Building suitable sets for locally compact groups by means of continuous selections

Building suitable sets for lo cally compact groups b y means of con tin uous selections ∗ Dmitri Shakhmato v Dedicated t o the memory of Jan P elan t Abstract If a discrete subset S of a topolo gical group G with the identit y 1 g enerates a dense subgroup of G and S ∪ { 1 } is closed in G , then S is called a suitabl e set for G . W e apply Michael’s selection theorem to offer a direct, self-co ntained, purely topolo g ical pro of o f the result of Hofmann and Morris [8] on the existence of suitable sets in lo cally compact groups. Our approach uses only elementary facts from (top ological) group theory . All top ologi cal groups considered in this p ap er are assum ed to b e Hausdorff, and all top olo gical spaces are assu med to b e T ychonoff. 1 Motiv ating bac kground Let G b e a group . W e use 1 G to denote the identit y element of G . If X is a su bset of G , then h X i will denote the smallest subgroup of G con taining X , and w e sa y that X (alge braically) gener ates h X i . Definition 1. [2 , 13, 8] A subset X of a top ologic al group G is called a suitable set for G pro vided that: (i) X is discrete, (ii) X ∪ { 1 G } is closed in G , (iii) h X i is dense in G . Suitable sets were considered fir s t in the early sixties b y T ate in the framewo rk of Galois cohomology (see [2]). T ate pr o ved 1 that every pr o finite group has a suitable set. This result has later b een pro v ed also b y Mel’nik o v [13]. Later on, Hofmann and Morris disco v ered the follo wing fund ame nt al theorem: ∗ MSC Subj. Class. : Primary: 22D05; Secondary: 22A05, 22C05, 54A25 , 54B05, 54B35, 54C60, 54C 65, 54D30, 54D45, 54H11. Keywor ds and phr ases : locally compact group, generating rank, suitable set, low er semicon tinuous, set- v alued map, selection, conv erging seq uence, compactly generated. Mailing addr ess: D ivisi on of Mathematics, Ph ysics and Earth Sciences, Graduate School of Science and Engineering, Ehime Universit y , 790-8577, Japan. E-mail addr ess: dmitri@dpc.ehim e-u.ac.jp . T o app ear in: T op ol ogy and its App l ications . The author gratefully ac knowl edges partial financial sup p ort from the Gran t - in-Aid for Scientific Research no. 1954009 2 by th e Ja pan So ciet y for the Promotion Science (JSPS). 1 This pro of is extremely condensed. Detailed p roofs can b e found in [16] and [7]. 1 Theorem 2. [8, T h eo rem 1.12] E very lo c al ly c omp act gr oup has a suitable set. Let u s briefly outline main p oin ts of the pro of fr o m [8]. The authors first pr o ve the existence of su ita ble sets in compact connected Ab elian groups. This is accomplished b y using the full strength of the theory of free compact Ab elian groups [6]. The theorem f o r compact connected groups then follo ws from the Ab elian compact connected case and the result of Kur a nishi [11] that eve ry compact connected simple group has a dense sub g roup generated by t w o elemen ts. Sin ce compact totally d isconn e cted group s hav e suitable sets by the resu lt s of T ate and Mel’nik ov (cited ab o ve ), the authors of [8] then com bine connected and totally disconnected cases together to get the conclusion f o r all compact groups by deplo ying a theorem of Lee [12]: Ev ery compact group G conta ins a closed totally discon- nected su b group K such that G = c ( G ) · K , where c ( G ) is the connected comp onen t of G . Ha ving prov ed the r e sult in compact case, Hofmann and Morris then p roceed to d e duce the general case from the compact case using some structure theorems for lo cally compact groups. The main pur pose of th i s article is to offer a dir e ct, self-con tained, pu rely top olog ical pro of of Theorem 2 based on Mic hael’s select ion th e orem. Our pro of is in th e spirit of [18, 17], and uses only elemen tary facts from (topological) group theory . Theorem 2 allo wed Hofmann and Morris [8] to introd uce the gener ating r ank s ( G ) = min {| X | : X is a suitable set for G } of a lo ca lly compact group G . (F or pr o finite groups, s ( G ) has b een already defin ed by Mel ′ nik o v [13].) As witnessed by the fact th a t the whole Chapter 12 of the monograph [9 ] b y Hofmann and Morris is dev oted to the study of this cardinal fun c tion (and its relation to the w eigh t), s ( G ) is und o ubtedly one of the most imp ortan t cardinal inv arian ts of a (lo cally) compact group G . Let G b e a top olog ical group . F ollo wing [1] defin e the top olo gic al ly ge ner ating weight tg w ( G ) of G by tg w ( G ) = min { w ( F ) : F is closed in G and h F i is dense in G } , where w ( X ) = min {|B | : B is a base of X } + ω is th e weig ht of a space X . The t wo principle results of [1] are su mmarize d in the follo wing Theorem 3. L et G b e a c omp act gr oup. Then: (i) tg w ( G ) = s ( G ) whenever s ( G ) is infinite, and (ii) tg w ( G ) = w ( G/c ( G )) · ω p w ( c ( G )) , wher e c ( G ) is the c onne cte d c omp onent of G and ω √ τ is define d to b e the smal lest infinite c ar dinal κ such that κ ω ≥ τ . The pr oof of this theorem in [1] is essen tially topological and completely self-con tained with the only exc eption of Theorem 2 w hic h is still necessary . Our presen t manuscript completes the job s t arted in [1] by pro viding a self-con tained, pu rely top olog ical pro of of Theorem 2. It is w orth mentio ning that in [1, Section 9] Theorem 3 has b een used to dedu ce (as straigh tforward co rollaries) a series of ma jor results f rom Chapter 12 of the monograph [9] by Hofmann and Morris. 2 2 Necessary facts In this sectio n w e collec t (mostly) w ell-kno wn facts that w ill b e used in the pro of. Recall that a map f : X → Y is: (i) op en p r o vided that f ( U ) is op en in Y for every op en subset U of X , (ii) close d p r o vided that f ( F ) is closed in Y for ev ery close d subset F of X , (iii) p erfe ct if f is a closed map and f − 1 ( y ) is compact for ev ery y ∈ Y . F act 4. [3, Prop osit ion 3.7.5] Assume that f : X → Y and g : Y → Z ar e c ontinuous surje ctions and the map g ◦ f : X → Z i s p erfe ct. Then g is also p erfe ct. F or ev er y i ∈ I let f i : X → Y i b e a map. Th e diagona l pr o duct △{ f i : i ∈ I } of th e family { f i : i ∈ I } is a map f : X → Q { Y i : i ∈ I } whic h assigns to eve ry x ∈ X the p oin t { f i ( x ) } i ∈ I of the Cartesian pro duct Q { Y i : i ∈ I } . (More p recise ly , f assigns to eac h x ∈ X the p oin t f ( x ) ∈ Q { Y i : i ∈ I } d efined b y f ( x )( i ) = f i ( x ) for all i ∈ I .) F act 5. [3, Theorem 3.7.10] F or every i ∈ I let f i : X → Y i b e a c ontinuous p erfe ct map. Then the diagonal pr o duct △{ f i : i ∈ I } is also a c ontinuous p erfe ct map. F act 6. [5, Chapter I I, Theorem 5.1 8] If N is a c omp act normal sub gr oup of a top olo gi c al gr oup G , then the quotient map fr om G onto its quotient gr oup G/ N is p erfe ct. F act 7. L et π : G → H b e a c ontinuous gr oup homomorphism fr om a top olo gi c al gr oup G onto a top olo gic al gr oup H . If π is a q uo tient map, then π is also an op en map . In p articular, if π is a p e rfe ct map, then π is an op en map. Pr o of. The fir st statemen t follo ws from [5, C hapter I I, Theorem 5.17]. T o pro ve the second statemen t note that a prefect map is a closed map, and ev ery closed map is a quotien t map [3, Corollary 2.4 .8]. F act 8. [5, Chapter I I, T h eo rem 5.11] A lo c al ly c omp act sub g r oup G of a top olo gic al gr oup H is close d in H . Recall that a top ologica l group G is c omp actly gener ate d provided that there exists a compact su bset K of G suc h that G = h K i . F act 9. [10] If U is an op en subse t of a c omp actly gener ate d, lo c al ly c omp act gr oup G , then ther e exists a c omp act normal sub gr oup N ⊆ U of G such that G/ N has a c ountable b ase. W e note that in [5, Chapter I I, Theorem 8.7] one finds a pur e ly top ological , elemen tary pro of of F act 9 that do es not use the structure theory of lo call y compact groups. Definition 10. If D is an infi n ite set, then S ( D ) = D ∪ {∗} will denote the one-p oin t compactificatio n of the discrete set of size | D | . (Here ∗ 6∈ D .) That is, all p oin ts of D are isolated in S ( D ), and the family { S ( D ) \ F : F is a fin ite subset of D } consists of op en neigh b ourho ods of a single non-isolated p o in t ∗ . Note th a t S ( D ) can b e c haracterized as a compact Hausdorff sp a ce of size | D | h a vin g precisely one non-isolated p oin t. The r e lev ance of this space to our topic can b e seen fr o m the follo wing folklore 3 F act 11. If X is an infinite suitable set for a c omp act gr oup G , then the subsp ac e X ∪ { 1 G } of G is c omp act and home omorphic to the sp ac e S ( X ) . Pr o of. Indeed, X ∪ { 1 G } is closed in G b y item (ii) of Definition 1 . Sin c e G is compact, so is X ∪ { 1 G } . S ince X is an infi nite discrete subset of G by item (i) of Definition 1 , the p oin t 1 G cannot b e isolate d in X ∪ { 1 G } (ot herwise X ∪ { 1 G } w ould b ecome an infinite d i screte compact space). Hence, X ∪ { 1 G } is a compact sp a ce with a single n on-iso lated p oin t 1 G , and thus X ∪ { 1 G } is h o meomorphic to S ( X ). F act 12. A ssume that X is a c omp act sp ac e with a single non-isolate d p oint x and f : X → Y is a c ontinuous surje ction of X onto an infinite sp ac e Y . Then Y is a c omp act sp ac e with a single non-isolate d p oint f ( x ) . Pr o of. W e are going to sho w first that Y \ V is fin ite for eve ry op en subset V of Y con taining f ( x ). In deed, since f : X → Y is con tinuous, U = f − 1 ( V ) is an op en subset of X con taining x . Sin ce ev ery p o in t of X d i fferen t fr o m x is isolated, X \ U consists of isolated p oints of X . Since X is compact, w e conclude that the set X \ U is finite. Therefore, the set Y \ V m ust b e fin it e as w ell. Sin c e Y is an infin ite set, V m ust b e infin it e. Th us, f ( x ) is a n on-iso lated p oin t of Y . Let us sho w n ext that Y is compact. Let V b e an op e n co v er of Y . There exists V ∈ V su c h that f ( x ) ∈ V . F or ev ery y ∈ Y \ V c h oose V y ∈ V with y ∈ V y . No w { V y : y ∈ Y \ V } ∪ { V } is a fin it e sub co v er of V . Finally , let y ∈ Y \ { f ( x ) } . Since Y is Hausdorff, there exist op en su b sets W and V of Y such that y ∈ W , f ( x ) ∈ V and W ∩ V = ∅ . Then W ⊆ Y \ V , and hence W is fi nite. Since ev ery singleton is a closed subs et of Y , it no w follo ws that y is an isolated p oin t of Y . Our n ext lemma, whic h is in a certain sense the “con v erse” of F act 11, is the key to building su ita ble sets in (compact-lik e) top ologi cal groups. Lemma 13. Supp ose that G is a top olo gic al gr oup, X is an infinite set and f : S ( X ) → G is a c ontinuous ma p such tha t f ( ∗ ) = 1 G and h f ( S ( X )) i is dense in G . Then S = f ( S ( X )) \ { 1 G } is a suitable set for G such that S ∪ { 1 G } is c omp act. Pr o of. Supp ose first that f ( S ( X )) is a finite set. Then S is discrete, S ∪ { 1 G } is compact and closed (b eing finite), and h S i = h S ∪ { 1 G }i = h f ( S ( X )) i is dense in G . T h erefore, S is a suitable set for G . Assume now that f ( S ( X )) is infin it e. As an infi nite conti nuous image of the compact space S ( X ) w it h a single non-isolated p oin t ∗ , the s pac e f ( S ( X )) is also a compact space with a single non-isolated p oin t f ( ∗ ) = 1 G (F act 12). Therefore, S = f ( S ( X )) \ { 1 G } is a discrete set and S ∪ { 1 G } is compact (and th us closed in G ). Moreo ve r, h S i = h f ( S ( X ) \ { 1 G } ) i = h f ( S ( X )) i . Since the latter set is dens e in G , w e conclude that S is a s u ita ble s et for G . Note that S ( N ) is (homeomorphic to) a non-trivial conv ergence sequence toget her with its limit. T h e next f a ct is a key ingredient in ou r p roof, so to make our man uscript self- con tained we include its pr oof adapted f rom [4]. F act 14. [4] L et G b e a c omp actly gener ate d metric gr oup. Then ther e exists a c ontinuous map f : S ( N ) → G such that f ( ∗ ) = 1 G and h f ( S ( N )) i is dense in G . 4 Pr o of. Fix a local base { V n : n ∈ N } at 1 G suc h that V 0 = G and V n +1 ⊆ V n for all n ∈ N . Let G = h K i , where K is a compact subset of G . One can easily see that G is separable, so let D = { d n : n ∈ N } b e a coun table dense su b set of G . Fix n ∈ N . Since { xV n +1 : x ∈ G } is an open co ver of G and K is a compact su bset of G , K ⊆ S { xV n +1 : x ∈ F n } for some finite set F n . No w w e hav e G = h K i ⊆ D [ { xV n +1 : x ∈ F n } E ⊆ h F n ∪ V n +1 i . (1) By in duction on n we will defin e a sequence { E n : n ∈ N } of finite subsets of G with the follo wing prop erties: (i n ) E n ⊆ V n , (ii n ) G ⊆ h E 0 ∪ E 1 ∪ · · · ∪ E n ∪ V n +1 i , and (iii n ) d n ∈ h E 0 ∪ E 1 ∪ · · · ∪ E n i . T o b egin with, note that the set E 0 = F 0 ∪ { d 0 } satisfies all three conditions (i 0 )–(iii 0 ). Supp ose that we ha ve already defined finite sets E 0 , E 1 ,. . . , E n − 1 suc h that conditions (i 0 ), . . . , (i n − 1 ), (ii 0 ), . . . , (ii n − 1 ) and (iii 0 ), . . . , (iii n − 1 ) are satisfied. Condition (ii n − 1 ) implies that F n ∪ { d n } ⊆ h E 0 ∪ E 1 ∪ · · · ∪ E n − 1 ∪ V n i , and since F n is fi nite , we can fin d a finite set E n ⊆ V n suc h that F n ∪ { d n } ⊆ h E 0 ∪ E 1 ∪ · · · ∪ E n − 1 ∪ E n i . (2) Conditions (i n ) and (iii n ) are clear, and (ii n ) follo ws from (1) and (2). F r o m (i n ) for n ∈ N it follo w s that th e s e t S = S { E n : n ∈ N } form s a sequence con verging to 1 G . Since (iii n ) h ol ds for eve ry n ∈ N , w e get D ⊆ h S i , and s o h S i is dense in G . No w tak e an y b ijec tion f : N → S and defin e also f ( ∗ ) = 1 G . Recall that a set-value d map is a m a p F : Y → Z which assigns to ev ery p oin t y ∈ Y a non-empt y closed subset F ( y ) of Z . This set-v alued m ap is lower semic ontinuous if V = { y ∈ Y : F ( y ) ∩ U 6 = ∅} is op en in Y for ev ery op en su b set U of Z . A (single-v alued) map f : Y → Z is cal led a sele ction of F pro vided that f ( y ) ∈ F ( y ) for all y ∈ Y . W e finish this section with the follo wing sp ecial case of Mic h ae l’s selection theorem [14, Theorem 2] (see also [15]). F act 15. A lower semic ontinuous set-value d map F : Y → Z fr om a zer o-dimensional (p ar a)c omp act sp ac e Y to a c omplete metric sp ac e Z has a c ontinuous sele ction f : Y → Z . 3 Lifting lemmas based on Mic hael’s selection theorem Lemma 16. Supp ose that K 0 , K 1 ar e top olo gic al gr oups, N i s a sub gr oup of the pr o duct K 0 × K 1 , and for e ach i = 0 , 1 let q i = p i ↾ N : N → K i b e the r estriction to N of the pr oje ction p i : K 0 × K 1 → K i onto the i th c o or dinate. Assume also that: (1) K i = p i ( N ) for e ach i = 0 , 1 , (2) q 0 is an op en map, (3) q 1 is a close d map, 5 (4) K 1 is a c omp ete metric sp ac e , (5) Y is a (p ar a)c omp act zer o-dimensional sp ac e and h : Y → K 0 is a c ontinuous map. Then ther e e xist s a c ontinuous map g : Y → N such that h = q 0 ◦ g . Pr o of. F or y ∈ Y define F ( y ) = { z ∈ K 1 : ( h ( y ) , z ) ∈ N } . Note that N ∩ ( { h ( y ) } × K 1 ) is a closed su b set of N , and so the set F ( y ) = q 1 ( N ∩ ( { h ( y ) } × K 1 )) must b e closed in q 1 ( N ) = p 1 ( N ) = K 1 b y (1) and (3). Since h ( y ) ∈ K 0 = p 0 ( N ) by (1) and (5), it f ollo ws that F ( y ) 6 = ∅ . Therefore F : Y → K 1 is a s et-v alued map. W e cla im that F is low er semicont inuous. I n deed, let U b e an op en subset of K 1 . Since N ∩ ( K 0 × U ) is an op en sub set of N , q 0 ( N ∩ ( K 0 × U )) is an op en sub se t of q 0 ( N ) = p 0 ( N ) = K 0 b y (1) and (2). S ince h : Y → K 0 is a contin uous map by (5), V = h − 1 ( q 0 ( N ∩ ( K 0 × U ))) is an op en subset of Y . No w note that V = { y ∈ Y : F ( y ) ∩ U 6 = ∅} b y definitions of F and V . In view of (4), th e assumptions of F act 15 are satisfied if one tak es K 1 as Z . Let f : Y → K 1 b e a (single-v alued) con tinuous selection of F whic h exists b y th e conclusion of F act 15. Define g : Y → K 0 × K 1 b y g ( y ) = ( h ( y ) , f ( y )) for y ∈ Y . S ince b oth h and f are con tinuous, so is g . If y ∈ Y , then g ( y ) = ( h ( y ) , f ( y )) ∈ { h ( y ) } × F ( y ) b ecause f is a selection of F , which yields g ( y ) ∈ N b y the d e finition of F ( y ). Therefore, g ( Y ) ⊆ N . The equalit y h = q 0 ◦ g is obvio us from our defin ition of g . In the sequel we will only need a particular case when the p revio us lemma is applicable: Lemma 17. Supp ose that G is a lo c al ly c omp act gr oup, K 0 is a top olo gic al gr oup, K 1 is a metric gr oup, χ i : G → K i is a c ontinuous gr oup homom orphism for i = 0 , 1 , χ = χ 0 △ χ 1 : G → K 0 × K 1 is the diagonal pr o duct of maps χ 0 and χ 1 , and N = χ ( G ) . Assume also that: (a) K i = χ i ( G ) for e ach i = 0 , 1 , (b) e ach χ i is a p erfe ct map, (c) Y is a (p ar a)c omp act zer o-dimensional sp ac e and h : Y → K 0 is a c ontinuous map. Then ther e exi sts a c ontinuous map g : Y → N such that h = q 0 ◦ g , wher e q 0 = p 0 ↾ N : N → K 0 is the r estriction to N of the pr oje ction p 0 : K 0 × K 1 → K 0 . Pr o of. It suffices to c hec k that N , Y and h satisfy all the assump t ions of Lemma 16. (1) follo ws f rom (a). Let i = 0 , 1. S ince b oth χ : G → N an d q i : N → K i are sur jec tions, χ i = q i ◦ χ and χ i is a p erfect map by item (b ) , q i is a p erfect map (F act 4), and so also an op en map (F act 7). T his yields b oth (2) and (3). Being an op en conti nuous image of a lo c ally compact space G , K 1 is lo ca lly compact. Since a lo c ally compact metric sp ace admits a complete metric, w e get (4). Finally , (5) coincides with (c). No w the conclusion of our lemma follo ws from the conclusion of L e mma 16. 6 4 Pro of of T heore m 2 If G and H are groups and f : G → H is a group homomorphism, th e n k er f = { x ∈ G : f ( x ) = 1 H } denotes the kernel of f . Obvio usly , ker f is a normal subgroup of G . W e are no w ready to pro ve a sp ecific version of Theorem 2. Our pr oof is based on represent ing a compactly generated, lo c ally compact grou p as a limit of some inv erse sp ectra (ak a a pr o jecti v e limit in the terminology of algebraists) of lo ca lly compact separable metric groups. In order to mak e an exp osition easier to comprehend for readers not familiar with in v erse (ak a pro jectiv e) limits, we ha ve c hosen the presentati on using diagonal pro ducts of maps, thereby allo wing for a m u c h simp ler visualiziat ion of such a limit. Theorem 18. L et G b e a top olo gic al gr oup gener ate d by its op en subset with c omp act closur e. Then G has a suitable set S such that S ∪ { 1 G } is c omp act. Pr o of. Fix a lo cal b ase { U α : α < τ } at 1 G . If τ ≤ ω , then G is a compactly generated metric group, and hence G has the desir ed suitable set b y F act 14 and Lemma 13. F r o m now on we will assum e that τ ≥ ω 1 . Let X b e a s e t with | X | = τ . F or eve ry ordinal α < τ , ap p ly F act 9 to choose a compact norm a l subgrou p N α of G su ch that N α ⊆ U α and H α = G/ N α has a coun table base, and let ψ α : G → H α b e the quotien t map. F or every ord inal α satisfying 1 ≤ α ≤ τ defin e ϕ α = △ { ψ β : β < α } : G → Q { H β : β < α } and G α = ϕ α ( G ). F or 1 ≤ β ≤ α ≤ τ let  α β : Q { H γ : γ < α } → Q { H γ : γ < β } b e the natural pro jection, and d efine π α β =  α β ↾ G α : G α → G β to b e the restriction of  α β to G α ⊆ Q { H γ : γ < α } . Note that π α β is a s u rjectio n. By our construction, ϕ α ◦ π α β = ϕ β and π α γ = π β γ ◦ π α β whenev er 1 ≤ γ ≤ β ≤ α ≤ τ . (3) Claim 19. ϕ α is a p erfe ct map for every α with 1 ≤ α ≤ τ . Pr o of. Eac h ψ β is a p erfect m a p by F act 6, so the map ϕ α = △ { ψ β : β < α } is also p erfect b y F act 5. By transfinite recursion on α , for ev ery ordinal α satisfying 1 ≤ α ≤ τ we will d e fine a con tinuous map f α : S ( X ) → G α satisfying the f o llo win g prop erties: (i α ) f β = π α β ◦ f α whenev er 1 ≤ β < α , (ii α ) f α ( ∗ ) = 1 G α , (iii α ) |{ x ∈ X : f α ( x ) 6 = 1 G α }| ≤ ω · | α | , (iv α ) h f α ( S ( X )) i is dense in G α . T o motiv ate these cond itions, w e mention that (ii α ) and (iv α ) guarante e that f α ( S ( X )) \ { 1 G α } is a suitable s e t for G α (Lemma 13). T he other tw o conditions (i α ) and (iii α ) are tec hn ical and needed only for carr y in g out the recursion construction. W e start our r ec ursion with α = 1. First of all note that ϕ 1 = ψ 0 and G 1 = H 0 . Being a contin uous homomorphic imag e of a compactly generated group G , G 1 itself is compactly generated. Let N b e a coun table subset of X . Since S ( N ) and S ( N ) are homeomorphic, applying F act 14 we can fin d a con tin u ous map f : S ( N ) → G 1 suc h that f ( ∗ ) = 1 G 1 and h f ( S ( N )) i is dense in G 1 . W e extend this map to th e con tin u ous map f 1 : S ( X ) → G 1 b y 7 defining f 1 ( x ) = 1 G 1 for ev ery x ∈ X \ N and f 1 ( y ) = f ( y ) for y ∈ S ( N ). No w note that f 1 satisfies pr o p erties (i 1 )–(iv 1 ). Supp ose no w th a t α is an ordinal with 1 < α ≤ τ . Assume also that a con tinuous map f β : S ( X ) → G β satisfying prop erties (i β )–(iv β ) has b een already defined for every ordin al β suc h that 1 ≤ β < α . W e are going to d efi ne a con tin uous map f α : S ( X ) → G α satisfying prop erties (i α )–(iv α ). As usual, we consider tw o cases. Case 1. α = β + 1 is a successor ord inal . Clearly , a subspace Y β = { x ∈ X : f β ( x ) 6 = 1 G β } ∪ {∗} (4) of S ( X ) is closed in S ( X ). Hence, Y β is a compact space w i th at most one non-isolated p oin t. In p a rticular, Y β is zero-dimensional. W e claim th at K 0 = G β , K 1 = H β , χ 0 = ϕ β , χ 1 = ψ β , N = G α , Y = Y β and h = f β ↾ Y β satisfy the assump ti ons of Lemma 17. Ind e ed, χ = χ 0 △ χ 1 = ϕ β △ ψ β = ϕ α , and so N = G α = ϕ α ( G ) = χ ( G ). (a) holds trivially . T he map χ 0 = ϕ β is p erfect by Claim 19, while χ 1 = ψ β is a p erfect map b y F act 6. This pro ves (b). Since f β is a conti nuous map, so is h = f β ↾ Y β . Thus esta blishes (c). Let g : Y β → G α b e a con tin uous map s atisfying f β ↾ Y β = π α β ◦ g which exists according to the conclusion of Lemma 17. Define g ′ : Y β → G α b y g ′ ( y ) = g ( y ) · g ( ∗ ) − 1 for y ∈ Y β . Clearly , g ′ is a con tin uous map an d g ′ ( ∗ ) = 1 G α . If y ∈ Y β , then π α β ◦ g ( ∗ ) = f β ↾ Y β ( ∗ ) = f β ( ∗ ) = 1 G β b y (ii β ), and so π α β ( g ′ ( y )) = π α β ( g ( y ) · g ( ∗ ) − 1 ) = π α β ( g ( y )) · π α β ( g ( ∗ )) − 1 = f β ↾ Y β ( y ) · (1 G β ) − 1 = f β ↾ Y β ( y ) b ecause π α β is a group homomorphism. This giv es π α β ◦ g ′ = f β ↾ Y β . (5) Since β ≥ 1, from (3) w e ha ve k er π α β = ϕ α (k er ϕ β ) ⊆ ϕ α (k er ψ 0 ) ⊆ ϕ α ( N 0 ). Since N 0 is compact, so is ϕ α ( N 0 ). Being a closed sub space of ϕ α ( N 0 ), ker π α β m ust b e compact. Since k er π α β ⊆ { 1 G β } × H β and H β has a coun table base, ker π α β is a compact metric group. Note that | Y β | ≤ ω · | β | < τ b y (iii β ), and since τ ≥ ω 1 , w e can c ho ose a coun table set Z β ⊆ X with Y β ∩ Z β = ∅ . Since Z β ∪ {∗} is naturally homeomorphic to S ( N ), F act 14 allo ws us to find a con tinuous map θ : Z β ∪ {∗} → k er π α β ⊆ G α suc h that θ ( ∗ ) = 1 G α and h θ ( Z β ) i is dense in ker π α β . No w define the map f α : S ( X ) → G α b y f α ( x ) =    g ′ ( x ) if x ∈ Y β , θ ( x ) if x ∈ Z β , 1 G α if x ∈ S ( X ) \ ( Y β ∪ Z β ) . Since b oth g ′ and θ are con tin uous maps, on e can easily c hec k that the map f α is con tin uous as well. Claim 20. f β = π α β ◦ f α . Pr o of. If y ∈ Y β , then π α β ( f α ( y )) = π α β ( g ′ ( y )) = f β ↾ Y β ( y ) = f β ( y ) by (5). Supp ose now that x ∈ S ( X ) \ Y β . W e claim that π α β ( x ) = 1 G β . Indeed, if x ∈ Z β , then π α β ( f α ( x )) = π α β ( θ ( x )) = 1 G β b ecause θ ( x ) ∈ θ ( Z β ) ⊆ ke r π α β . If x ∈ S ( X ) \ ( Y β ∪ Z β ), then f α ( x ) = 1 G α , and so π α β ( f α ( x )) = π α β (1 G α ) = 1 G β . Finally , (4 ) and (ii β ) yields f β ( x ) = 1 G β = π α β ( f α ( x )) for x ∈ S ( X ) \ Y β . 8 Let us c hec k no w conditions (i α )–(iv α ). (i α ) Su p pose that 1 ≤ γ < α = β + 1. I f γ = β , then Claim 20 applies. S upp ose no w that 1 ≤ γ < β . T hen π α γ ◦ f α = π β γ ◦ π α β ◦ f α = π β γ ◦ f β = f γ b y (3), Claim 20 and (i β ). (ii α ) f α ( ∗ ) = g ′ ( ∗ ) = 1 G α . (iii α ) F rom the definition of f α one h as { x ∈ X : f α ( x ) 6 = 1 G α } ⊆ Y β ∪ Z β , and so |{ x ∈ X : f α ( x ) 6 = 1 G α }| ≤ | Y β | · | Z β | ≤ ω · | β | · ω ≤ ω · | α | by (iii β ). (iv α ) Let F b e th e closure of h f α ( S ( X )) i in G α . W e n eed to sho w that F = G α . Observe that h f α ( Z β ) i ⊆ h f α ( S ( X )) i ⊆ F . Since h f α ( Z β ) i is d e nse in k er π α β , it n ow follo ws that ker π α β ⊆ F . Since b ot h ϕ α and π α β are surjections and π α β ◦ ϕ α = ϕ β is a p erfect map b y (3) and Claim 19, F act 4 allo ws us to conclude that π α β is a p erfect (and hence also closed) m a p. Therefore, π α β ( F ) is a closed subset of G β . F rom (i α ) one gets π α β ( f α ( S ( X ))) = f β ( S ( X )), and since π α β is a group homomorphism, one also has h f β ( S ( X )) i = π α β ( h f α ( S ( X )) i ) ⊆ π α β ( F ). According to (iv β ), the set h f β ( S ( X )) i is d e nse in G β , and sin c e π α β ( F ) is closed in G β , this yields π α β ( F ) = G β . Since F is a subgroup of G α satisfying b oth ker π α β ⊆ F and π α β ( F ) = G β = π α β ( G α ), one obtains F = G α . Case 2. α is a limit ord i nal . Define L α = n h ∈ Y { H β : β < α } : h ↾ β ∈ G β whenev er 1 ≤ β < α o . (6) Claim 21. Supp ose that H ⊆ L α and { h ↾ β : h ∈ H } is dense in G β whenever 1 ≤ β < α . Then H is dense in L α . Pr o of. Let U b e an op en subset of the pro duct Q { H β : β < α } suc h that U ∩ L α 6 = ∅ . Pic k arbitrarily g ∈ U ∩ L α . There exist n ∈ ω , pairwise distinct ord inals γ 0 , γ 1 , . . . , γ n < α and an op en subset V i of H γ i for ev ery i ≤ n suc h that g ( γ i ) ∈ V i for all i ≤ n and n h ∈ Y { H β : β < α } : h ( γ i ) ∈ V i for all i ≤ n o ⊆ U. (7) Since α is a limit ordinal, β = max { γ i : i ≤ n } + 1 < α . Note th at W = n h ∈ Y { H γ : γ < β } : h ( γ i ) ∈ V i for all i ≤ n o (8) is an op en sub se t of Q { H γ : γ < β } and g ↾ β ∈ W . Since g ∈ L α , one has g ↾ β ∈ G β b y (6). It follo ws that g ↾ β ∈ W ∩ G β 6 = ∅ . By the assumption of our claim, there exists some h ∈ H such that h ↾ β ∈ W . No w from (7), (8) and the c hoice of β we get h ∈ U . Thus h ∈ H ∩ U 6 = ∅ . Claim 22. G α ⊆ L α and G α is dense in L α . Pr o of. Let h ∈ G α . T hen h = ϕ α ( g ) for some g ∈ G . F or ev ery ordinal β satisfying 1 ≤ β < α one has h ↾ β = ϕ β ( g ) ∈ G β , wh ic h yields h ∈ L α b y (6). Thus, G α ⊆ L α . Assume th a t β is an ordinal satisfying 1 ≤ β < α . Let h ′ ∈ G β . Then h ′ = ϕ β ( g ) for some g ∈ G . No w h = ϕ α ( g ) ∈ G α and h ↾ β = ϕ β ( g ) = h ′ . This yields G β ⊆ { h ↾ β : h ∈ G α } . The con verse inclusion { h ↾ β : h ∈ G α } ⊆ G β is trivial. Th is shows that { h ↾ β : h ∈ G α } = G β . Therefore, G α (tak en as H ) satisfies the assumptions of Caim 21, so G α m ust b e dense in L α b y the conclusion of this clai m. 9 Claim 23. G α = L α . Pr o of. The map ϕ α is op en by Claim 19 and F act 7. As an op en contin uous image of a lo c ally compact group G , the grou p G α = ϕ α ( G ) is also lo cally compact. S ince L α is a top olo gical grou p conta ining G α (Claim 22), G α m ust b e closed in L α (F act 8). Since G α is also dense in L α (Claim 22), the conclusion of our claim follo ws. W e are now r e ady to define f α : S ( X ) → G α . Let x ∈ S ( X ) b e arbitrary . Since (i β ) holds for ev ery ord inal β satisfying 1 ≤ β < α , th e re exists a unique h x ∈ L α suc h that h x ↾ β = f β ( x ) for all β with 1 ≤ β < α . No w h x ∈ G α b y Claim 23, and so we can define f α ( x ) to b e this unique h x . Let u s c heck n o w cond itions (i α )–(iv α ). Condition (i α ) clearly h o lds. Since eac h f β is a contin uous map, so is f α . (ii β ) for 1 ≤ β < α trivially implies (ii α ). Similarly , (iii β ) for 1 ≤ β < α yields (iii α ). T o chec k (iv α ) it su ffice s to sho w, in view of Claim 23, that H = h f α ( S ( X )) i ⊆ G α = L α satisfies the assump tions of Claim 21. Indeed, assume 1 ≤ β < α . S ince π α β is a group homomorphism, f rom (i α ) one h a s { h ↾ β : h ∈ H } = { π α β ( h ) : h ∈ H } = π α β ( h f α ( S ( X )) i ) =  π α β ( f α ( S ( X )))  = h f β ( S ( X )) i , and the latter set is dense in G β b y (iv β ). The recurs ive construction h a s b een complete. According to (ii τ ), w e hav e f τ ( ∗ ) = 1 G τ . According to (iv τ ), h f τ ( S ( X )) i is dense in G τ . F r o m Lemma 13, we conclud e that S = f τ ( S ( X )) \ { 1 G τ } is a suitable set for G τ suc h that S ∪ { 1 G τ } is compact. No w observ e that ker ϕ τ ⊆ T { N α : α < τ } ⊆ T { U α : α < τ } = { 1 G } , and hence ϕ τ : G → G τ is an algebraic isomorphism. F urthermore, ϕ τ is a p erfect m a p b y Claim 19. Finally , note that a one-to-one con tin u ous p erfect map is a homeomorph ism. Thus, G and G τ are isomorph i c as topological groups. Pro o f of Theorem 2: Let H b e a lo call y compact group. T ak e an op en neigh b ourh oo d U of the identit y 1 H that has a compact closure U in H . Then G = h U i is an op en (and th us closed [5, Chapter I I, Theorem 5. 5]) subgroup of H . In particular, U ⊆ G = G , and so G is generated b y its op en sub set U with compact closure (in G ). 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