∼ 1 . F or A, B ∈ L ′ ( X ) we shall say that A ∼ 2 B if f ( A ) = f ( B ) for every f ∈ C b ∼ ( L ′ ( X )). Then ∼ 2 is an e q uiv alence relation on L ′ ( X ) a nd by definition A ∼ 1 B implies A ∼ 2 B . On Q ( X ) := L ′ ( X ) / ∼ 2 we shall consider tw o topolo g ies: the quotien t top ology , τ Q , defined by the quotient map Q : L ′ ( X ) − → Q ( X ) when L ′ ( X ) is endow ed with the τ s top ology and the completely reg ular top ology τ C R given by the functions of C b ∼ ( L ′ ( X )) co nsidered as functions on Q ( X ). Obviously τ C R ⊂ τ Q ; it will follow from subsequent results that the question of equality b e t ween these tw o topo logies parallels the situation betw een ( γ ( X ) , τ cr ) and ( γ ( X ) , τ q ). Let f ∈ C b ( X ); then f is co ns tan t on every closed limit subs et of X . Define f L on L ′ ( X ) by f L ( S ) = f ( x ) where x is any p oint of S ∈ L ′ ( X ). Then f L is τ w -contin uo us of L ′ ( X ), thus f L ∈ C b ∼ ( L ′ ( X )). Indeed, if D is an o pen s ubs et o f C then U := { x ∈ X | f ( x ) ∈ D } is op e n hence { S ∈ L ′ ( X ) | f L ( S ) ∈ D } = { S ∈ L ′ ( X ) | S ∩ U 6 = ∅} is in τ w . W e hav e a c o n verse to the sta tement ab out the contin uity of f L but first we need a lemma a bout τ s -conv ergence in L ′ ( X ). It is included in [16, L emma H.2] but we give b elo w its simple pro of for the sake o f self sufficiency . 4 ALDO J. LAZAR Lemma 2.1. Le t { S α | α ∈ A} b e a net in L ′ ( X ) that τ s -c onver ges to S ∈ L ( X ) . If x α ∈ S α for α ∈ A and { x α } c onver ges to x ∈ X then x ∈ S . Pr o of. Ass uming that x / ∈ S we let K b e a compact neighbourho o d of x disjoint from S . Thus S ∈ { T ∈ L ( X ) | T ∩ K = ∅ } , hence even tually S α ∩ K = ∅ and { x α } cannot conv er ge to x , a con tradiction.  Theorem 2.2 . The m ap f − → f L is an isomorphism of C b ( X ) ont o C b ∼ ( L ′ ( X )) . Pr o of. W e hav e to prov e only the surjectivit y of the map. Let g ∈ C b ∼ ( L ′ ( X )) and define f on X b y f ( x ) = g ( S ), S b eing an y element in L ′ ( X ) such that x ∈ S , for instance the closure of { x } . Then f is w ell defined and we are going to show that it is contin uous. Once this will b e done we clear ly shall hav e f L = g a nd the pro of will be finished. Let D be an op en subse t of C a nd x ∈ f − 1 ( D ). W e claim that there is a neighbour hoo d of x contained in f − 1 ( D ). If no t then there is a net { x α | α ∈ A} that co n verges to x but f ( x α ) / ∈ D for every α ∈ A . W e c ho ose S α ∈ L ′ ( X ) s uc h that x α ∈ S α so that f ( x α ) = g ( S α ) for each α ∈ A . The ne t { S α } has a subnet { S α ′ } tha t τ s -conv erges to some S ∈ L ( X ). By Lemma 2.1 x ∈ S hence S 6 = ∅ and g ( S ) = f ( x ) ∈ D . The con tinuit y of g implies that eventu ally f ( x α ′ ) = g ( S α ′ ) ∈ D contradicting our choice of the net { x α } .  R emark 2.3 . It follo ws from the τ w -contin uity of η X , the definition of f L for f ∈ C b ( X ) and the preceding pro of that in the definition of C b ∼ ( L ′ ( X )) we can substitute τ w -contin uity for τ s -contin uity . W e can now define a o ne-to-one map χ from γ ( X ) ont o Q ( X ) as follows: for x ∈ X we let χ ( q ( x )) := Q ( S ) wher e S is any closed limit set that contains x . It rea dily follows from Theorem 2 .2 that the ma p is well defined and it has the stated pr oper ties. As a direct consequence o f the definitions we hav e for every g ∈ C b ( γ ( X )) that ( g ◦ q ) L = g ◦ χ − 1 ◦ Q . It is clear tha t χ is a homeomo rphism of ( γ ( X ) , τ cr ) on to ( Q ( X ) , τ C R ). Prop osition 2 .4. The map χ define d ab ove is a home omorphism fr om ( γ ( X ) , τ q ) onto ( Q ( X ) , τ Q ) . QUOTIENT SP ACES DETERMINED BY ALGEBRAS OF CONTINUOUS FUNCTIONS 5 Pr o of. Let O ⊂ Q ( X ) b e o pen in the q uotien t to p olog y . Then U := q − 1 ( χ − 1 ( O )) is the unio n of all the element s of Q − 1 ( O ) a nd w e claim that U is op e n. Other wise, there are x ∈ U a nd a net { x α } in X \ U that conv er g es to x . F or e a c h index α w e choose S α such that x α ∈ S α . This compels ea c h S α to b elong to the clo sed subset L ′ ( X ) \ Q − 1 ( O ) o f L ′ ( X ). By pas sing to a subnet if necessary we may supp ose tha t { S α } τ s -conv erges to so me S ∈ L ( X ). Lemma 2.1 yields x ∈ S thus S ∈ Q − 1 ( O ). On the o ther ha nd, S / ∈ Q − 1 ( O ) as the τ s -limit o f S α , a contradiction. Therefore U is a n op e n subse t of X , χ − 1 ( O ) is a τ q -op en subset of γ ( X ) and the contin uity of χ is esta blished. Suppo se no w that V ⊂ γ ( X ) is τ q -op en. T he n Q − 1 ( χ ( V )) =  S ∈ L ′ ( X ) | S ⊂ q − 1 ( V )  =  S ∈ L ′ ( X ) | S ∩ q − 1 ( V ) 6 = ∅  is τ w -op en hence τ s -op en. Thus χ ( V ) is τ Q -op en.  Before stating the main result of this section we ne e d a lemma. Lemma 2.5. If X is σ -c omp act then ( L ′ ( X ) , τ s ) is σ -c omp act to o. Pr o of. Let X = ∪ ∞ n =1 K n where each K n is compact and s et L n = { S ∈ L ′ ( X ) | S ∩ K n 6 = ∅} . Clearly L ′ ( X ) = ∪ ∞ n =1 L n and w e are going to show that each L n is τ s -compact. Let { S α } b e a net in L n and without loss o f generality we shall suppo se that it τ s -conv erges to some S ∈ L ( X ). F or each α choose x α ∈ S α ∩ K n . By passing to a subnet w e may supp ose that { x α } conv erge s to so me x ∈ K n . Lemma 2.1 yields x ∈ S thus S ∈ L n .  Theorem 2.6 . If X is σ -c omp act then τ cr = τ q and γ ( X ) is p ar ac omp act. Pr o of. In view of Prop osition 2.4 and the remarks preceding its statement it will suffice to show that τ C R = τ Q and tha t Q ( X ) is paraco mpact. Now, L ′ ( X ) with the τ s top ology is lo cally compa c t Hausdo rff and σ -co mpact by Lemma 2.5 hence Lindel¨ of. Its quotien t space ( Q ( X ) , τ Q ) is Hausdo rff since the real v a lued b ounded contin uous functions on Q ( X ) separate its p oint s. It follows fr om Theorem 1 of [14] that Q ( X ) with its τ Q top ology is a paraco mpact space. In particular it is also completely regular, hence τ C R = τ Q and w e are done. 6 ALDO J. LAZAR  R emark 2.7 . In all of the ab o ve we could have used the space ML s ( X ) instead of L ′ ( X ). W e are g o ing to treat a no ther situation when the tw o to polog ies o n γ ( X ) coincide but fir st we have to introduce a new r elation on the space X that w as considered in [2] for the primitive ideal space o f a C ∗ -algebra . F o r x , y ∈ X we s hall w r ite x ∼ H y if x and y ca nnot be separa ted by disjoint open subsets o f X . This is the same a s saying that there is a closed limit subset of X to which b oth x and y belo ng. In general this is not a transitive rela tion. Clearly , if ∼ H is an equiv a le nce r e lation then each equiv alence class for it is the union o f all the elements in an equiv alence cla ss with resp ect to ∼ 1 on L ′ ( X ) and ea c h such union is an equiv alence class for ∼ H . The following r esult is the same as Prop osition 3.2 of [2] when X is the primitive ideal space of a C ∗ -algebra but the pr oo f b elo w differs in part from that g iv en there. Prop osition 2 . 8. Supp ose ∼ H is an op en e quivalenc e r elation. Then e ach e qu iva- lenc e cla ss is a maximal limit set and e ach maximal limit set of X is an e quivalenc e class for ∼ H . The r elations ∼ and ∼ H ar e the same, τ q = τ cr , the quotient map q is op en and γ ( X ) is a lo c al ly c omp act Hausdorff sp ac e. Pr o of. Let S b e an equiv alence class for ∼ H . W e are going to show that for ev ery finite set { x i | 1 ≤ i ≤ n } ⊂ S and every neighbourho o d V i of x i , 1 ≤ i ≤ n , w e hav e ∩ n i =1 V i 6 = ∅ ; by [4, Lemme 9 ] this will imply that S is a limit set. The c laim is obviously v alid for any pair o f points of S . Suppos e that it is true for any s ubset of n − 1 points of S and let x i ∈ S with a n a rbitrary neig h bo urho o d V i , 1 ≤ i ≤ n . Let U b e the op en sa turation of V n for ∼ H . Then x i ∈ U and U i := V i ∩ U is a neighbourho o d of x i , 1 ≤ i ≤ n − 1. By the induction hyp othesis W := ∩ n − 1 i =1 U i is a no n-v oid o pen set. Cho ose x ∈ W ; there is y ∈ V n such that x ∼ H y . By the definition of ∼ H we have ∩ n i =1 V i ⊃ W ∩ V n 6 = ∅ and the claim is established. No w if S ′ is a limit set with S ′ ⊃ S then each point o f S ′ is ∼ H -equiv alent to e ac h point of S hence S ′ = S . Thus S is a maximal limit set. Since all the p oin ts of a max imal limit set a r e ∼ H -equiv alent no t wo differen t maximal limit sets can intersect and each maximal limit set is a n equiv alence class for ∼ H . QUOTIENT SP ACES DETERMINED BY ALGEBRAS OF CONTINUOUS FUNCTIONS 7 F ro m here o n we follow the pro of of [2 , Pro position 3.2]. If S 1 and S 2 are tw o different ∼ H -classes a nd x i ∈ S i then x 1 and x 2 hav e tw o disjo in t op en neighbour- ho ods V 1 and V 2 , resp ectiv ely . No p oint of V 1 can b e ∼ H -equiv alent to an y po int of V 2 bec ause V 1 ∩ V 2 = ∅ . Hence the op e n quotient map o f X onto X/ ∼ H maps V 1 and V 2 onto tw o disjoint neighbo urhoo ds of S 1 and S 2 resp ectively , which means that the quotient spac e is Hausdor ff. Since the quotient map is op en the quotient space is also lo c ally compact. Clearly if x ∼ H y then x ∼ y . By the complete regularity of X/ ∼ H we conclude that if x and y are no t ∼ H -equiv alent they are also not ∼ - equiv alent. Thus ∼ H and ∼ are iden tical. Since ( γ ( X ) , τ q ) and ( γ ( X ) , τ cr ) ha ve the same b ounded contin uous functions and b oth ar e completely reg ular, the identit y map is a ho meomorphism.  3. C R -sp aces In this section we shall discuss a class of lo cally compact second co un table spaces. A lo cally compact s e cond countable space has a c oun table base consis ting of inte- riors of compact subsets since the family of the in terior s of all the compact subsets is a base and as suc h it m ust co n tain a coun table base b y [10, Pr oblem 1 .F]. Th us such a space X is σ -compact and by Theo rem 2.6 w e ha ve only one topolo gy on γ ( X ) that will be o f in tere st for us. O f course, this is true also for Q ( X ). As we rema rk ed in the prev io us section, the real v alued b ounded contin uous functions on γ ( X ) separate the p oints o f this space. It turns out that when X is second count able a count able family of suc h functions will suffice. Prop osition 3.1 . If X is se c ond c ountable then ther e is a c ountable family of r e al value d b ounde d c ontinuous fu n ctions on γ ( X ) that sep ar ates the p oints of γ ( X ) . Pr o of. F rom Prop osition 2.4 we gather that it will be enough to show tha t such a countable family of functions ex ists on Q ( X ). Recall that ( L ′ ( X ) , τ s ) is loca lly compact Hausdorff a nd sec ond c o un table, in particular σ -compact. Let { K n } be an incr easing sequence of τ s -compact subsets tha t covers L ′ ( X ). Then C ( K n ), the algebra o f a ll rea l contin uous functions on the compac t metriza ble s pace K n is sepa- rable. Hence C b ∼ ( L ′ ( X )) | K n ⊂ C ( K n ) is also s eparable. Thus there is a co un table family { f m n | 1 ≤ m < ∞} ⊂ C b ∼ ( L ′ ( X )) such that { f m n | K n | 1 ≤ m < ∞} is dense 8 ALDO J. LAZAR in C b ∼ ( L ′ ( X )) | K n . Now, if g and h are rea l b ounded co n tin uous functions on Q ( X ) then sup {| g ( y ) − h ( y ) | | y ∈ Q ( K n ) } = sup {| g ◦ Q ( S ) − h ◦ Q ( S ) | | S ∈ K n } . Hence, viewing the elements of C b ∼ ( L ′ ( X )) as functions on Q ( X ), the family { f m n | 1 ≤ m < ∞} s eparates the p oints of Q ( K n ) and { f m n | 1 ≤ m, n < ∞} sepa- rates the points of Q ( X ).  Definition 3.2. A seco nd countable lo cally compact space X will b e called a C R - space if γ ( X ) is a s econd coun table loca lly compact Hausdorff space. In [7] a separable C ∗ -algebra w ho se primitive ideal space is a C R -s pa ce in the ab o ve terminology was called a C R -algebr a . There the conditions imp osed o n the quotient of the primitive ideal space were for the top ology τ cr . Not every separ a ble C ∗ -algebra is a C R -algebra thus not every second countable lo cally compact space is a C R -space. An exa mple is given in [3 , Ex ample 9.2]. The cla ss of C R - a lgebras was found in [7] and [8] useful for the study of c e rtain C ∗ -dynamical systems and the corres p onding crossed pro ducts. It was remarked in [8 ] that for a sec ond co un table lo cally co mpa ct spa c e X each o f the following prop erties is sufficient to ensure that it is a C R -space: X is Hausdorff, X is compac t, ∼ H is an op en equiv alence re lation on X . Of course, this was done in [8 ] only for the primitive ideal space of a C ∗ - algebra so we s ha ll r epro duce a nd adapt the ar gumen ts for the gener al situation. The case of a Hausdorff space is tr ivial. If X is c o mpact then γ ( X ) is compact to o. Recall that L ′ ( X ) is τ s -compact. Theorem 2.2 and the definition o f the quotient top ology on γ ( X ) yield an isomorphism of the algebra C ( γ ( X )) in to the separable algebra C ( L ′ ( X )) hence C ( γ ( X )) and γ ( X ) is second coun table. Alternatively , w e can use the Prop osition 3.1 to infer that γ ( X ) is metriza ble. Now suppose that X is a second countable lo cally compact spac e for which ∼ H is an op en equiv alence relation. Then b y Prop osition 2.8, γ ( X ) is lo cally co mpa ct Hausdorff. Since the quotient map is co n tin uous and o p en it is easily seen that γ ( X ) is se c ond countable. W e give b elow a characteriza tion of the C R -spaces. QUOTIENT SP ACES DETERMINED BY ALGEBRAS OF CONTINUOUS FUNCTIONS 9 Theorem 3.3. L et X b e a se c ond c ountable lo c al ly c omp act sp ac e. The fol lowing c onditions ar e e quivalent: (i) X is a C R -sp ac e; (ii) γ ( X ) is lo c al ly c omp act; (iii) γ ( X ) is first c ount able. Pr o of. (i) ⇒ (ii). This is immediate. (ii) ⇒ (iii). By assumption γ ( X ) is lo c a lly compact Hausdorff and σ -compact since X is σ -compact. Then, by [6 , Theo rem 7.2], there is an increasing sequence o f op en sets { U n } in γ ( X ) such that γ ( X ) = ∪ n =1 U n , U n is compact and U n ⊂ U n +1 for ev ery n . Pro positio n 3.1 yields a sequence { g k } of cont inuous functions fro m γ ( X ) to the interv al [0 , 1] that separates the p oint s of γ ( X ). The restr ictions of the functions { g k } to the c o mpact Hausdorff spa ce U n allow us to define a homeomorphism of U n int o [0 , 1] ℵ 0 . Hence U n is metrizable and each op en set U n is second countable. W e co nclude that γ ( X ) is second co un table. W e a ctually prov ed tha t (i) follows fro m (ii) which is appa ren tly more than we needed. (iii) ⇒ (i). F rom Prop osition 2.4 and the hypothesis it fo llo ws that Q ( X ) is first countable. W e know that it is a Hausdorff space. It has been noted ab ov e that L ′ ( X ) is loca lly compact Hausdorff a nd under the pre sen t hypothesis on X it is also s econd countable. Thus Q ( X ) is a Hausdorff quo tien t of a second countable lo cally compac t Hausdor ff space. Then [15, Theorem 3] implies that Q ( X ) is lo cally compact and second countable and the same prop erties are shared b y γ ( X ) by Prop osition 2.4.  Another characterization of C R - spaces can b e given in terms of the quo tien t ma p q . A con tinuous map ϕ fro m a topo logical space Y onto a top ological space Z was called in [13] a bi-quotient map if for every z ∈ Z and every o pen c o ver of ϕ − 1 ( z ) there ar e finitely man y se ts { U i } in the co ver suc h that the interior of ∪ i =1 ϕ ( U i ) is a neighbourho o d of z . It follo ws from [13, Pro positio n 3.3(d) and P rop osition 3.4 ] that whenever ϕ is a quotient map o f the second co un table lo cally compact spa ce Y onto the Hausdorff spac e Z then Z is lo cally compact and second c o un table if and only if ϕ is bi-quotient. If w e adapt this general result to our situation we get 10 ALDO J. LAZAR Prop osition 3.4 . The se c ond c oun t able lo c al ly c omp act sp ac e X is a C R -sp ac e if and only if the quotient map q is bi-quotient. References 1. R. J. Arc hbold, T op olo gie s for primal ide als , J. London M ath. So c. 3 6 (1987), 524–542 . 2. R. J. Arch b old and D. W. B. Somerset, Q uasi-st anda r d C ∗ -algebr as , Math. Pro c. 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Michael, T op olo gies on sp ac es of subsets , T rans. Amer . Math. So c. 71 (1951 ), 152–182. 13. E. Mic hael, Bi-q uotient maps and c artesian pr o ducts of quotient maps , A nn. Inst. F ourier 18 (1968), 287–302. 14. K. Morita, On de c omp osition sp ac e s of lo c al ly c omp act sp ac es , Pro c. Japan Acad. 32 (1956), 544–548. 15. A. H. Stone, Metrisability of de c omp osition sp ac e s , Pro c. Amer . Math. Soc. 7 (1956), 690–700. 16. D. P . Williams, Cr osse d pr o ducts of C ∗ -algebr as , Mathematical Surve ys and Monographs, vol. 134, A merican Mathematical Society , Pr o vi dence , RI, 2007. School of Mat hemat ical Sciences, Tel A v iv University, Tel A v iv 69778, Israel E-mail addr ess : aldo@p ost.tau.ac.i l