Hyperspaces of Closed Limit Sets

HYPERSP A CES OF CLOSED LIMIT S E TS ALDO J. LAZAR Abstract. W e study Mi c hael’s low er semifinite topology and F ell’s topology on the co llection of all closed limit subsets of a topological s pace. Special atten tion is giv en to the subfamily of all maximal l imit sets. 1. Introduction The collection of all closed subsets of a top o lo gical spa ce has bee n for long of int erest to topo logists a nd functional a nalysts. It seems that the mo dern inv esti- gation of the sub ject b e gan with [8]. It is well known that there is a one-to- one corres p o ndenc e be tw een the closed tw o -sided ideals o f a C ∗ -algebra and the closed subsets of its primitive ideal space a s detailed in [4, Pr o p o sition 3.2 .2]. Natura lly , this co rresp ondence a ttracted the interest of o pe r ator algebra ists in the h yp erspace of the clos ed subs ets o f a top ologica l space. It led F ell to the definition in [6] of a top ology on this hyperspa ce that is of significance in top olog y and several branches of analysis. Moreov er, a ccording to [2, Prop osition 3.2], when one restr icts this corres p o ndenc e to the closed limit subs e ts of the primitive ideal space, a very in- teresting class of ideals is obtained. The wealth of information given in [1] on this class of ideals stimulated the present investigation a nd a significant p ortio n of the results that a ppe ar her e were pr ov ed in [1] for this special fa mily of ideals of a C ∗ -algebra . How e ver, no knowledge of the theor y of C ∗ -algebra s is req uired for the understanding of the following; w e discuss the pr op erties of tw o topolo gies on the collection of all the closed limit subsets of a top ologica l space. All the definitions beyond the common knowledge of a top ologis t or an analyst are given in the next section. Of course, all our results ar e significa nt only for no n Ha us dorff spaces, as the primitive idea l spaces often a re. 1991 Mathematics Subje c t Classific ation. Primary: 54B20; Secondary: 54D45, 46L05. Key wor ds and phr ases. the hyperspace of the closed subsets, the lo wer semifinite topology , F ell’ s topology . 1 2 ALDO J. LAZAR In section 3 we study the Michael’s low er s emifinite top olo gy o n the fa mily of all closed limit sets. W e establish that with this top ology this h yp erspa c e is a lo cally compact Baire space. W e restrict the discussion to the collection of all maxima l limit sets in section 4. The F ell top ology and the low e r semifinite top olog y coincide on this h yp erspa c e. This hyperspa c e is a lso a Baire spa ce and if the initial s pace is second co untable and lo cally co mpact then the hyper space of maximal limit sets is a G δ subspace in the space of all closed limit sets eq uipp ed with the F ell topolog y . 2. Prelimina ries F or a top ological space X we sha ll denote by F ( X ) the hyperspace of all its closed subsets and F ′ ( X ) will stand for the collection o f all the no n-void closed subsets of X . A s ubset L of X is called a limit set if there is a net that co nverges to all the po int s of L . By [5, Lemme 9 ], L ⊂ X is a limit se t if and only if every finite family o f o p en subse ts that intersect L has a non- void intersection. The collection of all the closed limit sets o f X will b e denoted by L ( X ) a nd we set L ′ ( X ) := L ( X ) ∩ F ′ ( X ). It easily follows from the lemma quoted above and Zorn’s lemma that each L ∈ L ( X ) is contained in a ma ximal limit set. Obviously , every maximal limit set is c lo sed and non-void. M L ( X ) will denote the collec tio n of all maximal limit sets. There is a natural map η X : X → L ′ ( X ) defined by η X ( x ) := { x } . This ma p is one to one if and only if X is a T 0 space. Some of the results b elow are v alid under the r estriction that the top ologica l space X is lo cally compact that is, each point in X ha s a fundamental system o f compact neighbourho o ds . Such s paces were called lo cally quasi-co mpact in [3, I, 9, Ex. 29]. F or C b e a co mpact subset and Φ a finite family of op en subsets of X let U ( C, Φ) := { A ∈ F ( X ) | A ∩ C = ∅ , A ∩ O 6 = ∅ , O ∈ Φ } . The collection o f all s uc h U ( C, Φ) forms a base for a top o logy on F ( X ) that was defined by F ell in [6] a nd which will be denoted her e b y τ s . It was shown in [6] that with this top olo gy F ( X ) is a compact space that is Hausdorff if X is lo cally compact. If X is lo cally compac t a nd has a co unt able bas e then ( F ( X ) , τ s ) is metrizable, see [5, Lemme 2]. HYPERSP ACES OF CLOSED LIMIT SETS 3 The collectio n of all U ( ∅ , Φ ) when Φ runs through all the finite families o f o p en subsets of X is the base of a T 0 top ology on F ( X ), weaker than τ s , which we shall denote by τ w . It was called the lo wer semifinite topo logy in [8, Definition 9.1] and was further discussed in [7]. It is eas ily see n that if B is a bas e for the top ology of X then the collection of all U ( ∅ , Φ) when Φ runs through a ll the finite subfamilies o f B is a base for ( F ( X ) , τ w ). Thus, if X is second countable then ( F ( X ) , τ w ) is a lso second coun table. Clearly F ′ ( X ) = U ( ∅ , { X } ) hence F ′ ( X ) is τ w -op en in F ( X ). The o nly τ w -op en subset of F ( X ) to which the empt y subset of X be lo ngs is F ( X ) itself so F ′ ( X ) is τ w -dense in F ( X ) and L ′ ( X ) is τ w -dense in L ( X ). Obviously , ML ( X ) is a ls o τ w -dense in L ( X ). F or every A ∈ F ( X ) the τ w -closure o f { A } is { B ∈ F ( X ) | B ⊂ A } a nd this en tails the T 0 separatio n prop erty for( F ( X ) , τ w ) . The map η X is τ w contin uous ; it is a ho meomorphism o n its image if X is T 0 . Generalizing [1, Pr op osition 3.1], we claim that alwa ys the τ w -closure of η X ( X ) is L ( X ). Indeed, it is easily seen that A ∈ F ( X ) is in the τ w -closure of η X ( X ) if and only if every finite fa mily of op en subsets that intersect A has a non-void int ersec tio n that is, if a nd o nly if A ∈ L ( X ). In particula r , L ( X ) is τ w -closed hence also τ s -closed. Th us ( L ( X ) , τ s ) is a compact Hausdorff space. F rom the τ w -density o f η X ( X ) in L ( X ) it follows that ( L ( X ) , τ w ) is co nnected when X is connected. How ever, triv ial exa mples sho w that ( L ( X ) , τ s ) need not b e co nnected if X is co nnec ted. Concerning the τ s -conv ergence of nets the following was prov e d in [9, Lemma H.2]: Prop ositio n 2.1. L et { A ι } b e a n et of close d subsets of the t op olo gic al sp ac e X and A ∈ F ( X ) . The net τ s -c onver ges to A if (a) given x ι ∈ A ι such that t he net { x ι } c onver ges t o x , then x ∈ A , and (b) if x ∈ A then ther e is a subnet { A ι κ } and p oints x ι κ ∈ A ι κ such that { x ι κ } c onver ges t o x . When X is lo c al ly c omp act the c onverse is true to o : the net { A ι } τ s -c onver ges to A only if the c onditions ( a) and (b) hold. The character iz ation o f the τ s -conv ergence of nets given below is in line with our attempt to in vestigate the links b etw een the t wo topolo gies on the hyper space of closed subsets no ted ab ov e. A net in a top ological space w as ca lled by F ell primitive 4 ALDO J. LAZAR in [6] if the set of a ll its limits equals the set o f all its cluster p oints. With this definition we have Prop ositio n 2.2. L et X b e a top olo gic al sp ac e. If { A ι } is a primitive net in ( F ( X ) , τ w ) and the set of al l its τ w -limits is { B ∈ F ( X ) | B ⊂ A } wher e A ∈ F ( X ) then { A ι } τ s -c onver ges to A . If X is lo c al ly c omp act then the c onverse holds: a net { A ι } that is τ s -c onver gent to A in F ( X ) is primitiv e in ( F ( X ) , τ w ) and the set of al l its τ w -limits is { B ∈ F ( X ) | B ⊂ A } . Pr o of. Supp os e { A ι } is a τ w -primitive net in F ( X ) and the s et of a ll its limits is { B | B ⊂ A } . Let U ( C, Φ) b e a basic τ s -neighbourho o d of A . If we assume that { A ι } is not even tually in U ( C , Φ) then, by passing to a s ubnet and relab elling, we hav e A ι ∩ C 6 = ∅ for each ι . W e choose po ints x ι ∈ A ι ∩ C . There is a subnet { x ι κ } that conv erges to a p oint x in the compact s et C . W e claim tha t { A ι κ } τ w -conv erges to { x } . Indeed, let Φ 1 be a finite family of op en subsets o f X all of which intersect { x } that is, such that x be lo ngs to the intersection V of all the sets in Φ 1 . Then x ι κ is even tually in V . Thus, for κ larg e enough A ι κ ∩ V 6 = ∅ and the claim is established. Thus { x } is a τ w -cluster p oint of the primitive net { A ι } hence { x } ⊂ A . W e g ot A ∩ C 6 = ∅ , a contradiction. Suppo se no w that X is lo cally co mpa ct and the net { A ι } τ s -conv erges to A . It follows readily fro m the definition o f the top olog ies on F ( X ) that { A ι } τ w -conv erges to every clo sed subset B of X which is a subset of A . Assume tha t there is a s ubnet { A ι κ } that τ w -conv erges to some B ∈ F ( X ) with B \ A 6 = ∅ and let x ∈ B \ A . There is a co mpact set C ⊂ X such that x ∈ I nt ( C ) ⊂ C ⊂ X \ A . U ( C, { X } ) is a τ s -neighbourho o d of A hence A ι κ ∩ C = ∅ even tually . O n the o ther hand, U ( ∅ , { I nt ( C ) } ) is a τ w -neighbourho o d of B hence A ι κ ∩ I nt ( C ) 6 = ∅ a nd w e got a contradiction. W e hav e proved that each τ w -cluster p o int o f { A ι } is a subset of A and we are do ne.  3. the topology τ w First we w ant to e stablish the lo ca l co mpa ctness of F ( X ), F ′ ( X ), L ( X ), and L ′ ( X ) with their τ w -top ology when the space X is lo cally compac t. The r esult for the fir st tw o spaces is lik ely to be known but w e hav e no referenc e for it. The HYPERSP ACES OF CLOSED LIMIT SETS 5 lo cal compactness of L ( X ) and L ′ ( X ) was establis hed when X is the pr imitive ideal space of a C ∗ -algebra in [1, Theor em 3.7] by using sp ecial prop erties of such spaces . Lemma 3.1. L et C 1 , . . . C n b e c omp act subsets of the top olo gic al sp ac e X . Then S := { A ∈ F ( X ) | A ∩ C i 6 = ∅ , 1 ≤ i ≤ n } is τ w -c omp act. Pr o of. L et { M α | α ∈ A} b e a net in S and x i α ∈ M α ∩ C i . By pas sing to successive subnets w e ma y suppos e that each of the nets  x i α | α ∈ A  , 1 ≤ i ≤ n , con verges to a p oint x i ∈ C i . Denote by M the clos ure o f { x 1 , . . . x n } and supp ose E := { U 1 , . . . U p } is a finite family of o pen subsets o f X s uc h that M ∈ U ( ∅ , E ). T hen for each k , 1 ≤ k ≤ p , there is 1 ≤ i k ≤ n such that x i k ∈ U k . Hence there is α 0 ∈ A such that for all 1 ≤ k ≤ p and α > α 0 we ha ve x i k α ∈ U k . Th us, if α > α 0 then M α ∩ U k 6 = ∅ , 1 ≤ k ≤ p . W e hav e established that { M α } con verges weakly to M and clearly M ∈ S .  Theorem 3.2 . If X is a lo c al ly c omp act sp ac e then F ( X ) , F ′ ( X ) , L ( X ) , and L ′ ( X ) ar e lo c al ly c omp act sp ac es with t heir τ w top olo gy. Pr o of. Supp os e X is a lo ca lly compact space and let A b e a closed subset of X . F or a basic τ w -neighbourho o d U ( ∅ , { U i } n i =1 ) of A we choos e x i ∈ A ∩ U i , 1 ≤ i ≤ n . Let V i be a compact neighbourho o d of x i contained in U i and W i := I nt ( V i ). Then A ∈ U ( ∅ , { W i } n i =1 ) ⊂ V := { B ∈ F ( X ) | B ∩ V i 6 = ∅ , 1 ≤ i ≤ n } ⊂ U ( ∅ , { U i } n i =1 ) . Thu s V is a neighbo urho o d of A that is compact by the pr eceding lemma. W e have prov ed tha t ( F ( X ) , τ w ) is lo cally co mpact. As remar ked a bove, F ′ ( X ) is τ s -op en in F ( X ), L ( X ) is τ w -closed, L ′ ( X ) = L ( X ) ∩ F ′ ( X ) is relativ ely op en in L ( X ) and the conc lus ion follows.  The next res ult was stated in [1, Pro po sition 3.4 ] for the primitive ideal space o f a C ∗ -algebra . How e ver, the pr o of given there is v alid for a ny top ologica l spa ce and we repro duce it her e. Prop ositio n 3. 3 . If X is a Bair e top olo gic al sp ac e then ( L ( X ) , τ w ) and ( L ′ ( X ) , τ w ) ar e Bair e sp ac es. 6 ALDO J. LAZAR Pr o of. F or e ach natural n umber n let U n be a τ w -dense op en subset of L ( X ). Since η X ( X ) is τ w -dense in L ( X ) and η X is τ w -contin uo us, η − 1 X ( U n ) is an op en dens e subset of X . F rom the h yp othesis it follows that ∩ n ≥ 1 η − 1 X ( U n ) is dense in X . But then η X ( ∩ n ≥ 1 η − 1 X ( U n )) = η X ( X ) \ ( ∩ n ≥ 1 U n ) is τ w -dense in Li ( X ). In particular, ∩ n ≥ 1 U n is τ w -dense in L ( X ). L ′ ( X ) is an o p en dense s ubset of ( L ( X ) , τ w ) so it is a Ba ire space to o.  Prop ositio n 3.4. If X has a b ase c onsisting of op en and c omp act sets then t he same is tr ue for ( F ( X ) , τ w ) and its subsp ac es F ′ ( X ) , L ( X ) , and L ′ ( X ) . Pr o of. Supp os e B is a ba se for the top ology of X co nsisting of op en and compact sets. Then the collectio n of a ll the families U ( ∅ , Φ) where Φ runs through all the finite subfamilies of B is a base for ( F ( X ) , τ w ). Each U ( ∅ , Φ) is τ w -compact by Lemma 3.1. W e g et a base for F ′ ( X ) by r equiring Φ to run through the nonempty finite subfamilies of B . In tersecting ea ch of the elements of the bases we got for F ( X ) and F ′ ( X ) with the τ w -closed set L ( X ) we ge t bas es as needed for L ( X ) and L ′ ( X ), r esp ectively .  4. The hypersp ace ML ( X ) The next r esult generalizes [1, Theorem 4.2] where the fr a mework is that o f a certain family of ideals of a C ∗ -algebra and the pro of uses C ∗ -algebra ic metho ds. The ”if” part of the statemen t is a lso a conseq uence o f [5, Lemme 1 5]. Theorem 4. 1. The identity map ( L ( X ) , τ w ) → ( L ( X ) , τ s ) is c ontinuous at A ∈ L ( X ) if and only if A ∈ ML ( X ) . Pr o of. Supp os e A is a maximal limit set. Let C b e a compac t subset of X and Φ a finite family of op en subse ts o f X suc h that A ∈ U ( C, Φ). W e c la im that there is a finite family Ψ ⊃ Φ of o pen subsets o f X each of which has a nonempty intersection with A and s uch that U ( ∅ , Ψ) ∩ L ( X ) ⊂ U ( C, Φ) ∩ L ( X ). This, of cour se, will establish the co ntin uity of the iden tity map a t A . HYPERSP ACES OF CLOSED LIMIT SETS 7 Assume there is no such Ψ. Then for each finite family Ψ ⊃ Φ of op en subsets of X s uch that every set in Ψ ha s a nonempty int erse c tio n with A there is B Ψ ∈ ( U ( ∅ , Ψ) \ U ( C, Φ)) ∩ L ( X ). Denote the co lle ction of all such families Ψ by Λ and order it by inclusion. Clear ly Ψ ∈ Λ implies B Ψ ∩ C 6 = ∅ . Choose x Ψ ∈ B Ψ ∩ C . The net { x Ψ } has a c onv er ging subnet to some p oint x ∈ C . W e hav e x / ∈ A hence A ∪ { x } % A . W e shall show that A ∪ { x } is a limit set hence A ∪ { x } ∈ L ′ ( X ), and this will yield a con tradiction to the ma x imality of A . Let N be the family of all the open neighbourho o ds of x . W e order N × Λ b y defining ( V 1 , Ψ 1 ) ≺ ( V 2 , Ψ 2 ) if V 1 ⊃ V 2 and Ψ 1 ⊂ Ψ 2 ). Denote by Γ the co llection of all the pairs ( V , Ψ) ∈ N × Λ suc h that the finite family of open sets { V } ∪ Ψ has a nonempty intersection. F or ( V 1 , Ψ 1 ) , ( V 2 , Ψ 2 ) ∈ N × Λ there is ( V , Ψ) ∈ Γ such that ( V 1 , Ψ 1 ) ≺ ( V , Ψ) and ( V 1 , Ψ 2 ) ≺ ( V , Ψ). Indeed, V := V 1 ∩ V 2 is an op en neig hbo urho o d of x a nd Ψ 1 ∪ Ψ 2 ∈ Λ hence there is Ψ ∈ Λ that satisfies Ψ ⊃ Ψ 1 ∪ Ψ 2 and x Ψ ∈ V . Th us x Ψ ∈ V ∩ B Ψ and since B Ψ is a limit set that belo ngs to U ( ∅ , Ψ), the family o f o pen sets { V } ∪ Ψ has a nonempty intersection by the prev iously quoted Le mme 9 of [5]. W e go t ( V , Ψ) ∈ Γ as needed. In particular , Γ is a dir ected set with this or de r restr icted to it. F or each ( V , Ψ) ∈ Γ we cho ose y ( V , Ψ) in the in tersectio n of the family { V } ∪ Ψ. The net { y ( V , Ψ) } co nv er ges to every p oint of { x } ∪ A . It is clear that the net co nv er ges to x . L e t now y b e a po int of A and W a n op en neig h b ourho o d o f y . With Ψ 0 := { W } ∪ Φ w e hav e ( X, Ψ 0 ) ∈ N × Λ . By the o rder prop erty of Γ pr ov ed ab ov e there is ( V 1 , Ψ 1 ) ∈ Γ such that ( X , Ψ 0 ) ≺ ( V 1 , Ψ 1 ). Clear ly if ( V , Ψ) ∈ Γ and ( V 1 , Ψ 1 ) ≺ ( V , Ψ) then W ∈ Ψ hence y ( V , Ψ) ∈ ∩{ O | O ∈ Ψ } ⊂ W . W e hav e pr ov ed that the net { y ( V , Ψ) | ( V , Ψ) ∈ Γ } co nv er g es to y as cla imed. Let no w L b e a non-ma ximal clos ed limit set of X . There are z ∈ X \ L and a net that converges to all the p oints o f L ∪ { z } . The set L b elong s to the τ s -op en set U ( { z } , { X } ) but no τ w -neighbourho o d of L in L ( X ) is contained in U ( { z } , { X } ) th us the identit y map fro m ( L ( X ) , τ w ) to ( L ( X ) , τ s ) is not contin uous at L . Indeed, if U ( ∅ , Φ) is a ny basic τ w -neighbourho o d of L then L ∪ { z } ∈ U ( ∅ , Φ) ∩ L ( X ) but L ∪ { z } / ∈ U ( { z } , { X } ).  Corollary 4.2. The r estrictions of τ w and τ s to ML ( X ) c oincide. 8 ALDO J. LAZAR A po int y of a top olog ical spa ce Y is called separated in Y if, for every z ∈ Y \ { y } , y and z hav e disjoint neig hbo urho o ds; equiv a le n tly , { y } is a ma x imal limit set(see [5, D` efinition 1 6]). It is proved in [5, Th ` e o r´ eme 19] that if Y is a second countable lo cally compact Baire space then the s ubs et of all separated po in ts in Y is a dense G δ . F or any top ologica l space X the density of the set of all separated p oints in ( L ( X ) , τ w ) is an immediate corollar y of the next result. Theorem 4.3. L et X b e a top olo gic al sp ac e. A n element A of L ( X ) is sep ar ate d in ( L ( X ) , τ w ) if and only if A is a maximal limit set. Pr o of. I f A ∈ L ( X ) is not maximal then there is A 1 ∈ L ( X ) such that A 1 % A . Then A 1 do es not b elong to the τ w -closure o f { A } in L ( X ). How e ver, A is in the τ w -closure of { A 1 } in L ( X ) hence A and A 1 cannot be separ a ted by disjoint τ w -op en sets. Suppo se now that A is a maxima l limit s e t and A 1 ∈ L ( X ) do es not belong to the τ w -closure o f { A } that is, A 1 is not included in A . Then A ∪ A 1 ∈ F ( X ) \ L ( X ). By [5, Lemme 9] there is a finite family Φ of op en subsets o f X s uch that each of them ha s a nonempty in tersection with A ∪ A 1 but the int ersec tio n of all the sets in Φ is void. Let Ψ b e the subfamily of Φ co nsisting o f those sets that ha ve a nonempty intersection with A . Since A 1 ∈ L ( X ) we must hav e, by the ab ov e quoted lemma of Dixmier, Ψ 6 = ∅ . Similarly , Ψ 1 := Φ \ Ψ is not empty since A ∈ L ( X ). Now, U ( ∅ , Ψ ) ∩ L ( X ) is a τ w -neighbourho o d of A in L ( X ) and U ( ∅ , Ψ 1 ) ∩ L ( X ) is a τ w -neighbourho o d of A 1 in L ( X ). W e ha ve U ( ∅ , Ψ) ∩ U ( ∅ , Ψ 1 ) ∩ L ( X ) = ∅ hence A and A 1 can be separated by disjoint τ w -op en sets. Indeed, if the ab ov e equality do es no t hold and B ∈ U ( ∅ , Ψ) ∩ U ( ∅ , Ψ 1 ) ∩ L ( X ) then ∩{ V | V ∈ Φ } = ( ∩{ V | V ∈ Ψ } ) \ ( ∩{ V | V ∈ Ψ 1 } ) 6 = ∅ since B ∈ L ( X ), a cont radic tio n.  The following t wo prop ositions were stated and pro ved in [1 ] in the language of C ∗ -algebra s. W e only had to re w r ite the pro ofs to b e fit in a more general situation. HYPERSP ACES OF CLOSED LIMIT SETS 9 Prop ositio n 4.4 ([1, Prop os ition 4.9 ]) . M L ( X ) is a Bair e sp ac e if X is a Bair e sp ac e. Pr o of. L et {V n } be a sequence of τ w -op en subsets of L ( X ) s uch that ev ery U n := V n ∩ ML ( X ) is dense in M L ( X ). Since ML ( X ) is τ w -dense in L ( X ) we get that each V n is τ w -dense in L ( X ). By Pro po sition 3.3, ∩V n is τ w -dense in L ( X ). Let now U be an open set in ML ( X ). Then U = V ∩ ML ( X ), V b eing a τ w -op en set in L ( X ). There exist B ∈ ( ∩V n ) ∩ V and B 1 ∈ ML ( X ) with B ⊂ B 1 . Since B 1 belo ngs to any τ w -op en set of L ( X ) to which B be lo ngs, we have B 1 ∈ ( ∩V n ) ∩ V ∩ ML ( X ) = ( ∩U n ) ∩ U . Hence ∩U n is dense in ML ( X ).  Prop ositio n 4. 5 ([1, Corollar y 4.6]) . If X is a se c ond c ountable lo c al ly c omp act Bair e sp ac e then the fa mily { { x } | x is separ ated in X } is dense in ML ( X ) . Pr o of. As mentioned b efore, [5, Th ` eor´ eme 19] a sserts that the se t T := { x ∈ X | x is separated in X } is dense in X . Since η X ( X ) is τ w -dense in L ( X ) a nd η X is τ w -contin uo us w e can infer that η X ( T ) is τ w -dense in L ( X ). In particular, η X ( T ) = η X ( X ) ∩ M L ( X ) is dense in ML ( X ).  W e shall ha ve more to say ab out the set considered in the statement of Prop o- sition 4.5 in Corollary 4 .10 Theorems 4.1 and 4.3 give us s ome information ab out the wa y ML ( X ) is imbed- ded in L ( X ). Theo rem 4.8 will show us a nother asp ect of this imbedding when the space is seco nd co unt able. Firs t we need t wo lemma s. Lemma 4.6. L et Y b e a c omp act sp ac e, M ⊂ Y × Y and S ( M ) := { y ∈ Y | { y } × Y ⊂ M } . If M is op en then S ( M ) is op en and if M is a G δ set then S ( A ) is a G δ set to o. 10 ALDO J. LAZAR Pr o of. Supp os e M is an op en set. If y ∈ S ( A ) then, by using the co mpactness of Y , we can infer that there a re op en subsets { U i } n i =1 and { V i } n i =1 of Y such that { y } × Y ⊂ ∪ n i =1 ( U i × V i ) ⊂ M . Then y ∈ ∩ n i =1 U i ⊂ S ( M ) and ∩ n i =1 U i is op en. Suppo se now M is a G δ set, M = ∩ ∞ 1 M n with each M n op en in Y × Y . Since S ( M ) = ∩ ∞ 1 S ( M n ) and S ( M n ) is op en by the first par t of the pro of, the c o nclusion obtains.  Lemma 4.7. L et X b e a lo c al ly c omp act sp ac e. Then E := { ( A, B ) ∈ L ( X ) × L ( X ) | A ⊂ B } is ( τ s × τ s ) -close d in L ( X ) × L ( X ) . Pr o of. L et { ( A ι , B ι ) } be a net in E that ( τ s × τ s )-conv erges to ( A, B ). Given x ∈ A there ex ists, by P rop osition 2 .1, a subnet { A ι κ } o f { A ι } a nd po ints x ι κ ∈ A ι κ ⊂ B ι κ such that { x ι κ } co nv er ges to x . Aga in by Prop ositio n 2 .1, x ∈ B a nd we hav e shown A ⊂ B that is ( A, B ) ∈ E .  Theorem 4.8. If X is a se c ond c ountable lo c al ly c omp act sp ac e then ML ( X ) is a G δ subset of ( L ( X ) , τ s ) . Pr o of. Se t D := { ( A, A ) | A ∈ L ( X ) } , E := { ( A, B ) ∈ L ( X ) × L ( X ) | A ⊂ B } , and T := L ( X ) × L ( X ) \ ( E \ D ) . Then for A ∈ L ( X ) w e have A ∈ ML ( X ) if and only if { A } × L ( X ) ⊂ T .  R emark 4 .9 . If X is a seco nd countable lo cally co mpact spa ce then ML ( X ) is a Baire space since it is a G δ subset of the compact metrizable space ( L ( X ) , τ s ). HYPERSP ACES OF CLOSED LIMIT SETS 11 Corollary 4 . 10. If X is a se c ond c ount able lo c al ly c omp act sp ac e in which every close d subset is a Bair e sp ac e then { { x } | x is separ ated in X } is a G δ subset of L ( X ) ; it is also a dense subset of ML ( X ) . Pr o of. B y [5, Theor em 7], η X ( X ) is a G δ subset of F ( X ) hence it is a G δ subset of L ( X ). Then { { x } | x is sep ar ate d in X } = η X ( X ) ∩ M L ( X ) is a dense G δ subset of ML ( X ) by Pro p o s ition 4 .5 a nd a G δ subset of L ( X ) b y Theo rem 4 .8.  R emark 4.11 . The pr imitiv e ideal space of a s eparable C ∗ -algebra with its hull- kernel top ology s atisfies the hypothesis of Corollary 4.10. —————————————————————- References 1. R. J. Ar c hbold, T op olo gies for primal i de als , J. London Math. Soc. (2) 36 (1987), 524–542. 2. R. J. Ar c hbold and C. J. K. Batty , O n factorial states of op er ator algebr as, III , J. Op erator Theory 15 (1986), 53–81. 3. N. 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