Properness, Cauchy-indivisibility and the Weil completion of a group of isometries

PR OPERNE SS, CA UCHY-INDIVI SI B ILITY AND THE WEIL COMPLETION OF A GR OUP OF ISOMETRI ES A. MANOUSSOS AND P . STRANTZALOS Abstract. In this pa p er we intro duce a ne w class of metric ac- tions on separable (not necessarily connected) metric spaces called “Cauch y-indivisible” actions. This new class coincides with that o f prop er a c tio ns o n lo ca lly compact metric spa ces and, as examples show, it ma y b e different in general. The concept o f “Cauch y - indivisibility” follows a more genera l r e search direction prop o s al in whic h we inv estigate the impact o f basic notions in substa n- tial re sults, like the impact of lo c a l co mpactness and co nnectivity in the theory of pro pe r transfor ma tion gro ups. In or der to pr o - vide s o me basic theor y for this new class of actions we e m bed a “Cauch y-indivisible” a ction of a gro up of isometr ies of a separa ble metric space in a pr op er action of a semig roup in the completion of the underlying space. W e show that, in ca se this subgro up is a gr oup, the original group ha s a “W eil completion” and vice versa. Finally , in order to establish further connections b etw een “Cauch y-indivisible” actions on separ able metric spaces a nd pro p er actions on lo c ally compact metr ic spaces we inv estigate the rela - tion b etw een “ Borel se c tions” for “Cauchy-indivisible” a ctions and “fundamental sets” for prop er a ctions. Some op en questions are added. 1. Intr oduction W e b egin with a brief summary of a r esearch prop osal, that will b e published in mor e details elsewhere. The History of Mathematics indicates tha t during t he 19 th cen tury the mathematical creativit y lived a quantitativ e and -mainly- a qualita- tiv e explosion, whic h is initiated and dev elop ed b y a critical r eno v at ion 2010 Mathematics S ubje ct C lassific ation. Primary 37B05, 54H20; Secondary 54H15. Key wor ds and phr ases. Pr op er action, W eil completion, Cauch y-indivisibilit y , Borel s ection, fundamental set. During this research the fir st author was fully supp or ted by SFB 701 “Sp ektrale Strukturen und T op olo g ische Me tho de n in der Mathematik” at the Universit y o f Bielefeld, Germany . He is grateful for its genero sity and hospita lit y . 1 2 A. MANOUS SOS AN D P . STR ANTZALOS of notions and pro cedures that ha v e b eing initiated the foregoing cen- turies. Our p erio d can b e c har a cterized as one where w e understand deep er and bro ader the ach iev em en ts o f the 1 9 th cen tury , as it w as the case a few decades b efore the b eginning of the 19 th cen tury when the fundamental notions b egan to b e clarified. These remarks led to a researc h direction whic h -roughly sp eaking- amoun ts to the follo wing general requiremen t: W alking to w ards the origins of the 19 th cen tury , try to critically understand the impact of functionally crucial assump- tions in substantial results, preferably within comp osed frames, there- fore fr a mes whic h w ork broadly in Mathematics a nd their applications. Suc h a f rame is, certainly , tha t of T op ological T ransformat io n Groups with applications, fo r instance, in T op o lo gy , Geometry and Ph ysics . The ab o v e researc h direction led to [9], where the impact of “prop- ert y Z” (: ev ery compact subset o f the underlying space is con tained in a compact and connected one) in the main result of [1] is inv estigated. The main result of [9] is that, in almost all cases, under the restrictive “prop erty Z” w a s hidden that, without assuming this prop ert y , t here alw a ys exists a maximal zero-dimensional compactification of the un- derlying space whic h do es the same jo b as in [1], that ma y b e differen t from the (absolutely maximal) end-p o int compactification considered in [1], and coincides with it, among o t hers, if “ pr o p ert y Z” is satisfied. In the pap er at hand, within the same researc h direction a nd ha v- ing in mind the f ruitful theory of prop er t r a nsformation groups on lo cally compact and connected spaces, w e prop ose an analogous class of actions, not necessarily prop er, without assuming lo cal compactness and connectedness of the underly ing spaces. So, w e introduce a new, rather natural, class of m etric actions on s eparable (not necessarily connected) metric spaces called “Cauc h y-indivis ible”. Note that the isometric actions constitute no w adays an imp o r tan t part of the t heory of pro p er actions and that the g roup of isometries of a lo cally compact and connected metric space acts prop erly on it. As the followin g definition show s, “Cauc h y-indivisible” actions are c haracterized by an “ isotr opic ” b eha vior o f divergen t nets of the acting group with resp ect to the basic metric notion of a “Cauc h y sequence”. Recall that “ z i → ∞ in Z ” means that the net { z i } do es not ha v e any con v ergent subnet in the space Z . Definition 1.1. Let ( G, X ) b e a con tin uous action of a top ological group G on a metric space X . The action is said t o b e C auchy- indivisible if the fo llo wing holds: If { g i } is a net in G suc h that g i → ∞ in G and { g i x } is a Cauc h y net in X for some x ∈ X then { g i x } is a Cauc h y net for ev ery x ∈ X . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 3 It turns out that a Cauch y-indivisible action on a lo cally compact or complete metric space is prop er and vice v ersa (cf. § 3), and that in general the t wo notions ma y differ (cf. § 4). In bo th cases the un- derlying space is not assume d to b e connected. The omission of this assumption in the lo cally compact case, as w ell t he omission of the lo cal compactness in the main part of the pap er at hand is an adv an- tage coming from the fact that Cauc h y-indivisibilit y esse n tially r eflects the glob al b eha vior of selfmaps of X comparing to lo c al prop erties or the c onne cte d ness of t he underlying space. So w e can generalize the framew ork of prop er actions and go b ey ond, in accordance to our re- searc h direction prop osal, prov ided that this new fr amew ork leads (a) to in teresting results in the non lo cally compact case and (b) enables a b etter understanding of pro p er actions on lo cally compact spaces. Concerning requiremen t (b) w e note that in Theorem 3.3 we g iv e an answ er to the op en question of c haracterizing prop er actions on non-connected lo cally compact metric spaces and in Theorem 7.4 w e establish an in terconnection b etw een Borel sections ( whic h o ccur in Cauc h y-indivisible actions on separable spaces, cf. Prop osition 7.1) and fundamen tal sets that c haracterize prop er isometric actions. Recall that a se c tion o f an action ( G , X ) is a subset o f X whic h contains only one po in t from eac h orbit. A Bor el se ction is a section that is Borel subset of X (e.g. useful in measure theory). Theorem (Theorem 7.4) . L et G b e a lo c a l ly c omp act gr oup which a c ts pr op erly on a lo c al ly c omp act sp ac e X , a n d supp ose that the orbi t sp ac e G \ X is p ar ac om p act. L et S b e a se ction for the action ( G, X ) . Then (i) F or every op en neigh b orho o d U of S we c an c onstruct a clo s e d fundamental set F c and an op e n fundamental set F o such that F c ⊂ F o ⊂ U . (ii) If , in addition, ( X, d ) is a sep ar able metric sp ac e, in w hich c ase the action ( G, X ) is Cauchy-indivisib le, then ther e exists a B o r el se ction S B , wh i c h is also a fundamental s e t, such that S B ⊂ F c ⊂ F o ⊂ U . Note that S B in (ii) o f the ab o v e theorem, is a “minimal” fundamen- tal set, b ecause of its construction, and as suc h ma y lead to applica- tions. The new notion “like prop erness” seams to b e suitable for structure theorems, as our first results indicate. Concerning requiremen t (a) ab ov e, in § 5, whic h is the main part of the pap er at hand w e consider a separable metric space ( X , d ) suc h that the natural ev aluation action of the group of isometries I so ( X ) on X is Cauc h y-indivisible. L et b X denote t he completion of ( X , d ) and let E b e the El lis semig r oup of the 4 A. MANOUS SOS AN D P . STR ANTZALOS lifted group \ I so ( X ) in C ( b X , b X ), i.e. the p oin t wise closure of \ I so ( X ) in C ( b X , b X ). Let H = { h ∈ C ( b X , b X ) | there exists a sequence { g n } ⊂ I so ( X ) with g n → ∞ in I so ( X ) a nd b g n → h in C ( b X , b X ) } , X l = { hx | h ∈ H , x ∈ X } , X p = { hx | h ∈ H ∩ I so ( b X ) , x ∈ X } . With this notat io n and the previous men tioned assumptions, among other results w e sho w that Theorem (Theorem 5.12) . The set X ∪ X p is the maxi m al subset of X ∪ X l that c ontains X such that the map ω : E × ( X ∪ X p ) → ( X ∪ X p ) × b X with ω ( f , y ) = ( y , f y ) , f ∈ E and y ∈ X ∪ X p is pr op er. The in terest on this theorem lies on the fact that an action ( G, X ) is prop er if the map G × X → X × X defined b y ( g , x ) 7→ ( x, g x ) is prop er, cf. [2, Definition 1, p.250]. W e remind that a top ological group has a Weil c ompletion with re- sp ect t o the unifo rmit y of p oin t wise con v ergenc e if it can b e embedded densely in a complete group with respect to its left uniform structure. Prop osition (Proposition 5.16) . The fol lowing a r e e quivalent: (i) The map ω : E × ( X ∪ X l ) → ( X ∪ X l ) × b X is pr op er. (ii) E is a gr oup (pr e cisely a close d sub gr oup of I so ( b X ) ). (iii) I so ( X ) has a Weil c omp letion. Corollary (Corolla r y 5.18) . I f E is a gr oup the action ( I so ( X ) , X ) is em b e dde d densely in the p r op er action ( E , X ∪ X l ) such that the fol lowing e quivariant diagr am c ommutes ( I so ( X ) , X )   / / X   ( E , X ∪ X l ) / / b X wher e X → X ∪ X l is the inclusion m ap an d the map I so ( X ) → E is define d by g 7→ b g for every g ∈ I so ( X ) . By the wor d “densely” we me an that X is dense in X ∪ X l and \ I so ( X ) is dense in E . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 5 The ab o v e result may lead to further structure theorems, cf. Ques- tion 5 .19. Prop osition (Prop osition 7.1) . If the El lis semigr oup E is a gr oup then the action ( E , X ∪ X l ) has a B or el s e ction. As Theorem 7.4( ii) mentioned ab o v e indicates, the notion of a Bo rel section, whic h according to the ab ov e r esult is a feature of the Cauc h y- indivisible actions on separable metric spaces, is remark ably related to that o f a fundamen tal set in the lo cally compact case a nd ma y b e, sim- ilarly , used for structural theorems. So, it is in teresting to ask whether the existing Borel section for the action ( E , X ∪ X l ) can b e reduced t o a Bo rel section for the initial action ( I so ( X ) , X ), cf. Question 7.6. In o r der to indicate or to exclude p ossible directions for further inv es- tigation concerning Cauc h y-indivis ible actions, w e study v arious exam- ples, cf. Examples 5 .13, 5.15, 5.2 1 and 7.3. Among them, an example that may b e of indep enden t in terest, is the fo llo wing (cf. § 6): Consider the action ( I so ( I so ( Z )) , I so ( Z )), where Z is the discrete space of the in tegers, and with suitable metrics on the acting group I so ( I so ( Z )) and the underlying space I so ( Z ). W e sho w that this action is prop er and Cauc h y-indivis ible while the Ellis semigroup is not a group and I so ( I so ( Z )) has no W eil completion. 2. Basic notions and not a tion F or what follow s, a nd in addition to the notatio n established in the in tro duction, ( X , d ) will denote a metric space with metric d and I so ( X ) will denote its group of (surjectiv e) isometries of X endo w ed with the top ology of p oint wise conv ergence. With this top ology I so ( X ) is a top ological gr o up [3, Ch. X, § 3.5 Corollary]. Let ( b X , b d ) stands for the completion of ( X , d ). F or a Cauc hy sequence { x n } in X let [ x n ] ∈ b X denote the limit p oin t of { x n } in b X . W e denote by b g and \ I so ( X ) the lift of g ∈ I so ( X ) and the lift of the group I so ( X ) resp ective ly in C ( b X , b X ), the space of the contin uous selfmaps of b X (whic h is considered with the top ology of p oin t wis e con v ergence). A contin uous action of a top ological group G on a top ological space X is a contin uous map G × X → X with ( g , x ) 7→ g x , g ∈ G , x ∈ X suc h that ( e, g ) 7→ x , for ev ery x ∈ X where e denotes the unit elemen t of G , and ( h, ( g , x )) 7→ ( hg ) x f or ev ery h, g ∈ G and x ∈ X . When the action map is kno wn w e will denote the action simply b y ( G, X ). Let U ⊂ X , then GU := { g x | g ∈ G, x ∈ U } . Esp ecially , if U = { x } then the set Gx := G { x } is called the orbit of x ∈ X under G . The subgroup G x := { g ∈ G | g x = x } of G is called the isotro p y group of x ∈ X . The 6 A. MANOUS SOS AN D P . STR ANTZALOS natural ev aluation action of I so ( X ) on X is the map I so ( X ) × X → X with ( g , x ) 7→ g ( x ), g ∈ I so ( X ) , x ∈ X and is denoted b y ( I so ( X ) , X ). If w e endo w I so ( X ) with the top ology of p oint wise con v ergence this action is alw a ys con tin uo us. As usual, S ( x, ε ) will denote the op en ball cen tered at x with r a dius ε > 0. Definition 2.1. A contin uous action ( G, X ) is (equiv alen tly to the Bourbaki-definition) prop er if J ( x ) = ∅ , for eve ry x ∈ X , where J ( x ) = { y ∈ X | there exist nets { x i } in X and { g i } in G with g i → ∞ , lim x i = x and lim g i x i = y } denotes the extended (prolonga tional) limit set of x ∈ X . It is easily seen that in the sp ecial case o f actions by isometries J ( x ) = L ( x ) holds for ev ery x ∈ X , where L ( x ) = { y ∈ X | there exists a net { g i } in G with g i → ∞ and lim g i x = y } , denote the limit set of x ∈ X under the action of G on X . Henc e an action b y isometries ( G, X ) is prop er if and only if L ( x ) = ∅ for ev ery x ∈ X . 3. Cauchy-indivisibility and pr oper actions on locall y comp act metric sp aces In t his section w e show that for gro up actions on lo cally compact metric spaces the no t ions of prop erness and Cauch y-indivisibilit y coin- cide. W e start with the fo llowing easily pro v ed observ a tion. Lemma 3.1. L et ( X , d ) b e a lo c al ly c omp act metric sp ac e and { g i } ⊂ I so ( X ) b e a net such that { g i x } is a Cauchy n e t for some x ∈ X . Then ther e exis ts a p oint y ∈ X such that g i x → y . Prop osition 3.2. L et ( X , d ) b e a lo c al ly c omp act metric sp ac e. The action ( I so ( X ) , X ) is pr op er if and only if it is Cauchy-indivis i b le. Pr o of . Assume that ( I so ( X ) , X ) is Cauc h y-indivis ible. W e will sho w that the limit sets L ( x ) are empty fo r ev ery x ∈ X . Assume the con trary , that is there exist a net { g i } in I so ( X ) and x, y ∈ X suc h that g i → ∞ a nd g i x → y . W e will sho w that g i → h for some h ∈ I so ( X ), whic h is a contradiction to the assumption g i → ∞ . Since ( I so ( X ) , X ) is Cauc hy-indiv isible then { g i x } is a Cauch y net, for ev ery x ∈ X . Therefore, b y the previous lemma, there is a map h : X → X defined by h ( x ) := y suc h that g i → h p oint wise on X and h preserv es the metric d . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 7 Observ e that g − 1 i y → x since d ( g − 1 i y , x ) = d ( y , g i x ). Applying Cauc h y- indivisibilit y for the action ( I so ( X ) , X ) and the previous lemma a g ain, w e conclude that there exis ts a map f : X → X suc h that g − 1 i → f p oin t wise on X and f preserv es the metric d . Ob viously f is the in v erse map of h , hence h ∈ I so ( X ). The con v erse implication follows easily in a similar w ay .  If X is lo cally compact and G acts prop erly on X (hence G is a lo cally compact group), it is we ll kno wn, see e.g. [8], that there exists a G - inv ar ia n t compatible metric on X . Compatible means that t his metric induces the top olo gy of X . Hence, the previous prop osition states the follow ing result that c haracterizes the prop erness of actions on lo cally compact metric spaces indep enden tly of the connectedness of the underlying space. Theorem 3.3. L et ( X , d ) b e a lo c al ly c omp act me tric sp ac e. A n action ( G, X ) is pr op e r if an d only if it is Cauchy-ind i v isible. R emark 3.4 . The previous theorem also holds, and can b e similarly pro v ed, if we replace the full group of isometries of X b y a closed sub- group of it or if w e replace the lo cal compactness of X b y completeness. 4. Cauchy-indivisibility vs pr operness In this section w e pro vide examples sho wing that Cauc h y-indivisibili- t y and prop erness a r e distinct notions for isometric actions on separable and non lo cally compact metric spaces. W e also provide some criteria for the co existence of Cauc h y-indivisible and prop er actions on the basis of the dynamical b eha vior of the lifting of the action ( I so ( X ) , X ) in the completion of the underlying space. R emark 4.1 . The example in § 6 shows that the t w o notions ma y co exist also in the case when X is neither lo cally compact nor complete. The follo wing example sho ws that the action ( I so ( X ) , X ) can b e prop er and not Cauc h y-indivis ible. Example 4.2. Let X b e the set Q of the rational n um b ers endo w ed with the Euclidean metric. It is easy to see that the action ( I so ( X ) , X ) is pro p er. T ak e a sequence of rational n um bers { q n } suc h that q n → a , where a is an ir r ational. Let { g n } ⊂ I so ( X ) with g n x := ( − 1) n x + q n for ev ery x ∈ X , then g n → ∞ in I so ( X ). S ince g n 0 = q n for ev ery n ∈ N , the sequence { g n 0 } is Cauch y . But for x 6 = 0 the sequence { g n x } has tw o limit p oints in R hence cannot b e a Cauc h y sequenc e. The follo wing example sho ws that the action ( I so ( X ) , X ) can b e Cauc h y-indivisible and not prop er. 8 A. MANOUS SOS AN D P . STR ANTZALOS Example 4.3. Let X b e the set Q + √ 2 N endo w ed with the Euclidean metric. Its group of isometries is Q acting b y t r a nslations (reflections are excluded b ecause of the addend √ 2 N ). Therefore, ( I so ( X ) , X ) is Cauc h y-indivisible. Ho w ev er, the action ( I so ( X ) , X ) is no t prop er. T o see that tak e a sequence of rational num b ers { q n } such that q n → √ 2. Let { g n } ⊂ I so ( X ) with g n x := x + q n . Observ e that g − 1 n √ 2 → 0 / ∈ X . Therefore g n → ∞ in I so ( X ). Since g n √ 2 → 2 √ 2 ∈ X the limit set L ( √ 2) is not empt y , so the action ( I so ( X ) , X ) is not prop er. Motiv ated b y these examples w e giv e nece ssary and sufficien t con- ditions for a Cauc hy-indivis ible action ( I so ( X ) , X ) to be prop er and vice ve rsa: Prop osition 4.4. L e t I so ( X ) b e Cauchy-indivi sible. The fol lo w ing ar e e quivalent: (i) The action ( I so ( X ) , X ) is pr op er. (ii) If h is in the p oi n twise closur e of \ I so ( X ) in C ( b X , b X ) then either h ( X ) ⊂ X or h ( X ) ⊂ b X \ X . Pr o of . Assume that the action ( I so ( X ) , X ) is prop er and h is in the p oin t wise closure of \ I so ( X ) in C ( b X , b X ). Then there is a net { b g i } in \ I so ( X ) suc h that b g i → h p oin t wis e in b X . If h ( X ) ∩ X 6 = ∅ then there is some x ∈ X suc h that b g i x → hx ∈ X . Since the action ( I so ( X ) , X ) is prop er the net { g i } has a con vergen t subnet in I so ( X ). Then it is easy to see that h ∈ \ I so ( X ), hence h ( X ) ⊂ X . Assume now that condition (ii) holds. W e will sho w that the limit sets L ( x ) are empty for ev ery x ∈ X , hence the action ( I so ( X ) , X ) is prop er. W e will pro ceed b y contradiction. Assume that t here exist x, y ∈ X a nd a net { g i } in I so ( X ) with g i x → y and g i → ∞ in I so ( X ). Since { g i x } is a Cauc h y net in X and I so ( X ) is Cauc hy-indivis ible then { g i x } is a Cauc h y net for ev ery x ∈ X , hence { g i x } con v erges in b X for ev ery x ∈ X . So, w e can define a map h : X → b X by letting hx := lim b g i x . It is easy to see that h preserv es the metric b d on X . Th us, if w ∈ b X a nd { x n } ⊂ X is a sequence in X such that x n → w in b X then { hx n } is a Cauch y sequence in X , hence it conv erges to a p o in t in b X whic h is indep enden t of the c hoice of the sequence { x n } . Then, b y [2, Ch. I, § 8.5 Theorem 1], the map h : X → b X has a unique con tin uous extension on b X . It is easy to see tha t b g i → h p oint wise on b X , th us h is in the p oint wise closure of \ I so ( X ) in C ( b X , b X ). Since g i x → y then hx = y where x, y ∈ X . So using our h ypo thesis h ( X ) ⊂ X . Since g i preserv es t he metric d then g − 1 i y → x . Using the same argumen t s PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 9 as b efore w e hav e that h ∈ \ I so ( X ) hence the net { g i } con v erges in I so ( X ), a contradiction to the assumption g i → ∞ in I so ( X ).  Prop osition 4.5. Assume that ( I so ( X ) , X ) is a pr op er action. Th e fol lowing ar e e quivale n t: (i) I so ( X ) is C a uchy-indivisible. (ii) L et { g i } ⊂ I so ( X ) a net with g i → ∞ and { g i x } b e a Cauchy net for som e x ∈ X . If y ∈ X then the net { g i y } c an not have mor e than one limit p oint in the c omple tion b X of X . Pr o of . The direction f rom (i) to (ii) is trivial. If the con v erse implica- tion does not hold, then there is a Cauch y net { g i x } suc h that there is y ∈ X , an ε > 0 and subnets { g i k y } , { g i l y } of { g i y } suc h that d ( g i k y , g i l y ) ≥ ε for ev ery index k , l . Since { g i x } is a Cauc hy net in X then w e ma y assume that d ( g i k x, g i l x ) → 0. Hence, d ( g − 1 i k g i l x, x ) → 0 . W e can define a new net { h i,j } ⊂ I so ( X ) b y letting h i,j := g − 1 j g i for ev ery pair of indices ( i, j ), with direc tion defined by ( i 1 , j 1 ) ≤ ( i 2 , j 2 ) if and only if i 1 ≤ i 2 and j 1 ≤ j 2 . Therefore, h i k ,i l x → x . Since ( I so ( X ) , X ) is prop er there is a subnet { h i k m ,i l m } and some g ∈ I so ( X ) suc h that h i k m ,i l m → g . Hence { h i k m ,i l m y } is a Cauc hy net in X , there- fore f or ev ery ε ′ > 0 there exists an index m 0 suc h that d ( g − 1 i k m g i l m y , g − 1 i k n g i l n y ) < ε ′ for ev ery m, n ≥ m 0 . By taking m = n ≥ m 0 it is easy to see that { g i l m y } is a Cauc h y net and if w e follow the same pro cedure w e can also sho w that { g i k m y } is also a Cauc h y net. Since d ( g i k m y , g i l m y ) ≥ ε for ev ery index m the net { g i y } has tw o limit p oint in the completion b X o f X , a con tradiction to our hypothesis.  5. Cauchy-indivisible isometric a ctions on sep arable metric sp aces In this se ction ( X , d ) wil l de n ote a sep ar a b l e metric sp ac e such that the action ( I so ( X ) , X ) is C auchy-indivisible . Firstly , w e sho w the a d- equacy of sequences in the definition of Cauch y-indivisibilit y . Prop osition 5.1. In the definition of Cauchy-indivisibi l i ty fo r iso m et- ric actions nets c a n b e r eplac e d by se quenc es. Pr o of . Assume t hat if { g n } is a sequence in I so ( X ) suc h that g n → ∞ and { g n x } is a Cauc hy sequenc e in X fo r some x ∈ X then { g n x } is a Cauc h y sequence fo r ev ery x ∈ X . Let { f i } b e a net in I so ( X ) suc h that f i → ∞ and { f i x } is a Cauc h y net in X for some x ∈ X . W e will sho w that { f i x } is a Cauc hy net in X for ev ery x ∈ X . W e a rgue 10 A. MANOUS SOS AN D P . STR ANTZALOS b y con tradiction. Suppose that there exists y ∈ X suc h that { f i y } is not a Cauc h y net. Hence, there is an ε > 0 and subnets { f i k } , { f i l } suc h that d ( f i k y , f i l y ) ≥ ε for ev ery k , l . Since { f i x } is a Cauc h y net in X there is a p oin t z ∈ b X suc h that b f i x → z . Hence, the subnets { c f i k x } , { c f i l x } also conv erges to z . So w e may find seque nces { d f i k n x } and { c f i l n x } suc h that d f i k n x → z a nd c f i l n x → z . Therefore, { f i k n x } and { f i l n x } are Cauc h y sequences in X a nd d ( f i k n y , f i l n y ) ≥ ε for ev ery n ∈ N . Let { h n } ⊂ I so ( X ) with h 4 n − 3 = f i k 2 n − 1 , h 4 n − 2 = f i l 2 n − 1 , h 4 n − 1 = f i l 2 n and h 4 n = f i k 2 n , n = 1 , 2 . . . . It is easy to see that c h n x → z , hence { h n x } is a Cauch y sequence in X . Moreo v er, { h n y } is not a Cauch y sequence in X since d ( f i k n y , f i l n y ) ≥ ε for ev ery n ∈ N a nd fo r the same reason h n → ∞ in I so ( X ), whic h is a con tradiction to our h yp o thesis.  Definition 5.2. Fix a dense sequence D = { x i } ⊂ X in b X . Since the metric b d 1+ b d is an equiv alent metric to b d on X ( also giv es the same groups of isometries on X and b X and the same Cauch y sequences) w e may assume that b d is bo unded by 1. W e define δ : I so ( b X ) × I so ( b X ) → R + b y δ ( f , g ) = ∞ X i =1 1 2 i b d ( f x i , g x i ) for ev ery f , g ∈ I so ( b X ). It is easy to see that δ is a left-in v arian t metric on I so ( b X ). Prop osition 5.3. The uniformity of p ointwise c onver genc e, the left uniformity and the unifo rmity induc e d by δ on I so ( b X ) an d I so ( X ) c oincide , indep ende n tly of Cauchy-indivi s i b ility. Pr o of . The pro of is similar to the pro o f of L emma 2.11 in [6].  Prop osition 5.4. The p ointwis e closur es of I so ( X ) in C ( X, X ) and of I so ( b X ) in C b X , b X ) endowe d with the m e tric δ ar e sep ar abl e metric sp ac es. Pr o of . It fo llo ws easily using the same argumen ts as in the pro of of Lemma 2.11 in [6] and [3, Ch. X, § 3 Exercise 6 (b), p. 327].  The follo wing lemma will b e used often in the sequel. Lemma 5.5. L et { g n } b e a se quenc e in I so ( X ) such that { g n x } is a Cauchy se q uen c e in X for some x ∈ X and g n → ∞ . T hen PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 11 (i) { g n x n } is a Cauchy se quenc e for every Cauchy se q uen c e { x n } in X . (ii) If { x k } is Cauchy se quenc e in X then b g n [ x k ] → [ g k x k ] in b X . Pr o of . (i) The pro of follows b y the triangle inequalit y a nd the fact that { g n x n 0 } is a Cauc h y sequence, fo r suitable n 0 ∈ N . (ii) By item ( i) , { g k x k } is a Cauch y sequence in X , hence [ g k x k ] ∈ b X . The rest of the pro of is similar t o that of item (i).  Corollary 5.6. If { g n } is a se quenc e in I so ( X ) such that g n → ∞ and { g n x } is a Cauchy se quenc e in X for some x ∈ X , then { b g n } c o n ver ges p ointwise on b X to some h ∈ C ( b X , b X ) which pr eserves the metric b d . In addition, if { g − 1 n y } is a Cauchy se quenc e for s ome y ∈ X , then { b g n } c onver ges p ointwise on b X to some h ∈ I so ( b X ) . Pr o of . The pro of is an immediate consequence of Lemma 5.5 (ii) if w e set h : b X → b X with h [ x k ] := [ g k x k ] for eve ry [ x k ] ∈ b X .  Corollary 5.6 enables t he following equiv alent expressions of t he cor- resp onding sets defined in the in tro duction: Notation. H = { h ∈ C ( b X , b X ) | ther e exists a s e quenc e { g n } ⊂ I so ( X ) with g n → ∞ in I so ( X ) , { g n x } is a Cauchy se quenc e for so m e x ∈ X and b g n → h in C ( b X , b X ) } . X l = { y ∈ b X | ther e exists a se quenc e { g n } ⊂ I so ( X ) with g n → ∞ in I so ( X ) , such that { g n x } is a Cauchy se quenc e for some x ∈ X an d y = [ g k x ] } , denotes the set of the limit p oints of the action ( I so ( X ) , X ) in b X . X p = { y ∈ b X | ther e exists a se quenc e { g n } ⊂ I so ( X ) with g n → ∞ in I so ( X ) , such that { g n x } and { g − 1 n x } ar e Cauchy se q uen c es for s o me x ∈ X and y = [ g k x ] } , denotes the set of the sp e c i a l lim it p oints of ( I so ( X ) , X ) in b X . Prop osition 5.7. If { g n } is a s e quenc e in I so ( X ) such that g n → f on X fo r some f in C ( X, X ) , then b g n → b f on b X and b f ∈ I so ( b X ) . Pr o of . If { g n } has a con v ergen t subsequenc e { g n k } to some p oin t g ∈ I so ( X ) then f = g on X . W e will show that b g n → b g point wise on b X . 12 A. MANOUS SOS AN D P . STR ANTZALOS T ak e some y ∈ b X and an ε > 0. Then there exis ts some x ∈ X suc h that b d ( x, y ) < 1 3 ε . Since g n x → g x , there is a p ositiv e in teger n 0 suc h that d ( g n x, g x ) < 1 3 ε for ev ery n ≥ n 0 . Hence, b d ( b g n y , b g y ) < ε , for ev ery n ≥ n 0 . Assume, no w, that g n → ∞ in I so ( X ). Since g n → f on X , then { g n x } is a Cauc h y sequence, for ev ery x ∈ X . Hence, b y Corollar y 5.6, { b g n } conv erges point wise to some h ∈ C ( b X , b X ). Since b f x = f x = hx for ev ery x ∈ X , then b f = h on b X . Note that if [ x k ] ∈ b X then b f [ x k ] = h [ x k ] := [ g k x k ]. Next, w e sho w that b f is surjectiv e. Let [ y k ] ∈ b X . Since g n → f on X and f x ∈ X f o r ev ery x ∈ X , then g − 1 n f x → x . Hence, b y Lemma 5.5 (i) a nd the Cauc h y-indivisibilit y of the action ( I so ( X ) , X ) w e hav e that [ g − 1 k y k ] ∈ b X . Therefore, b y Lemma 5 .5 (ii), b d ( b f [ g − 1 k y k ] , [ y k ]) = lim n b d ( b g n [ g − 1 k y k ] , [ y k ]) = b d ([ g k g − 1 k y k ] , [ y k ]) = b d ([ y k ] , [ y k ]) = 0. This finishes the pro of.  With the not a tion establishe d in the in tro duction, w e hav e Prop osition 5.8. The set E is (i) the union \ I so ( X ) ∪ H , (ii) c omplete with r esp e ct to the uniformity of p ointwise c onver genc e on b X , and (iii) a semigr oup, the El lis sem i g r oup of \ I so ( X ) in C ( b X , b X ) , i. e . the p ointwise closur e o f \ I so ( X ) in C ( b X , b X ) . Pr o of . (i) T ake a sequence { b g n } in \ I so ( X ) suc h that b g n → h for some h ∈ C ( b X , b X ). If { g n } has a conv ergen t subseque nce to some g ∈ I so ( X ) then, b y Prop osition 5.7, h = b g ∈ \ I so ( X ). Le t g n → ∞ in I so ( X ) a nd tak e some x ∈ X . Since b g n x → hx , then { g n x } is a Cauc h y sequence in X therefore h ∈ H . Items (ii) and (iii) follow from Lemmata 2.1 0 and 2.11 in [6] b y noticing that a sequence { g n } in I so ( X ) is Cauc hy with resp ect to the left uniformit y of I so ( X ) if a nd only if { g n x } is Cauc h y in X for ev ery x ∈ X .  R emark 5.9 . As the example in § 6 sho ws , the Ellis semigroup E is not in general a group. Ho w ev er Prop osition 5.10. The El l i s semigr oup E is a gr oup if and only if X l = X p . Pr o of . Assume that E is a group and let y ∈ X l . Hence, there is a seque nce { g n } ⊂ I so ( X ) with g n → ∞ in I so ( X ) and a map h ∈ C ( b X , b X ) suc h that b g n → h p oint wise on b X and y = hx for some x ∈ X . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 13 Since E is a group then h has an in v ers e h − 1 . Thus b g n − 1 → h − 1 . The last implies that { g − 1 n x } is a Cauc h y sequence in X , therefore y ∈ X p . T o show the con v ers e implication, assume that X l = X p and take some h ∈ E . By Prop o sition 5.8 (i), h ∈ \ I so ( X ) ∪ H . So, if h ∈ \ I so ( X ) ob viously it has an in v erse. Assume that h ∈ H . Hence, there is a sequence { g n } ⊂ I so ( X ) with g n → ∞ in I so ( X ) suc h that b g n → h p oin t wise on b X . So [ g n x ] ∈ X l for ev ery x ∈ X . But X l = X p , hence { g − 1 n x } is a Cauc h y sequence. Applying Corollar y 5.6, h ∈ I so ( b X ) so it has an in v erse in E .  Lemma 5.11. The set X ∪ X l is E -invariant. Pr o of . It is easy to v erify that X a nd X l are \ I so ( X )-in v a rian t. W e will sho w that are also H -in v arian t. L et h ∈ H and x ∈ X . By the definition of H there is some seque nce { g n } in I so ( X ) such that g n → ∞ in I so ( X ) and b g n → h p oin t wise on b X . If [ f n x ] ∈ X l , for some sequence { f n } ⊂ I so ( X ) and x ∈ X t hen, b y Coro llary 5.6, h [ f n x ] = [ g n f n x ]. If the sequence { g n f n } has a con v ergen t subsequence in I so ( X ) then the Cauc h y sequence { g n f n x } has a con v ergent subsequenc e in X , so it con v erges in X . So h [ f n x ] = [ g n f n x ] ∈ X . O therwise g n f n → ∞ and h [ f n x ] = [ g n f n x ] ∈ X l .  Theorem 5.12. The set X ∪ X p is the maximal subset o f X ∪ X l that c ontains X s uch that the map ω : E × ( X ∪ X p ) → ( X ∪ X p ) × b X with ω ( f , y ) = ( y , f y ) , f ∈ E and y ∈ X ∪ X p is pr op er. Pr o of . W e firstly sho w tha t the map ω : E × ( X ∪ X p ) → ( X ∪ X p ) × b X is prop er. Since the ev aluation map E × ( X ∪ X p ) → b X is isometric and action- lik e, according to § 2, it suffices to sho w t hat the limit sets L ( x ) are empt y for eve ry x ∈ X ∪ X p . Let { f n } b e a sequence in E suc h that f n y → z for some y ∈ X ∪ X p and z := [ z k ] ∈ b X . Case I. Assume t ha t y ∈ X . If { f n } has a subsequence { f n k } in \ I so ( X ) then either the restriction of { f n k } on X has a conv ergen t subsequen ce in I so ( X ) hence, b y Prop osition 5.7, the sequence { f n k } con v erg es p oin t wise to some p oin t of \ I so ( X ) ⊂ E , o r f n → ∞ in I so ( X ). In this case, since { f n y } is a Cauc hy seq uence in X , the sequenc e { f n } con v erges p o in t w ise to some p oin t of H ⊂ E b y Corolla ry 5.6. Assume, now , that { f n } is in H and consider the dense sequence D = { x i } in X whic h we used t o define the metric δ ; cf. D efinition 5.2. So, there is a sequence { x i n } in D suc h that x i n → y . By the definition 14 A. MANOUS SOS AN D P . STR ANTZALOS of H and Pro p osition 5.3, there is a sequenc e { g n } in I so ( X ) suc h tha t δ ( b g n , f n ) < 1 i n 2 i n . (5.1) Hence, using the form of the metric δ , w e conclude that b d ( b g n x i n , f n x i n ) < 1 i n . Moreo v er, b d ( b g n y , z ) ≤ b d ( b g n x i n , f n x i n ) + b d ( f n x i n , f n y ) + b d ( f n y , z ) = b d ( b g n x i n , f n x i n ) + b d ( x i n , y ) + b d ( f n y , z ) . Therefore, g n y → z . Arguing as in the b eginning of the pro of, { g n } has a con v ergent subsequenc e to a p oin t of E , hence by equation 5.1, the same holds for the seq uence { f n } . Case II. Assume that y ∈ X p . Hence, there exist a sequence { p k } ⊂ I so ( X ) with p k → ∞ in I so ( X ), an isometry h 1 ∈ I so ( b X ) suc h that b p k → h 1 p oin t wise on b X and h 1 x := [ p k x ] = y for some x ∈ X . If { f n } has a subsequence { f n k } in \ I so ( X ) then either the restriction of { f n k } on X has a con v ergen t subsequence in I so ( X ) hence, by Prop osition 5.7, the sequence { f n k } con v erges p oin t wise to some p oin t of \ I so ( X ) ⊂ E , or f n → ∞ in I so ( X ). If the later holds, then w e will sho w that there is a Cauc h y sequence of t he form { f n i p k i x } in X for some subsequences { f n i } and { p k i } of { f n } and { p k } respective ly ( t he problem is t ha t w e do not know if { f n x } or { f n p n x } is a Cauc h y seq uence in X for some x ∈ X ). Let i b e a p ositiv e in teger. Since f n [ p k x ] → z a nd z := [ z k ] ∈ b X , there is a p ositiv e in teger n 0 that depends only on i suc h tha t b d ( f n [ p k x ] , [ z k ]) < 1 i for ev ery n ≥ n 0 ( i ). Therefore lim k d ( f n p k x, z k ) := b d ( f n [ p k x ] , [ z k ]) < 1 i for ev ery n ≥ n 0 ( i ). Hence, using induction, we may find strictly increasing seq uences of p ositiv e in tegers { n i } and { k i } such that d ( f n i p k i x, z k i ) < 1 i (5.2) for ev ery p ositiv e in teger i . Since { z k i } is a Cauch y seq uence then by equation 5.2, { f n i p k i x } is a Cauc h y sequence in X . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 15 No w, either { f n i p k i } has a con vergen t subsequence in I s o ( X ) (with- out loss of g enerality and for the economy of the pro of w e may assume that { f n i p k i } con v erges in I so ( X )) or f n i p k i → ∞ in I so ( X ). In b oth cases, b y Corolla r y 5.6 and Prop osition 5.7, there is h 2 ∈ C ( b X , b X ) suc h that c f n i c p k i = \ f n i p k i → h 2 p oin t wise on b X . W e will show t ha t c f n i → h 2 h − 1 1 p oin t wise on b X . T ak e w ∈ b X . Since h 1 ∈ I so ( b X ), there is some u ∈ b X suc h that h 1 ( u ) = w . Hence b d ( f n i w , h 2 h − 1 1 w ) = b d ( f n i h 1 u, h 2 u ) ≤ b d ( f n i h 1 u, \ f n i p k i u ) + b d ( \ f n i p k i u, h 2 u ) = b d ( h 1 u, c p k i u ) + b d ( \ f n i p k i u, h 2 u ) whic h con v erges to 0, since c p k i → h 1 and \ f n i p k i → h 2 p oin t wise on b X . Hence { f n x } is a Cauc h y sequence for eve ry x ∈ X . Since w e assumed that f n → ∞ in I so ( X ) then, by Corollary 5.6, { f n } conv erges p oin t wise on b X to h 2 h − 1 1 ∈ E . T o finish the pro of of t he second case a ssume t ha t { f n } is in H . Then arguing a s in the first case w e can show that { f n } has a con v ergent subseque nce to a p oin t of E . Next, w e sho w t ha t if Y is a subset of X ∪ X l that con ta ins X suc h that the map ω : E × Y → Y × b X is prop er then Y ⊂ X ∪ X p . T o see tha t tak e a p oint [ g k x ] ∈ Y \ X . This means that { g k } is a sequence in I so ( X ) suc h that g k → ∞ in I so ( X ) and { g k x } is a Cauc h y sequence in X . By Lemma 5 .5 (ii), b g n x → [ g k x ] and, by Coro llary 5 .6, { b g n } conv erges p oint wise on b X to some h ∈ C ( b X , b X ). Note that x ∈ X ⊂ Y . Since b g n x → [ g k x ] and b g n preserv es the metric b d then b g n − 1 [ g k x ] → x , Hence, by the prop erness of ω , w e ma y assume that { b g n − 1 } has a subsequence { c g n k − 1 } that con v erges p oin t wise to some f ∈ E . This mak es h a surjection, hence h ∈ I so ( b X ). Therefore, [ g k x ] ∈ X p , so Y \ X ⊂ X p .  Note that, as the follo wing example sho ws, it may happ en that X p = X l 6 = ∅ , X ∪ X p 6 = b X and the set X ∪ X p is no t the maximal s ubse t of b X such that the ac tion ( E , X ∪ X p ) is pr op er . Example 5.13. T ak e X := { ( x, y ) ∈ R | x ∈ Q + √ 2 N , y > 0 } , 16 A. MANOUS SOS AN D P . STR ANTZALOS endo w ed with the Euclidean metric. Its g roup of isometries is the additiv e group of the rat ional num b ers acting b y horizontal tr a nsla- tions. Therefore, ( I so ( X ) , X ) is Cauc h y-indivisible . Ob viously b X is the closed upp er half plane, X p = X l 6 = ∅ , X ∪ X p is the op en upper half plane and E is the additiv e gr oup of the r eal n um b ers acting b y horizon tal translations on b X . Hence E acts prop erly on b X . R emark 5.14 . The sets X p and X l constructed in Theorem 5.12 a r e optimal in the sense that if one ma y think to replace the sets X p and X l with the following more general sets X ∗ l = { y ∈ b X | there exist a sequence { g n } ⊂ I so ( X ) and some x ∈ X suc h that g n → ∞ in I so ( X ) , { g n x } is a Cauc h y sequ ence and y = [ g k x k ] , for some [ x k ] ∈ b X } , and X ∗ p = { y ∈ b X | there exist a sequence { g n } ⊂ I so ( X ) and some x ∈ X suc h that g n → ∞ in I so ( X ) , { g n x } and { g − 1 n x } are Cauc hy sequence s and y = [ g k x k ] , for some [ x k ] ∈ b X } and ask if the set X ∪ X ∗ p is the maximal subse t of the completion b X suc h that the map ω ∗ : E × ( X ∪ X ∗ p ) → ( X ∪ X ∗ p ) × b X with ω ∗ ( f , y ) = ( y , f y ), f ∈ E and y ∈ X ∪ X ∗ p is prop er this is not true in gener al . This follows from the following assertion and Example 5.15, whic h sho ws t ha t there is a metric space X suc h that ( I so ( X ) , X ) is Cauc h y-indivisible, X p 6 = ∅ and t he map ω ∗ as ab o v e is not prop er. Assertion. If X ∗ p 6 = ∅ (equiv alently X p 6 = ∅ ) then X ∗ p = b X . Pr o of . Let y = [ x k ] in b X . By assumption, there exist a sequence { g n } ⊂ I so ( X ) and a p oin t x ∈ X suc h that g n → ∞ in I so ( X ) and the sequence s { g n x } , { g − 1 n x } are Cauch y . By Lemma 5.5, { g n x n } is a Cauc h y sequ ence in X and g − 1 k [ g k x k ] → [ g − 1 k g k x k ] = [ x k ] = y . Hence, y ∈ X ∗ p .  Example 5.15. The example is a comb ination of Example 4.3 and of a 3-dimensional v ariation of the “riv er metric” [5, Example 4.1 .6]. Let X = { ( x, y , z ) | x ∈ Q + √ 2 N , y ∈ Q + √ 2 N , z > 0 } . F or eve ry pair of po ints w 1 = ( x 1 , y 1 , z 1 ), w 2 = ( x 2 , y 2 , z 2 ) ∈ X define d ( w 1 , w 2 ) :=  | y 1 − y 2 | + | z 1 − z 2 | , if x 1 = x 2 | y 1 | + | y 2 | + | x 1 − x 2 | + | z 1 − z 2 | , if x 1 6 = x 2 . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 17 W e can easily ve rify than d is a metric on X . The group of isometries I so ( X , d ) consists of all the maps g : X → X of the form g ( x, y , z ) = ( x + p, y + q , z ) , p, q ∈ Q . The action ( I so ( X ) , X ) is Cauc h y-indivisible since X do es no t con tain the xy -plane ( the last co ordinate of the po in ts of X is p ositiv e). Then X p = { ( x, y , z ) | x ∈ Q + √ 2 N , y ∈ R , z > 0 } . T o see t ha t take x ∈ Q + √ 2 N , y ∈ R a nd z > 0 a nd choose k ∈ N suc h that y − √ 2 k / ∈ Q . Let { q n } b e a sequenc e of rationa l n um bers suc h that q n → y − √ 2 k . Hence, if we let { g n } ⊂ I so ( X ) with g n ( x, y , z ) := ( x, y + q n , z ) then g n ( x, √ 2 k , z ) = ( x, q n + √ 2 k , z ) → ( x, y , z ). Hence ( x, y , z ) ∈ X p . Observ e that b X = { ( x, y , z ) | x ∈ Q + √ 2 N , y ∈ R , z ≥ 0 } , and E consists of all the maps g : b X → b X with g ( x, y , z ) = ( x + p, y + r , z ) , p ∈ Q , r ∈ R . Ho w eve r, t he map b ω : E × b X → b X × b X with b ω ( f , w ) = ( w , f w ), f ∈ E and w ∈ b X is not prop er since if w e t a k e a sequence of rational n um b ers { p n } suc h that p n → √ 2 and let { g n } ⊂ E with g n ( x, y , z ) = ( x + p n , y , z ) then g n ( x, 0 , 0) → ( x + √ 2 , 0 , 0) for eac h x ∈ Q + √ 2 N . The sequence { g n } div erges in E since for instance the distance of the p o ints g n ( √ 2 , √ 2 , 1 ) = ( q n + √ 2 , √ 2 , 1 ) from an y p o int of X is ev entually at least √ 2. Hence the limit set L (( x, 0 , 0)) is not empt y . A question that ar ises natura lly from Theorem 5.12 is if the action of the Ellis semigroup E on X ∪ X l is prop er. Surprisingly , a s the fol- lo wing prop osition sho ws, this is equiv alen t to the existence of a W eil completion (with resp ect to the uniformit y of p o in t wis e con v ergence) for the group I so ( X ). Before w e giv e t he statemen t let us recall a f ew things ab out the W eil completion of I so ( X ), defined in the intro duc- tion. The uniformity o f p oint wise con v ergence on X coincides with the left uniformit y of I so ( X ) (cf. [2, Ch. I I I, § 3.1 and Ch. X, § 3 Exercise 19 (a), p. 33 2]) a nd I so ( X ) has W eil completion with resp ect to this uniformit y if the left and the right uniformities coincide; cf. [2, Ch. I I I, § 3.4 a nd § 3 Exerc ise 3, p. 306]. Note that the left completion of I so ( X ) do es not dep end on the c hoice of a left-inv aria nt metric on I so ( X ); cf. Lemma 2.9 in [6]. Prop osition 5.16. The fol lowing ar e e quivalent: 18 A. MANOUS SOS AN D P . STR ANTZALOS (i) The map ω : E × ( X ∪ X l ) → ( X ∪ X l ) × b X is pr op er. (ii) E is a gr oup (pr e cisely a close d sub gr oup of I so ( b X ) ). (iii) I so ( X ) has a Weil c o mpletion with r esp e ct to the uniformity of p ointwise c onve r genc e (in this c ase E is the Weil c ompletion of I so ( X ) ). Pr o of . W e sho w t hat item (i) implies item (ii) and vice vers a. Supp ose that ω is prop er. T ak e some h ∈ E . Since E is a semigroup, cf. Prop osition 5.8 (iii), w e ha v e only to show that h has an in v erse in E . If h = b g ∈ \ I so ( X ) for some g ∈ I so ( X ) , then d g − 1 is the in v erse of h in \ I so ( X ) ⊂ E . If h ∈ H there is a seque nce { g n } in I so ( X ) suc h that g n → ∞ in I so ( X ) and b g n → h p oint wise on b X . Hence, if x ∈ X then b g n x → hx . Since b g n preserv es the metric b d then b g n − 1 hx = d g − 1 n hx → x . By Lemma 5.11, hx ∈ X ∪ X l , hence, b y the prop erness of ω , { d g − 1 n } has a con v ergen t subseq uence { d g − 1 n k } to some f ∈ E . This mak es h a surjection, hence h ∈ I so ( b X ) and h has an in v erse in E . T o sho w the con vers e implication note that if E is a group t hen X l = X p ; cf. Prop osition 5.10. Hence, b y Theorem 5.12, the map ω is pro p er. T o finish the pro o f o f t he prop osition let us sho w that item (iii) im- plies item (ii) and vice v ersa. Note that I so ( X ) ha s a W eil completion if and only if the map with g 7→ g − 1 for ev ery g ∈ I so ( X ) maps Cauc h y sequence s of I so ( X ) to Cauc h y sequence s; cf. [2, Ch. I I I, § 3.4 Theorem 1]. It is easy to c hec k t hat in case when I so ( X ) is Cauch y-indivisible this is equiv alent to X l = X p . Equiv alen tly , b y Prop osition 5.10, E is a g roup.  R emark 5.17 . In case when I so ( X ) is a lo cally compact group, e.g. if X is a lo cally compact space and I so ( X ) a cts properly on it (as it is kno wn in the case X is connected), then by [2, Ch. II I, § 3 Exercis e 8 , p. 307], I so ( X ) has a lo cally compact completion hence E is a lo cally compact group. W e summarize with the following PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 19 Corollary 5.18. If E is a gr o up the action ( I so ( X ) , X ) is emb e dde d densely in the pr op er action ( E , X ∪ X l ) such that the fo l lowing e q ui- variant diagr am c ommutes ( I so ( X ) , X )   / / X   ( E , X ∪ X l ) / / b X wher e X → X ∪ X l is the inclusion m ap an d the map I so ( X ) → E is define d by g 7→ b g for every g ∈ I so ( X ) . By the wor d “densely” we me an that X is dense in X ∪ X l and \ I so ( X ) is dense in E . Question 5.19 . The ab ov e embedding of a Cauch y-indivisible action as a dense sub-action of a prop er one establishes a remark a ble connec- tion b et w een Cauc h y-indivis ible and pro p er actions. And at the same time propo ses an intere sting question: Is there an y analo g y with the situation of embedding o f a pro p er action (on a lo cally compact and connected space) in an appropriate zero-dimensional compactification, lik e in [1] and [9]? Namely , can w e o btain any structurally informative corresp ondence b et w een div ergen t nets in I so ( X ) and suitable subsets of X l ? R emark 5.20 . As w e will see in the example describ ed in § 6 it may b e happ en that X p 6 = X l and X ∪ X l = b X ; cf. Prop osition 6.8. In view of po ssible questions for refineme n ts of Corollary 5.18 w e note that it may happ en that X ∪ X p = b X and E is not dense in I so ( b X ), a s the follow ing example sho ws: Example 5.21. There is a separable metric space ( X , d ) suc h that ( I so ( X ) , X ) is Cauc h y-indivisible, pro p er, X ∪ X p = b X and I so ( X ) has a W eil completion whic h do es not coincide with the group I so ( b X ). Pr o of . Let X b e the set Q + √ 2 N endo w ed with the Euclidean metric; cf. Example 4.3. It is easy to che c k tha t X ∪ X p = X ∪ X l = R , see also Example 5.15, hence by Prop ositions 5.10 and 5 .16, I s o ( X ) has a W eil completion (or just observ e that I so ( X ) is an ab elian group and use [2, Ch. II I, § 3.5 Theorem 2]). But all the reflections of the space are excluded, hence the p oin t wis e closure E o f \ I so ( X ) does not coincide with I so ( R ).  20 A. MANOUS SOS AN D P . STR ANTZALOS 6. An example of a proper Cauchy-indiv isible action of a gro up which has no Weil completion In this section w e sho w that there is a separable metric space X such that the action ( I so ( X ) , X ) is prop er, Cauch y-indivisible and the Ellis semigroup E is not a group. Equiv alently , in view of Prop osition 5.16, I so ( X ) has no W eil completion. Consider the space o f the integers Z with the discrete metric d , that is if m, n ∈ Z then d ( m, n ) = 0 if m = n and d ( m, n ) = 1 otherwise. The group of isometries I so ( Z ) consists o f all the self bijections of Z and is an example of a top o logical group that has no W eil completion. T o see tha t ta k e f n : Z → Z with f n z = z for − n < z < 0, f n ( − n ) = 0 and f n z = z + 1 otherwise. Then it is easy t o v erify t ha t f n → f , where f z = z for z < 0, and f z = z + 1 for z ≥ 0. Hence { f n z } is a Cauc h y sequence in Z for ev ery z ∈ Z , therefore { f n } is a Cauc h y sequence in I so ( Z ) with resp ect t o the uniformity of p oint wise con vergence on Z . But { f − 1 n 0 } = {− n } is not a Cauch y sequence , th us { f − 1 n } to o . So, b y [2, Ch. I I I, § 3.4 Theorem 1], I so ( Z ) has no W eil completion. The problem is that the action ( I s o ( Z ) , Z ) is not Cauch y-indivisible. T o see that no t ice that { f − 1 n 1 } = { 0 } but { f − 1 n 0 } = {− n } is not a Cauc h y sequence. Neve rtheless, t he group I so ( I so ( Z )) is Cauc h y-indivisible and has no W eil completion as w e sho w in the follo wing. T ak e an enume ration A = { z i } of Z and equip I so ( Z ) with the metric  ( f , g ) = ∞ X i =1 1 3 i d ( f z i , g z i ) for f , g ∈ I so ( Z ). In view of Prop osition 5.3 the unifor mity o f p o in t wis e con v ergence, the left uniformit y and the uniformit y induced b y  on I so ( Z ) coincide ( the c hoice of 1 3 instead of 1 2 in Definition 5 .2 will b e clarified in the pro of of Lemma 6.1). Note that ( I so ( Z ) ,  ) is a separable metric space. W e will show that I so ( I so ( Z )) is Cauch y-indivisible but has no W eil completion. Lemma 6.1. If T ∈ I so ( I so ( Z )) and f , g ∈ I so ( Z ) then d ( T ( f ) z , T ( g ) z ) = d ( f z , g z ) for every z ∈ Z . Pr o of . Since  ( T ( f ) , T ( g )) =  ( f , g ) t hen ∞ X i =1 1 3 i d ( T ( f ) z i , T ( g ) z i ) = ∞ X i =1 1 3 i d ( f z i , g z i ) . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 21 Since the v alues of d are 0 or 1 then d ( T ( f ) z n , T ( g ) z n ) = d ( f z n , g z n ), for eve ry z n ∈ A = Z (here is t he role of the c hoice of 1 3 instead of 1 2 ).  Prop osition 6.2. If T ∈ I so ( I so ( Z )) and f ∈ I so ( Z ) then T ( f ) = T ( e ) ◦ f , w h e r e e is the unit element of I so ( Z ) . Pr o of . Let z k , z l b e tw o distinct in tegers and let g ∈ I so ( Z ) b e suc h that g z k = z l , g z l = z k and g z = z elsewhere. W e show that T ( g ) = T ( e ) ◦ g . If z 6 = z k , z l then, b y Lemma 6.1, d ( T ( g ) z , T ( e ) z ) = d ( g z , z ) = 0. Hence T ( g ) z = T ( e ) z = T ( e ) ◦ g z . Moreo v er d ( T ( g ) z k , T ( e ) z k ) = d ( g z k , z k ) = d ( z l , z k ) = 1 and, similarly , d ( T ( g ) z l , T ( e ) z l ) = 1. Sinc e T ( g ) z k 6 = T ( g ) z = T ( e ) z for z 6 = z k , z l and T ( e ) is surjectiv e then T ( g ) z k = T ( e ) z l = T ( e ) ◦ g z k and, similarly , T ( g ) z l = T ( e ) ◦ g z l . Therefore T ( g ) = T ( e ) ◦ g . Fix f ∈ I so ( Z ) and some z ∈ Z . If f z = z then T ( f ) z = T ( e ) z = T ( e ) ◦ f z since d ( T ( f ) z , T ( e ) z ) = d ( f z , z ) = 0. If f z 6 = z , let g ∈ I so ( Z ) with g z = f z , g f z = z a nd g w = w elsewhere. Since d ( T ( f ) z , T ( g ) z ) = d ( f z , g z ) = 0 then T ( f ) z = T ( g ) z . Us ing the result o f the previous paragraph, T ( f ) z = T ( g ) z = T ( e ) ◦ g z = T ( e ) ◦ f z . Since z was arbitrary then T ( f ) = T ( e ) ◦ f .  Corollary 6.3. L et L, T ∈ I so ( I so ( Z )) . Th e n L ◦ T ( e ) = L ( e ) ◦ T ( e ) and T − 1 ( e ) = ( T ( e )) − 1 . Pr o of . Since T ( f ) = T ( e ) ◦ f for ev ery T ∈ I so ( I so ( Z )) and f ∈ I so ( Z ), then L ◦ T ( f ) = L ( T ( f )) = L ( e ) ◦ T ( f ) = L ( e ) ◦ T ( e ) ◦ f . Hence, L ( e ) ◦ T ( e ) = L ◦ T ( e ). If I denote the iden tit y on I so ( I s o ( Z )), then f = I ( f ) = I ( e ) ◦ f . Hence I ( e ) = e and T − 1 ( e ) = ( T ( e )) − 1 .  Prop osition 6.4. The map B : I so ( I so ( Z )) → I so ( Z ) with B ( T ) = T ( e ) is a uniform gr oup isomorphism with r esp e ct to the uniformities of p oin twis e c o nver genc e on the underlying sp ac es I so ( Z ) and Z r esp e c- tively. Pr o of . By Prop osition 5.3 w e can equip I so ( I so ( Z )) with a left in- v ariant metric σ suc h that the uniformity of p oint wise conv ergence, the left uniformity and the unifor mity induced b y σ on I s o ( I so ( Z )) coincide. Let L n , T n ∈ I so ( I so ( Z )) suc h that σ ( L n , T n ) → 0 , hence σ ( T − 1 n L n , I ) → 0. Therefore T − 1 n L n → I p oin t wis e on I so ( Z ) so T − 1 n L n ( e ) → e , thus  ( L n ( e ) , T n ( e )) → 0. F o r the con v erse, note t ha t if  ( T − 1 n L n ( e ) , e ) → 0 then  ( T − 1 n L n ( e ) ◦ f , f ) → 0 for ev ery f ∈ I so ( Z ) 22 A. MANOUS SOS AN D P . STR ANTZALOS since t he map I so ( Z ) → I so ( Z ) with g 7→ g f is contin uous. Hence T − 1 n L n → I p oin t wise on I so ( Z ). Corollary 6 .3 implies that B is also group isomorphism.  Prop osition 6.5. T h e gr oup I so ( I so ( Z )) is Cauchy-indivisible an d has no Wei l c o mpletion. Pr o of . Let us show firstly that I so ( I so ( Z )) is Cauch y-indivisible. Let { T n } ⊂ I so ( I so ( Z )) and f ∈ I so ( Z ) suc h that { T n ( f ) } is a Cauc h y sequence in I so ( Z ). T ake some g ∈ I so ( Z ). Since { T n ( f ) } is a Cauc h y sequence in I so ( Z ) then, it is easy to see that { T n ( f ) z } is a Cauc h y sequence for eve ry z ∈ Z . Equiv alen tly , { T n ( f ) f − 1 g z } is a Cauc hy sequence for ev ery z ∈ Z . By Prop osition 6.2, T n ( f ) f − 1 g z = T n ( e ) ◦ f f − 1 g z = T n ( e ) ◦ g z = T n ( g ) z . Therefore { T n ( g ) } is a Cauch y seq uence in I so ( I so ( Z )) for ev ery g ∈ I so ( Z ), hence I so ( I so ( Z )) is Cauc hy-indivis ible. Since b y the previous prop osition the groups I so ( I so ( Z )) and I so ( Z ) are uniformly isomorphic and the group I so ( Z ) has no W eil completion then the same also holds for I so ( I so ( Z )).  Prop osition 6.6. The ac tion ( I so ( I so ( Z )) , I so ( Z )) is pr op er. Pr o of . Let f , g ∈ I so ( Z ) and { T n } ⊂ I so ( I so ( Z )) b e a sequence suc h that T n ( f ) → g . Hence, by Prop o sition 6.2, T n ( e ) ◦ f → g th us T n ( e ) → g f − 1 . Therefore { T n ( h ) } con v erges for ev ery h ∈ I so ( Z ). Since ( T n ( e )) − 1 → f g − 1 it is easy to v erify that { T n } con v erges in I so ( I so ( Z )) hence the action ( I so ( I so ( Z )) , I so ( Z )) is prop er.  R emark 6.7 . Notice that I so ( I s o ( Z )) is not lo cally compact since has no W eil completion ( I so ( Z ) is, of course, not lo cally compact). In the following prop osition w e giv e a precise desc ription of t he sets b X , X p , X l and E for the action ( I so ( I so ( Z )) , I so ( Z )). Prop osition 6.8. The fol lowing holds. (i) The p ointwise closur e of I so ( Z ) in the set of a l l selfmaps of Z c onsists of al l the inje c tive selfmaps of Z and c oincides with the c ompletion of ( I so ( Z ) ,  ) . (ii) The set X p is em pty and the set X l c onsists of al l the inje ctive selfmaps of Z which ar e not surje ctive. Mor e o v er, the c omple- tion of ( I so ( Z ) ,  ) c oin c ides with the set X ∪ X l . (iii) The El l i s semigr oup E is uniformly hom e omorphic to the semi- gr oup of al l inje ctive selfmaps of Z . PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 23 (iv) F or every non-surje ctive map f in the c ompletion of ( I so ( Z ) ,  ) (that is f do es not b elong to I so ( Z ) ) ther e exists a s e quenc e { T n } ⊂ E with T n → ∞ in E and T n ( f ) → f . Henc e, E do es not act pr op erly on the c omple tion of ( I so ( Z ) ,  ) . Pr o of . (i) Consider an injective map f : Z → Z . There is a strictly increasing sequence of sets A n ⊂ Z (with resp ect to the inclusion) suc h that eac h A n con tains finitely many p oin ts and con tains also the sets [ − n, n ] and f ([ − n, n ]), for every n ∈ N . Define g n ∈ I so ( Z ) suc h that the restriction of g n on A n is a permutation whic h coincides with f on the in terv al [ − n, n ] and g n is the identit y on the complemen t of A n . Hence, g n → f p oin t wis e. (ii) Assume that X p 6 = ∅ . Then there exist a sequence { T n } ⊂ I so ( I so ( Z )) and a map f ∈ I so ( Z ) suc h that { T n ( f ) } and { T − 1 n ( f ) } are Cauc h y sequence s in I so ( Z ) a nd T n → ∞ . The Cauc h y-indivisibilit y of the action ( I so ( I so ( Z )) , I so ( Z )) (cf. P rop osition 6.5) implies that { T n ( e ) } and { T − 1 n ( e ) } are Cauc h y sequences in I so ( Z ), where e de- note the unit elemen t of I so ( Z ). Hence , for ev ery z ∈ Z the se- quences { T n ( e )( z ) } and { T − 1 n ( e )( z ) = ( T n ( e )) − 1 ( z ) } are Cauc hy in Z (cf. Corollary 6.3). Th us, for ev ery z ∈ Z the sequences { T n ( e )( z ) } and { ( T n ( e )) − 1 ( z ) } a r e ev en tually constan t. So, there exist injectiv e selfmaps g , h of Z such that T n ( e ) → g a nd ( T n ( e )) − 1 → h . Ob viously , h = g − 1 . This sho ws that g ∈ I s o ( Z ). T he properness of the a ction ( I so ( I so ( Z )) , I so ( Z )) (cf. Prop o sition 6.6) implies that the sequence { T n } has a con vergen t subsequence in I so ( I so ( Z )) (actually , b y the pro of of Prop osition 6.6 , the whole sequence { T n } con v erges) whic h is a contradiction. Therefore X p = ∅ . By item (i) it follo ws that X l is con tained in the semigroup of a ll the injectiv e selfmaps of Z whic h are not surjectiv e and b y the same pro of it follo ws that if f is a selfmap o f Z whic h is not surjectiv e then there exists a sequence { g n } ⊂ I so ( Z ) suc h that g n → f p oint wise on Z . Set T n ∈ I so ( I so ( Z )) with T n ( e ) = g n (that is T n ( h ) = T n ( e ) ◦ h = g n h for ev ery h ∈ I so ( Z )). T hen T n → ∞ , { T n ( e ) } is a Cauch y sequence in I so ( Z ) a nd T n ( e ) → f , hence f ∈ X l . By item (i) a map in the completion of ( I so ( Z ) ,  ) is an injection Z → Z . If this map is also a surjection then it b elongs to X = I so ( Z ) and if it is not a surjection then it b elongs to X l . Hence, the completion of ( I so ( Z ) ,  ) coincides with t he set X ∪ X l . (iii) It follo ws easily b y Pro p osition 6.4. (iv) Let f b e in the completion of ( I so ( Z ) ,  ) but do es not b elong to I so ( Z ). Hence , f is not surjectiv e. T ake a p oint z / ∈ f ( Z ). The re is a strictly increasing sequence of sets A n ⊂ Z suc h that each A n 24 A. MANOUS SOS AN D P . STR ANTZALOS con tains finitely man y p oints and con tains strictly the sets { z } , [ − n, n ] and f ([ − n, n ]), for ev ery n ∈ N . Define g n ∈ I so ( Z ) suc h that the restriction o f g n on A n is a p ermutation with t he prop ert y g n is the iden tit y on the set f ( [ − n, n ]), g n z > n and g n is the iden tit y on the complemen t of A n . Hence, g n ◦ f → f p oint wise and since f is not surjectiv e then g n → ∞ in I so ( Z ). Set T n ∈ I so ( I so ( Z )), as ab ov e, with T n ( e ) = g n . Then T n → ∞ in E b ecause T n ( e ) = g n → ∞ in I so ( Z ) a nd T n ( f ) = g n ◦ f → f .  7. Se ctions, Bore l Sections, Fund ament al Sets and Ca uchy-indivisibility As it is indicated in the in tro duction a section of an action ( G, X ) is a subset of X whic h con tains only one point fr o m each orbit. If a section is a Borel subset of X it called a Borel section. Concerning the existence of Borel sections, if ( Y , d ) is a separable metric space and R is an equiv alence relation on Y suc h that the R -saturation of eac h op en set is Borel, then there is a Borel set S whose in tersection with each R - equiv a lence class which is complete with resp ect to d is nonempt y , and whose in tersection with eac h R - equiv alence class is at most one po in t; cf. [7, Lemma 2]. The problem of the existence of a Borel section for a con tin uous P olish a ctio n is of remark able significance b ecause the existence of a Borel section is equiv alen t to many interes ting facts, lik e that t he underlying space has o nly t rivial ergo dic measures, the orbit space has a standard Borel structure and it has no non- t r ivial ato ms. Recall t ha t an action ( G, X ) is called Polish if b oth G and X are P olish spaces, i.e. they are separable and metrizable b y a complete metric. Keeping the previous in mind w e hav e the following: Prop osition 7.1. I f the El lis semigr oup E is a gr oup then the action ( E , X ∪ X l ) has a B or el s e ction. Pr o of . Assume that the Ellis semigroup E is a group. Since b y Prop o- sition 5.10 w e hav e X l = X p and b y Propo sition 5.1 6 the map ω : E × ( X ∪ X l ) → ( X ∪ X l ) × e X is prop er then each orbit E x , x ∈ X ∪ X l is closed in e X . Hence, b y [7, Lemma 2] there exists a Borel set S ⊂ e X suc h that S ∩ ( X ∪ X l ) is a Bor el section (with resp ect to the relativ e top ology of X ∪ X l ) for t he action ( E , X ∪ X l ).  A ve ry useful notion in the theory of prop er a ctions on lo cally com- pact spaces with paracompact o rbit space is the notion of a fundamen- tal set. PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 25 Let G b e a top olog ical group which acts contin uously on a to p ological space X and A, B ⊂ X . Let us call the set G AB := { g ∈ G : g A ∩ B 6 = ∅} t he tr a n sp orter f rom A to B . Definition 7.2. A subset F of X is called a fundamen tal set for the action ( G , X ) if the follo wing holds. (i) GF = X . (ii) F or ev ery x ∈ X there exists a neigh b orho o d V ⊂ X of x suc h that the transp orter G V F of V to F has compact closure in G . F or lo cally compact spaces w e can replace condition ( ii) with the follo wing equiv alen t condition. (iia) The tr a nsp orter G K F from K to F has compact closure in G for ev ery non-empty compact subset K of X . Note that the exis tence of a fundamen tal set implies that the action group G is lo cally compact and the action ( G, X ) is pro p er. The notion of a fundamental set is relative to the notion of a section but it is differen t in general, in the sense that there are cases where a section is a fundamen tal set, a fundamen tal set fails to b e a section and cases where a section fails to be a fundamen tal set. A section ma y not b e Borel or ev en if it is Borel may not to b e con tained in an y fundamen tal set as the follo wing example sho ws. Example 7.3. The action ( Z , R ) with ( z , r ) 7→ r + z , z ∈ Z , r ∈ R is prop er and it has a Borel section whic h is not con tained in any fundamen tal set. Indeed , it is easy to see that the set S := ([0 , 1) \ S n ∈ N { 1 n } ) ∪ S n ∈ N { n + 1 n } is a section because t he interv al [0 , 1) is a section (and a f undamen tal set) for the action ( Z , R ). T ak e an op en ball B cen tered at 0 with radius ε > 0. Then there exists n 0 ∈ N suc h that 1 n < ε for every n ≥ n 0 . Let A b e a subset o f R that con tains S . Hence { n | n ≥ n 0 } is a subset of the transp orter Z B S ⊂ Z B A , so A can not b e a fundamental set. It is also p ossible to construct a section whic h is not Borel. T ake a set D ⊂ [0 , 1) whic h is not a Borel set and consider the set S 1 := D ∪ { x + 2 | x ∈ R \ D } . Obviously S 1 is a section whic h is not a Borel subset of the reals. Nev ertheless sections, Bo rel section and fundamen tal sets hav e a v ery strong connection a s the following theorem sho ws. Theorem 7.4. L et G b e a lo c al ly c omp act gr oup w hich acts pr op erly on a lo c al ly c omp act sp ac e X , and supp ose that the orbit sp ac e G \ X is p ar ac o m p act. L e t S b e a se ction for the action ( G, X ) . Then 26 A. MANOUS SOS AN D P . STR ANTZALOS (i) F or every op en neigh b orho o d U of S we c an c onstruct a clo s e d fundamental set F c and an op e n fundamental set F o such that F c ⊂ F o ⊂ U . (ii) If , in addi tion , ( X , d ) is a sep a r able metric sp ac e, in which c ase, by Th e or em 3.3 the action ( G, X ) is Cauchy-indiv isible, then ther e exists a Bor el se ction S B , which is also a fundame ntal set, such that S B ⊂ F c ⊂ F o ⊂ U . Pr o of . (i) Since U is op en it is a union of op en balls, let sa y S i , i ∈ I . Let p : X → G \ X b e the natural map x 7→ Gx . The n p ( S i ), i ∈ I is an op en co vering of the lo cally compact and paracompact space G \ X . Hence, there is a lo cally finite refinemen t { W j } , j ∈ J whic h consists of op en subsets of G \ X with compact closures suc h tha t W j ⊂ p ( S i j ), for some i j ∈ I . Now w e can follow the classic al pro of for the existence of fundamen tal sets; cf. [8, Lemma 2, p. 8]. Let { V j } b e an op en co v ering of G \ X suc h that V j ⊂ W j for ev ery j ∈ J . Fix an index j ∈ J and consider the restriction of the natural map p : X → G \ X o n the op en ball S i j . Since S i j is lo cally compact t hen there exist an o p en set U i j ⊂ S i j with compact closure a nd a compact set K i j ⊂ U i j ⊂ S i j suc h that p ( U i j ) = W j and p ( K i j ) = V j . Let F c := S j K i j and F o := S j U i j . The family { U i j } j ∈ J is lo cally finite in X hence the set F c is closed; cf. [2, Ch. I, § 1.5 Prop osition 4 ]. Moreov er, GF c = X . T ak e a p oin t x ∈ X and neighborho o d A of x with compact closure. Since the co v ering { W j } j ∈ J is lo cally finite, then the transp or t ers G AU i j from A to U i j are non-empt y f or only finitely man y j ∈ J . Since the sets A and U i j ha v e compact closure and the action ( G, X ) is prop er, then the transp orter G AF o of A to F o has compact closure in G . Th us, F c and F o are fundamental sets and b y construction F c ⊂ F o ⊂ U . (ii) Let F c a closed fundamen t al set for the action ( G, X ) like in item (i). Define a relat io n R on F c with x R y , x, y ∈ F c if and only if y ∈ Gx . W e will find a Borel sec tion for the closed fundamen t al set F c with resp ect to the previous na tural relation on F c and then w e will sho w that it is, also, a Bor el section for the action ( G, X ). Ob viously R is a n equiv alence relation on the separable metric space ( F c , d ) . Since the action ( G , X ) is prop er eac h orbit Gx is closed in X , for ev ery x ∈ X . The R -equiv alence class o f a p oint x ∈ F c is Gx ∩ F c , hence it is a closed subset of X , thus it is complete space with resp ect to the metric d . If U is an op en subset of F c with resp ect to the relativ e top ology of F c then the R - satura tion of U is the set GU ∩ F c whic h is op en in F c hence it is a Borel set. Therefore we can apply [7 , Lemma 2] to find a Bor el section S B ⊂ F c for the equiv alence relation R . Moreo ver, S B is a Borel section (and a fundamen ta l set) PROPERNESS, CAUCHY-INDIV ISIBILITY AN D THE WEIL COMPLETION 27 for the action ( G, X ), since it is contained in the closed fundamental set F c .  R emark 7.5 . Note that the assumption that the orbit space G \ X is paracompact is auto ma t ically satisfied for prop er isometric actions. Actually they a re metrizable by the metric defined b y ρ ( Gx, Gy ) := inf { d ( g x, hy ) | g , h ∈ G } = inf { d ( x, hy ) | h ∈ G } where G = I so ( X ) or E . So w e can apply Theorem 7 .4 in b oth cases. Question 7.6 . As Theorem 7.4(ii) indicates the notion of a Borel sec- tion is remark a bly related to that of a fundamen tal set in the locally compact case and may b e, similarly , used for structural theorems. Note that the Borel section S B , b ecause o f its construction, is a “minimal” fundamen tal set for the action ( G, X ), tha t is for eac h p oint x ∈ X the transp orter G { x } S B = g G x for some g ∈ G . So, it is in teresting to ask whether the existing Borel sec tion for the action ( E , X ∪ X l ) can b e reduced to a Borel section for the initial action ( I so ( X ) , X ). Reference s [1] H. Ab els, Enden von Ra¨ umen mit eigentlichen T ra nsformationsgrupp en , Com- men t. Math. Helv. 47 (1972 ), 457- 473. [2] N. Bourbak i, Elements of Mathematics. Gener al top olo gy , P art 1. Hermann, Paris; Addison- W esley P ublis hing Co., Rea ding , Mass.-Lo ndon-Don Mills, Ont. 196 6. [3] N. Bourbak i, Elements of Mathematics. Gener al top olo gy , P art 2. Hermann, Paris; Addison- W esley P ublis hing Co., Rea ding , Mass.-Lo ndon-Don Mills, Ont. 196 6. [4] E .G. Effros, Polish tr ansformation gr oups and classific ation pr oblems , Gene r al top ology and mo der n a nalysis (Pro c. Conf., Univ. Califor nia, Riverside, Calif., 1980), Academic Pres s, New Y o rk-Londo n, 1981 . [5] R. Eng elking, Gener al top olo gy , Second editio n, Sigma Series in P ure Mathe- matics 6 , Heldermann V erla g, Berlin, 1989. [6] G. Hjor th, An oscil lation the or em for gr oups of isometries , Geom. F unct. Anal. 18 (2 008), 489 - 521. [7] R.R. Kallman a nd R.D. Mauldin, A cr oss se ction the or em and an applic ation to C ∗ -algebr as , Pro c. Amer. Math. So c. 69 (1978 ), 57-6 1. [8] J .L . K oszul, L e ctur es on gr oups of tr ansfo rmations , T ata Institute of F unda- men tal Research, Bomba y , 1 9 65. [9] A. Mano ussos a nd P . Strantzalos, On emb e ddings of pr op er and e quic ontinuous actions in zer o-dimensiona l c omp actific ations , T ransa ctions of the A.M.S. 359 (2007), 559 3-560 9. 28 A. MANOUS SOS AN D P . STR ANTZALOS F akul t ¨ at f ¨ ur Ma thema tik, SFB 7 01, Universit ¨ at Bielefeld, Post- f ach 100131, D-33501 Bielefeld, Germany E-mail addr ess : am anous s@mat h.uni-bielefeld.de Dep ar tment of Ma thema tics, University o f A thens, P anepistimioupo- lis, GR-157 84, A thens, Greece E-mail addr ess : ps trant z@mat h.uoa.gr