Fr{e}chet-Urysohn fans in free topological groups

F r ´ ec het-Urysohn fans in free top ological groups T aras Banakh, Du ˇ san Rep o v ˇ s, Lyub o m yr Zdom skyy Octob er 26, 202 1 Abstract In this pap er we answ er the qu estio n of T. Banakh and M. Zaric hnyi constructing a cop y of the F r ´ ec het-Urysohn fan S ω in a top ological grou p G admitting a fun ctoria l em b edding [0 , 1] ⊂ G . The latt er means that eac h autohomeomorphism of [0 , 1] extends to a con tinuous homomorphism of G . This implies that many n atural fr ee top ological group constructions (e.g. the constructions of the Marko v free topological group, free ab elian top olo gical group, f ree totally b oun ded group , free compact group) applied to a T ychono v space X con taining a top ologi cal cop y of the space Q of rationals giv e top ologica l groups con taining S ω . In tro duction D. Shakhmatov noticed in [16] that the classic al L efsche tz-N¨ ob eling-P on tr yagin Theorem on em b eddings of n -dimensional compacta in to R 2 n +1 has no catego rical counterpart: one cannot em b ed ev ery finite-dimensional compact space X in to a finite-dimensional top ological group F X so that eac h contin uous map f : X → Y extends to a contin uous group homomor phism F f : F X → F Y . The pro of of this fa ct exploited Kulesz a ’s example o f a patholo gical 1- dimensional (non-metrizable) compact space that cannot b e em b edded in to a finite-dimensional top ological group [11]. How ev er, it w as discov ered in [4] that the problem lies already at the lev el of the unit in terv al [0 , 1]: it admits no functorial em b edding into an y metrizable or finite- dimensional gro up. So, eac h top ological group containing a functorially em b edded in terv al is non-metrizable and t hus ha s uncoun table character. In ligh t of this let us remark that the Mark ov free top ological gro up F M I o v er the in terv al I = [0 , 1] has c haracter χ ( F M I ) = d (see [14], [15 ]) , where d is the w ell-kno wn uncoun table small cardinal equal to the cofinalit y of the p oset ( ω ω , ≤ ). This cardinal is equal to the cardina lity of con tinuum c under Martin’s Axiom, but can also b e strictly smaller than c in some mo dels of ZF C (see [6], [18]). In this pap er w e sho w that the inequalit y χ ( F I ) ≥ d holds for man y other free top ological group constructions. First we give precise definitions. Let T b e a sub category of the category T op of top ological spaces and their con tinuous maps. By a functor of a f r e e top o l o gic al gr oup on T w e understand a pair ( F , i ) consisting of 0 The author s were suppor ted b y the Slovenian Res earch Age ncy grants P1-0 292-010 1-04 and BI-UA/04 -06- 007. Keywor ds and phr ases. (F ree) top ological gr o up, functorial em b edding, sequ ential fan, small cardinals. 2000 MSC. 18B30, 22A05, 5 4C20, 54E35, 54F15. 1 a co v arian t functor F : T → G from T into the category G of top ological groups and their con tin uous ho mo mo r phisms, and a na t ur a l transformation i = { i X } : Id → F of the iden tity functor Id : T → T in to the functor F whose comp onen ts i X : X → F X are top ological em b eddings for all spaces X ∈ T . The naturality of i means that for an y morphism f : X → Y in T w e ha v e the follo wing comm utative diagram: X i X − → F X f ↓ ↓ F f Y − → F Y i Y Therefore a functor of a free top olog ical group ( F , i ) to an y to polo gical space X ∈ Ob( T ) assigns a to polo gical group F X con ta ining a top olog ical cop y i X ( X ) ⊂ F X of X so that eac h morphism f : X → Y to another ob ject of T extends in a canonical w ay to a g r oup homomorphism F f : F X → F Y . A functor ( F , i ) is said to be minim al , if for ev ery space X ∈ Ob( T ) the group F X is algebraically generated b y i X ( X ). The functor of a free compact top ological group is a natural example of a non-minimal functor o f a f r ee top ological group, see [7, 8]. In the case when T has only one ob ject X and Mor ( X, X ) coincides with the set of all autohomeomorphisms of X , the embedding i X : X → F X is simply called [4] a functorial emb e d d ing of X in to the group G = F X . It was pro ven in [4] that if there exists a functorial em b edding o f the interv al I = [0 , 1] in to a top ological group G , then G is non-metrizable and infinite-dimensional. In this pap er w e shall mak e this result more precise b y sho wing t hat such a group G con tains a top ological cop y of the quotien t space I S ω = [0 , 1] × ω / { 0 } × ω , called the se quential he dgeho g b y analo gy with the sequen tial (or alternative ly F r´ ec het-Urysohn) fa n S ω = S 0 × ω / { 0 } × ω , where S 0 = { 0 } ∪ { 1 /n : n > 0 } is the con v ergen t sequence. This a nsw ers a question stated in [4], and implies that χ ( G ) ≥ d . Theorem 1. If ther e exists a functorial emb e dding o f the close d interval [0 , 1] in to a top olo gic al gr oup G , then G c ontains a c opy of the se quential he dgeho g I S ω . In particular, to polo gical gro ups fulfilling the requiremen ts of Theorem 1 do no t hav e the prop ert y α 4 in tro duced in [1]. A sub category T of the category T op of top olog ical spaces is said to b e ful l if a n y con tin uous map b et w een ob jects of T is a morphism in T . Theorem 2. L et ( F , i ) b e a functor of a fr e e top o l o gic al gr oup on a ful l sub c ate gory T of T op , c ontaining the interval I = [0 , 1] as an o b je ct. F or every sp ac e X ∈ Ob( T ) c ontaining a c opy of Q (r esp. I ), the top olo g i c al gr oup F X c ontains a c o py of the F r´ echet-Urysohn fan S ω (r esp. se quential he dgeho g I S ω ) and henc e has cha r acter χ ( F X ) ≥ d . It is inte resting t o remark that Theorem 2 is not true fo r the catego ry T = C (0 ) of zero- dimensional compacta and their contin uous maps. F or example, the functor F whic h assigns to eac h zero- dimensional compact space X the compact group Z Mor( X, Z 2 ) 2 con taining a diago nally em b edded copy of X , is a f unctor of a free top ological group (see [1 6]). F or any compact metrizable space X the group Z Mor( X, Z 2 ) 2 is metrizable and hence contains no copy of the F r ´ ec het- Urysohn fan. This sho ws that the inclusion I ∈ O b( T ) in Theorem 2 is ess ential. Problem 3. L et ( F , i ) b e a functor o f a fr e e top olo gic al gr oup on a ful l sub c ate gory I of T op . Assume that I ∈ Ob( T ) and X is an obje ct of T . D o es F X c on tain a c opy of S ω if X c on tains a non -trivial c o n ver gent se quenc e? (This is true for the functor of the Mark ov free top ological group). 2 W e recall that a to polo gical space X is said to b e sc atter e d , if ev ery non-empt y subspac e Y of X has an isolated p oin t. Combining Theorem 2 with the main result o f [3], w e shall deriv e the follo wing Corollary 1. L et ( F , i ) b e a minimal functor of a fr e e top olo gic al g r oup on a ful l c ate gory T of top olo gic al s p ac es such that I ∈ Ob( T ) . Supp ose that X ∈ Ob( T ) i s a m etrizable s e p ar able sp ac e that has a c omp actific ation ¯ X ∈ Ob( T ) . If F X is a k -sp ac e, then X is lo c al ly c omp act or sc atter e d. This corollary can be compared with a result [2] of Arkhangel’ski ˘ ı, Okunev, and P esto v who pro v ed that for a metrizable space X the Mark ov free top ological group F M X is a k - space if and only if X is either discrete or lo cally compact and separable. Pro of of T heorem 1 First w e shall describ e a copy of the sequen tial hedgehog I S ω in a top ological group G admitting a functorial em b edding of [0 , 1]. Here the crucial role b elongs to sp ecial trees consist- ing of closed in terv als and ordered b y the inv erse inclusion relation, whic h will b e called usual Cantor sc h emas throughout the pap er. W e shall use the fo llowing notations: { 0 , 1 } 0 and an op en neighb orho o d V of e such that for every x 1 , x 2 ∈ [0 , 1] w ith | x 1 − x 2 | < ε ( U ) the fol low ing in clusion holds: { x − 1 1 V x 2 , x 1 V x − 1 2 , x − 1 2 V x 1 , x 2 V x − 1 1 } ⊂ U 4 F o r ev ery n > 0 and f : X → X w e shall denote b y f n the comp osition f ◦ · · · ◦ f | {z } n . It will b e also con v enien t for us to denote b y f 0 the iden tity map on X . Giv en a n y strictly increasing function ϕ ∈ ω ω suc h that ϕ (0 ) > 0 and a usual Can tor sc heme J , define a usual Cantor sc heme ˜ J = { ˜ J s : s ∈ { 0 , 1 } <ω } letting ˜ J ∅ = J ∅ and ˜ J ( s 0 ,...,s n ) = J ( s ϕ 2 (0) 0 ˆ s ϕ 4 (0) − ϕ 2 (0) 1 ˆ ··· ˆ s ϕ 2 n +2 (0) − ϕ 2 n (0) n ) . Th us ˜ J m ⊂ J ϕ 2 m (0) for all m ∈ ω . Claim 9. L et p ∈ ω and ϕ 2 p (0) ≤ n < ϕ 2 p +1 (0) . T hen ∂ J n,ξ ⊂ { 0 , 1 } ∪ S s ∈{ 0 , 1 } ≤ p ˜ J md s for al l ξ > 3 n − ϕ ( n ) . Pr o of. Let us fix arbitr ary a ∈ ∂ J n,ξ and ξ > 3 n − ϕ ( n ) . Assum e, contrary to our claim, that a lies in the in terior of ˜ J s for some s ∈ { 0 , 1 } p +1 . Concerning a , t wo cases are p ossible. 1. a ∈ { left t (0) , right t (0) } for some t ∈ { 0 , 1 } n . Then a ∈ ∂ J t . Since n < ϕ 2 p +2 (0) a nd ˜ J s ∈ J ϕ 2 p +2 (0) , the inclusion a ∈ In t( ˜ J s ) is imp ossible. 2. There exists t ∈ { 0 , 1 } n suc h that a ∈ { left t ( ξ ) , right t ( ξ ) } . Without loss of generalit y , a = left t ( ξ ). Set b = min J t = left t (0). As it w as already prov en in the case 1 , b ∈ { 0 , 1 } ∪ S s ∈{ 0 , 1 } ≤ p ˜ J md s . Again, tw o sub cases are p ossible: ( i ) b is an end-p oin t o f ˜ J md s 0 for some s 0 ∈ { 0 , 1 } ≤ p , or b = 0. Then there exist s 1 ∈ { 0 , 1 } p and i ∈ { 0 , 1 } such that b ∈ ∂ ˜ J s 1 ˆ i . Since b = min J t , w e conclude that i = 0 and b = min ˜ J s 1 ˆ0 . Let r ∈ { 0 , 1 } ϕ 2 p (0) b e suc h that ˜ J s 1 = J r . The inequality n ≥ ϕ 2 p (0) com bined with J t ∈ J n , J r ∈ J ϕ 2 p (0) , and min J r = min J t = b , implies that a = left t ( ξ ) ≤ left r ( ξ ) ≤ left r (1) = max J r ˆ0 < min J r ˆ1 ≤ min ˜ J s 1 ˆ1 . In addition, a = le ft t ( ξ ) > left t (3 n − ϕ ( n ) ) = max J t ˆ 0 ϕ ( n ) − n +1 ≥ max J t ˆ 0 ϕ 2 p +2 (0) − n = max ˜ J s 1 ˆ0 . (The last equalit y immediately follo ws from b = min J t ˆ 0 ϕ 2 p +2 (0) − n = min ˜ J s 1 ˆ 0 and J t ˆ 0 ϕ 2 p +2 (0) − n , ˜ J s 1 ˆ0 ∈ J ϕ 2 p +2 (0) .) Therefore a ∈ (max ˜ J s 1 ˆ0 , min ˜ J s 1 ˆ1 ) = Int( ˜ J md s 1 ), a contradiction. ( ii ) b ∈ Int ˜ J md s 2 for some s 2 ∈ { 0 , 1 } ≤ p . Again, let r ∈ { 0 , 1 } ϕ 2 p (0) b e suc h that min J r = min J t = b . Since min J r ∈ In t( ˜ J md s 2 ), w e conclude that J r ⊂ In t( ˜ J md s 2 ). The inequalit y n ≥ ϕ 2 p (0) comb ined with min J r = min J t = b implies that J t ⊂ J r ⊂ Int( ˜ J md s 2 ). Therefore a ∈ Int( ˜ J md s 2 ) b eing an eleme nt of J t , whic h con tradicts the equation a = left t ( ξ ). In the sequel the notation Auth + ([ a, b ]) stands fo r the family of all increasing auto ho meo- morphisms of the in terv al [ a, b ]. Lemma 10. L e t ϕ and ˜ J b e as ab ove. Then for every indexe d fa m ily { U s : s ∈ { 0 , 1 } <ω } of op en neighb orho o ds of the identity e of G ther e exists a se quenc e ( h n ) n ∈ ω ⊂ Auth + ([0 , 1]) such that (1) h n ( ˜ J ) is a symmetric usual Cantor scheme; (2) h n +1 | S s ∈{ 0 , 1 } ≤ n ˜ J md s = h n | S s ∈{ 0 , 1 } ≤ n ˜ J md s for al l n ∈ ω ; a n d 5 (3) { π − [ h n ( ∂ J m,ξ ∩ ˜ J md s )] , π + [ h n ( ∂ J m,ξ ∩ ˜ J md s )] } ⊂ U s for al l s ∈ { 0 , 1 } ≤ n , m ≥ ϕ 2 n (0) , an d ξ > 0 . Pr o of. The followin g claim is the main building blo c k of our pro of. In order to shorten its pro of, some explanations similar to those made in the pro of of Claim 9 are omitted. Claim 11. L et J , G , e b e as in L em ma 10. L et also s ∈ { 0 , 1 } n 0 , m > 0 , c = max J s ˆ0 m , d = min J s ˆ1 m , and let U b e an op en neighb orho o d of e in G . Then ther e exists h ∈ Auth + ( J s ) such that ( i ) h | J s ˆ0 m and h | J s ˆ1 m ar e line ar; ( ii ) diam( h ( J s ˆ0 m )) = diam( h ( J s ˆ1 m )) ; and ( iii ) { π − ( h ( ∂ J n,ξ ∩ [ c, d ])) , π + ( h (( ∂ J n,ξ ∩ [ c, d ])) } ⊂ U for al l n ≥ n 0 and ξ ∈ (0 , 1] . Pr o of. Let us find some a, b ∈ J s suc h that a < b , the middle p oints of [ a, b ] and J s coincide, and | a − b | < ε ( U ). The la tter means that there exists an op en neigh b orho o d V of e suc h that u − 1 V v ∪ u V v − 1 ⊂ U for all u, v ∈ [ a, b ]. The con tinuit y of the group op eration on G gives us a sequence ( V n ) n ∈ ω of o pen neighborho o ds of e suc h that V 2 n n ⊂ V . Let h ∈ Auth + ( J s ) b e suc h that the following conditions are satisfied: h | [min J s , c ] and h | [ d, max J s ] are linear bijections on to [min J s , a ] and [ b, max J s ] resp ectiv ely , and diam ( h ( J t )) < ε ( V l ( t ) ) prov ided J t ⊂ ( c, d ). The existence of h follo ws from Lemma 7 . Giv en arbitrary n ≥ n 0 and ξ ∈ [0 , 1], write the family { J ∈ J n,ξ : J ⊂ [ c, d ] } in the form { J 1 , . . . , J q } suc h that J i is situated to the left of J j pro vided i < j . L et us note that eac h J ∈ J n,ξ is contained in some elemen t of J n +1 , and hence q ≤ 2 n +1 . It can b e easily deriv ed from the definitions of a usual Cantor sc heme and maps left − , right − that { c, d } ⊂ ∂ J n,ξ for all n, ξ suc h that ξ = 3 − ( n 0 + m − n − 1) or n ≥ n 0 + m , and { c, d } ∩ ∂ J n,p = ∅ o therwis e. In the first case w e ha v e π − [ h ( ∂ J n,ξ ∩ [ c, d ])] = a − 1 π + ( h ( ∂ J 1 )) π + ( h ( ∂ J 2 )) · · · π + ( h ( ∂ J q )) b ⊂ a − 1 V q n +1 b ⊂ a − 1 V b ⊂ U. In the second case n < n 0 + m and ξ 6 = 3 − ( n 0 + m − n − 1) . If ξ < 3 − ( n 0 + m − n − 1) , then ( c, d ) con tains all elemen ts of J n,ξ whose interse ction with [ c, d ] is nonempt y , a nd hence π − [ h ( ∂ J n,ξ ∩ [ c, d ])] = π − ( h ( ∂ J 1 )) π − ( h ( ∂ J 2 )) · · · π − ( h ( ∂ J q )) ⊂ V q n +1 ⊂ U. It sufficies to consider the case ξ > 3 − ( n 0 + m − n − 1) . Let u = left s ( ξ ) a nd v = righ t s ( ξ ). Then c < u < v < d and J i ⊂ ( u, v ) for all i ≤ q . Therefore π − [ h ( ∂ J n,ξ ∩ [ c, d ])] = u − 1 π + ( h ( ∂ J 1 )) π + ( h ( ∂ J 2 )) · · · π + ( h ( ∂ J q )) v ⊂ u − 1 V q n +1 v ⊂ u − 1 V v ⊂ U. V erification that π + ( h ( ∂ J n,ξ ∩ [ c, d ])) ⊂ U is similar, and th us condition ( iii ) is satisfied. Applying Claim 11 for the usual Can tor sc heme J , s = ∅ , m = ϕ 2 (0), and U = U ∅ , we get h 0 ∈ Auth + ( ˜ J ∅ ) satisfying the conditions ( i ) − ( iii ) of Claim 11 . Conditions ( i ) and ( ii ) ob viously imply (1), condition ( iii ) implies (3), while (2) is trivial. Assuming that w e ha v e already constructed h k satisfying (1) − (3) for all k < n , set h n | S s ∈{ 0 , 1 } ≤ n − 1 ˜ J md s = h n − 1 | S s ∈{ 0 , 1 } ≤ n − 1 ˜ J md s . Th us condition (2) is satisfied. In additio n, (3) holds for all s ∈ { 0 , 1 } ≤ n − 1 . It suffices to define h n on [0 , 1] \ S s ∈{ 0 , 1 } ≤ n − 1 ˜ J md s = S s ∈{ 0 , 1 } n ( ˜ J s \ ∂ ˜ J s ). Let us note, that for ev ery particular s ∈ { 0 , 1 } n the construction of h n | ˜ J s is similar to that of h 0 . G iv en an y s = ( s 0 , . . . , s n − 1 ) ∈ { 0 , 1 } n , set t = s ϕ 2 (0) 0 ˆ s ϕ 4 (0) − ϕ 2 (0) 1 ˆ · · · ˆ s ϕ 2 n (0) − ϕ 2 n − 2 (0) n − 1 6 and m = ϕ 2 n +2 (0) − ϕ 2 n (0). Th us J t = ˜ J s . Applying Claim 11 for J , t ∈ { 0 , 1 } ϕ 2 n (0) , m , and U s , we get h s ∈ Auth + ( J t ) satisfying conditions ( i ) − ( iii ) of Claim 11. Set h n | ˜ J s = h s ◦ h n − 1 | ˜ J s . Again, ( i ) and ( ii ) imply (1), a nd ( iii ) implies (3). The follo wing simple statemen t is due to M. Tk ac henk o (see [17, Lemma 1 .3]). Lemma 12. L e t G b e a top olo gic al gr oup and U b e the family of al l op en neighb orho o d of the neutr al element of G . Then for every U ∈ U ther e exists a de cr e asin g se quenc e ( U n ) n ∈ ω ⊂ U such that U σ (0) U σ (1) · · · U σ ( n ) ⊂ U for every bije ction σ : { 0 , . . . , n } → { 0 , . . . , n } . The pro of of the follo wing simple tec hnical lemma is left to the reader. Lemma 13. F or every function ϕ : ω → ω \ { 0 } ther e exis t strictly incr e asing functions ψ , θ : ω → ω such that ( i ) ϕ ( k ) < ψ ( k ) < θ ( k ) for al l k ∈ ω ; and ( ii ) ψ n +1 (0) = θ n (0) f o r al l n ∈ ω . Pro of of Lemma 6 . Throughout the pro of w e denote b y ˆ h : G → G some con tinuous homomorphism extending h ∈ Auth + ( J ∅ ). Let U b e the family o f all op en neigh b orho o ds of the neutral elemen t e of G . The con tin uity of the gro up o p eration of G yields the existence of an eleme nt U ∈ U suc h that U ∩ 0 − 1 U 1 = ∅ . By Lemma 12 there exis ts a sequence ( U n ) n ∈ ω ⊂ U suc h that U i 1 U i 2 · · · U i 2 n +1 − 1 ⊂ U for any n ∈ ω and ( i 1 , . . . , i 2 n +1 − 1 ) suc h that |{ j < 2 n +1 : i j = k }| = 2 k for all k ∈ { 0 , . . . , n } . Let ϕ : ω → ω b e a strictly increasing map with the prop erty 3 n − ϕ ( n ) < d n for ev ery n ∈ ω . Let us fix a sequence ( h n ) n ∈ ω ⊂ Auth + ([0 , 1]) satisfying conditions (1) − (3) of Lemma 10 with U s equal to U l ( s ) defined ab o v e, where s ∈ { 0 , 1 } <ω . Conditions (1), (2) imply that | h n ( t ) − h n +1 ( t ) | ≤ 2 − n − 1 , and hence t he sequence ( h n ) n ∈ ω is uniformly conv ergent to a monotone con tin uous surjectiv e function h : [0 , 1] → [0 , 1]. F or eve ry s ∈ { 0 , 1 } ≤ n w e hav e h | ˜ J md s = h n | ˜ J md s b y (2), consequen tly h | ˜ J md s is not constan t for ev ery s ∈ { 0 , 1 } <ω . Since S s ∈{ 0 , 1 } <ω ˜ J md s is dense in [0 , 1], w e conclude that h ∈ Auth + ([0 , 1]) as a monotone contin uous surjection whic h fails to b e constant on arbitrary op en subset of [0 , 1]. W e claim that ˆ h ( [ p ∈ ω [ ϕ 2 p (0) ≤ n<ϕ 2 p +1 (0) { z n, J ( ξ ) : ξ > 3 n − ϕ ( n ) } ) ⊂ 0 − 1 U 1 . Indeed, let us fix arbitrary p ∈ ω , ϕ 2 p (0) ≤ n < ϕ 2 p +1 (0), and ξ ∈ (3 n − ϕ ( n ) , 1]. Then by Claim 9 w e ha ve ∂ J n,ξ ⊂ { 0 , 1 } ∪ S s ∈{ 0 , 1 } ≤ p ˜ J md s . Therefore z n, J ( ξ ) = π − ( ∂ J n,ξ ) = π − ( { 0 } ∪ [ s ∈{ 0 , 1 } ≤ p ( ∂ J n,ξ ∩ ˜ J md s ) ∪ { 1 } ) . Com bining the equation ab o v e with (2) and (3 ) of Lemma 1 0 , w e conclude that ˆ h ( z n, J ( ξ )) = ˆ h [ π − ( { 0 } ∪ [ s ∈{ 0 , 1 } ≤ p ( ∂ J n,ξ ∩ ˜ J md s ) ∪ { 1 } )] = = π − [ h ( { 0 } ∪ [ s ∈{ 0 , 1 } ≤ p ( ∂ J n,ξ ∩ ˜ J md s ) ∪ { 1 } )] = = π − ( { 0 } ∪ h p ( ∂ J n,ξ ∩ ˜ J md s 1 ) ∪ h p ( ∂ J n,ξ ∩ ˜ J md s 2 ) ∪ · · · ∪ h p ( ∂ J n,ξ ∩ ˜ J md s 2 p +1 − 1 ) ∪ { 1 } ) = = 0 − 1 π δ 1 ( h p ( ∂ J n,ξ ∩ ˜ J md s 1 )) π δ 2 ( h p ( ∂ J n,ξ ∩ ˜ J md s 2 )) · · · π δ 2 p +1 − 1 ( h p ( ∂ J n,ξ ∩ ˜ J md s 2 p +1 − 1 )) 1 ⊂ ⊂ 0 − 1 U l ( s 1 ) U l ( s 2 ) · · · U l ( s 2 p +1 − 1 ) 1 ⊂ 0 − 1 U 1 , 7 where { s 1 , . . . , s 2 p +1 − 1 } is the en umeration of { 0 , 1 } ≤ p suc h that ˜ I md s i is situated to the left of ˜ I md s j pro vided 1 ≤ i < j < 2 p +1 , and δ i ∈ { + , −} . Then S p ∈ ω S ϕ 2 p (0) ≤ n<ϕ 2 p +1 (0) { z n, J ( ξ ) : 1 ≥ ξ > 3 n − ϕ ( n ) } ∩ ˆ h − 1 ( U ) = ∅ b y our c hoice of U . Let ψ , θ : ω → ω b e a n increasing n um b er sequences suc h as in Lemma 13, i.e. ϕ ( n ) ≤ ψ ( n ) ≤ θ ( n ) fo r all n ∈ ω and θ n (0) = ψ n +1 (0) fo r all n ≥ 1. It follows f rom the ab ov e that there are contin uous homomorphisms h ψ , h θ : G → G suc h that [ p ∈ ω [ ψ 2 p (0) ≤ n<ψ 2 p +1 (0) { z n, J ( ξ ) : 1 ≥ ξ > 3 n − ψ ( n ) } ∩ h − 1 ψ ( U ) = ∅ and [ p ∈ ω [ θ 2 p (0) ≤ n<θ 2 p +1 (0) { z n, J ( ξ ) : 1 ≥ ξ > 3 n − θ ( n ) } ∩ h − 1 θ ( U ) = ∅ , whic h implies S n ∈ ω { z n, J ( ξ ) : 1 ≥ ξ > 3 n − ϕ ( n ) } ∩ ( h − 1 ψ ( U ) ∩ h − 1 θ ( U )) = ∅ , and hence the fact that e 6∈ S n ∈ ω { z n, J ( ξ ) : 1 ≥ ξ > d n } is prov en. ✷ F o r a subset X of a top ological group G w e shall denote b y h X i the smallest subgroup of G containing X . Lemma 14. L et [0 , 1] ⊂ G b e a functorial emb e dding, { x i : 0 ≤ i ≤ n } ∪ { y j : 0 ≤ j ≤ m } ⊂ [0 , 1] , and y j 0 6∈ { y j : j 6 = j 0 } ∪ { x i : 0 ≤ i ≤ n } for some j 0 . Then x k 0 0 · x k 1 1 · · · · · x k n n 6 = y l 0 0 · y l 1 1 · · · · · y l m m for arbi tr ary in te gers k i , l i such that l j 0 ∈ {− 1 , 1 } . In p articular, for eve ry n ∈ ω and usual Cantor scheme J the map z n, J : [0 , 1] → G is an emb e d d ing, a nd z n, J ((0 , 1]) ∩ z m, J ((0 , 1]) = ∅ for al l m 6 = n . Pr o of. The second statemen t easily follo ws from the first one. T o pro ve the first statemen t, set x = x k 0 0 · x k 1 1 · · · · · x k n n , y = y l 0 0 · y l 1 1 · · · · · y l m m , and assume to the con trary t ha t x = y . Let h ∈ Auth + ([0 , 1]) b e suc h that h ( u ) = u for all u ∈ { y j : j 6 = j 0 } ∪ { x i : 0 ≤ i ≤ n } and h ( y j 0 ) 6 = y j 0 . Since the em b edding [0 , 1 ] ⊂ G is functorial, there exists a contin uous homomorphism ˆ h : G → G extending h . It follo ws from the a b ov e that ˆ h ( x ) = x = y and ˆ h ( y ) 6 = y , a con tradiction. Pro of of Prop osition 5. In ligh t of Lemmas 6 and 14 w e are left with the task of constructing a sequence ( d n ) n ∈ ω of p ositiv e reals with the prop erty z n ((0 , d n ]) ∩ S k >n z k ([0 , d k ]) = ∅ fo r all n ∈ ω . Let A = { ξ ∈ (0 , 1] : z 0 ( ξ ) ∈ S k > 0 z k ([0 , 1])) } . Since lim ξ → 0 z k ( ξ ) = e fo r all k ∈ ω , we conclude that A consists of ξ ∈ [0 , 1] such that ξ ∈ S k > 0 z k ([ c k , 0]) for some sequen ce ( c k ) k > 0 of p ositiv e reals. W e claim that 0 6∈ A . Indeed, assuming the con v erse we could find a sequence ( ξ n ) n ∈ ω of elemen ts of A conv erging to 0. It follo ws from the ab o ve that for every n ∈ ω there exists a sequence ( c n,k ) k > 0 of p ositiv e reals suc h that z 0 ( ξ n ) ∈ S k > 0 z k ([ c n,k , 1]) . Set c k = min { c n,k : n ≤ k } . Then e ∈ S k > 0 z k ([ c k , 1]) , whic h con tradicts Lemma 6. It follows from the ab o v e that A ⊂ ( d 0 , 1] fo r some d 0 > 0, and consequen tly z 0 ((0 , d 0 ]) ∩ S k > 0 z k ([0 , 1]) = ∅ . In the same w ay f or ev ery n > 0 we can find d n suc h that z n ((0 , d n ]) ∩ S k >n z k ([0 , 1]) = ∅ , whic h completes our pro of. ✷ 8 Remark. L et J b e a usual Cantor sc heme suc h that J ∅ ⊂ G is a functoria l em b edding. Let also C n = z n, J ([0 , d n ]) for a seque nce ( d n ) n ∈ ω fulfilling the requiremen ts of Prop osition 5. Then it can b e easily deriv ed from Lemma 14 that the map Y i ≤ n [0 , d n ] ∋ ( ξ 0 , . . . , ξ n ) 7→ Y i ≤ n z n, J ( ξ i ) is a homeomorphism, and w e denote its ima g e by D n . Thu s we hav e an increasing sequence D 0 ⊂ D 1 ⊂ · · · ⊂ D n ⊂ · · · , where eac h D n is a homeomorphic copy of the ( n + 1)-dimensional cub e 2 . Let ν b e the top ology of G a nd let τ b e the strongest top ology o n D = S n ∈ ω D n suc h tha t τ | D n = ν | D n for all n ∈ ω . It is easy to see that ( D , τ ) contains a homeomorphic cop y of R ∞ = lim → R n , whic h is homeomorphic to the Mark ov free top ological group ov er [0 , 1] (see [19]). But we do not kno w whether ν | D = τ | D . This leads us to the followin g question. Question 15. L et [0 , 1] ⊂ G b e a functorial emb e dding. Do es G c ontain a top olo gic al c opy of the Markov fr e e top olo gic al gr oup over [0 , 1] ? Mor e pr e ci s ely, do es the c onstruction describ e d ab ove yield a c opy of R ∞ in G ? Similarly to the pro of of Prop osition 16 b elow , the p ositiv e solution of the ab ov e question w ould imply that fo r ev ery functor F of a f ree top ological group a nd ev ery T ychono v space X con taining a top ological cop y of [0 , 1], the group F X contains a top ological cop y of the Mark o v free top ological gro up o ve r [0 , 1]. ✷ Theorem 2 and its generalizati on W e shall deriv e Theorem 2 from the follow ing sligh tly more general statemen t, where w e sp ecify the prop erties of the category T used in t he pro of of Theorem 2. Prop osition 16. L et ( F , i ) b e a functor of a fr e e top olo gic al gr oup on a c ate gory T of top olo gic al sp ac es such that I ∈ Ob( T ) and Mor( I , I ) ⊃ Auth + ([0 , 1]) . The gr oup F X ove r an obje ct X of T c ontains: (1) a c opy of the F r´ echet-Urysohn fan S ω pr ovide d ther e is a morphi s m f : X → I in T w hose r estriction f | Q onto some subsp ac e Q ⊂ X without isolate d p oints is an emb e dding of Q into I , and Mor( I , I ) c ontains al l c ontinuous map s φ : I → I ; and (2) a c opy of the se quential he d geho g I S ω pr ovide d ther e is a c opy Y ⊂ X of I , a surje c tive map f ∈ Mor( X, Y ) , and a home omorph i sm h ∈ Mor( Y , I ) . Prop osition 16 is a consequenc e of Theorem 1 a nd the follo wing Lemma 17. L et X b e a T ychonov sp ac e c on taining a top olo gic al c opy Q of the sp ac e of r ational numb ers and J b e a usual Cantor scheme with J ∅ = [0 , 1 ] . Then ther e exists a home omorphic c opy Q ′ ⊂ Q of Q , and a c o ntinuous map ψ : X → J ∅ such that ψ | Q ′ is a home omorphism b etwe en the sp ac es Q ′ and ∂ J = S s ∈{ 0 , 1 } <ω ∂ J s . 2 This gives an a lternativ e (but muc h lo nger) pro of of the fact [4] that ea c h top ological gr oup is infinite- dimensional provided it admits a functor ial embedding of the closed interv al. 9 Pr o of. Let I = [0 , 1]. It is w ell-kno wn that the diagonal map δ : X → I Mor( X,I ) is an em b edding, where Mor( X, I ) stands f o r the set of all contin uous maps from X to I . Assume that X con tains a subset Q ⊂ X ho meomorphic to the space of rational n um b ers. Using the fact that δ ( Q ) ⊂ I Mor( X,I ) has a coun table base, one can construct a coun table subset C ⊂ Mor( X, I ) suc h tha t t he restriction pr | δ ( Q ) of the pro jection pr : I Mor( X,I ) → I C is a homeomorphic em b edding. By the standard argumen t (see [10, Theorem 21.18]), w e can find a top olog ical cop y Q 1 ⊂ pr ◦ δ ( Q ) of Q whose closure Q 1 in the (metrizable) cub e I C is zero-dimensional, and hence is homeomorphic to the Can tor space { 0 , 1 } ω . Let e : Q 1 → [ ( s n ) n ∈ ω ∈{ 0 , 1 } ω \ n ∈ ω J ( s 0 ,...,s n ) b e the homeomorphism with the prop ert y e ( Q 1 ) = ∂ J (its existence follows from [5, P art 4, Th. 1] or the main result of [9]), and ¯ e : I C → I b e an extension of e to a con tinu o us map. Then the set Q ′ = Q ∩ (pr ◦ δ ) − 1 ( Q 1 ) is a top ological cop y of Q and the map f = ¯ e ◦ pr ◦ δ : X → I restricted to Q ′ is a homeomorphism b et wee n Q ′ and ∂ J . Observ e that for an arbitrary family { ( x n k ) k ∈ ω : n ∈ ω } of sequences of elemen ts of (0 , 1], the subspace { x n k × { n } : n, k ∈ ω } ∪ {{ 0 } × ω } of I S ω is homeomorphic to S ω pro vided lim k →∞ x n k = 0 for all n ∈ ω . Pro of of Prop osition 16. Throughout the pro of w e shall iden tify X with i X ( X ). W e presen t here o nly the pro of o f the first part. The pro of of the second one is analo g ous. Let f ∈ Mor( X, I ) b e suc h that f | Q is an em b edding for some homeomorphic cop y Q ⊂ X of Q . Lemma 17 imples that there exists a usual Can tor sc heme J , a con tinuous map g : I → I , and a cop y Q 1 ⊂ f ( Q ) of Q suc h that g ( Q 1 ) = ∂ J and g | Q 1 is an em b edding. Then h = g ◦ f b elongs to Mor ( X, I ) and h ( f − 1 ( Q 1 )) = ∂ J . Set r n,k = z n, J (3 − k ). Applying Prop osition 5 , w e can find a sequence ( k n ) n ∈ ω of na t ur a l n um b ers suc h that R = { e } ∪ { r n,k : n ∈ ω , k ≥ k n } is a homeomorphic copy of S ω . By our construction of the maps z n, J , for ev ery n, k ∈ ω w e can find elemen t s 0 = u n,k , 0 < u n,k , 1 < · · · < u n,k , 2 n +2 − 1 = 1 of ∂ J suc h that r n,k = u − 1 n,k , 0 u n,k , 1 u − 1 n,k , 2 · · · u n,k , 2 n +2 − 1 . In addition, u n,k ,p do es not dep end on k pro vided 4 divides p or p + 1, and lim k →∞ u n,k , 4 q + 1 = u n,k , 4 q , lim k →∞ u n,k , 4 q + 2 = u n,k , 4 q + 3 (2) for all q < 2 n − 1. Set C = ( f | Q ) − 1 ( Q 1 ), v n,k ,p = ( h | C ) − 1 ( u n,k ,p ), y n,k = v − 1 n,k , 0 v n,k , 1 v − 1 n,k , 2 · · · v n,k , 2 n +2 − 1 , and Y = { e F X } ∪ { y n,k : n ∈ ω , k ≥ k n } . Since h | C : C → ∂ J is a homeomorphism, the sequence ( y n,k ) k ∈ ω con v erges to e F X for all n ∈ ω by (2). In addition, the con tinuous homomorphism F h : F X → F I maps y n,k to r n,k b y our c hoice of v n,k ,p , and hence F h ( Y ) = R . By the definition, S ω is a union of a countable family of disjoin t conv ergent sequences whose limit p oin ts coincide endow ed with the strongest top olog y in whic h these sequ ences ar e still con v ergent. Th us the con tin uity of F h | Y implies that Y is homeomorphic to R . ✷ Pro of of Theorem 2. F ollows immediately from Propo sition 16, Lemma 17, and the fact that [0 , 1] is an absolute retract in the category of T yc hono v spaces. ✷ 10 Lemma 18. Under the assumptions of Cor ol lary 1 the image i X ( X ) is close d in F X . Pr o of. Throughout the pro of we shall iden tify Z ∈ Ob( T ) with i Z ( Z ). Let j : X → ¯ X b e the inclusion. It sufficies to show that X = ( F j ) − 1 ( ¯ X ). Assuming the con ve rse, by the minimality of F w e could find a finite subs et { x i : i ≤ n } ⊂ X and in tegers m i , i ≤ n , suc h that x ∗ := ( F j ) ( x m 0 0 · · · · · x m n n ) ∈ ¯ X \ X . Therefore the elemen ts x ∗ and x m 0 0 · · · · · x m n n of F ¯ X coincide. Let f : ¯ X → [0 , 1] b e a con tin uous map such that f ( x ∗ ) 6 = f ( x i ) for all i ≤ n . F rom the ab o v e it follo ws that f ( x ∗ ) = F f ( x ∗ ) = F f ( x m 0 0 · · · x m n n ) = f ( x 0 ) m 0 · · · · · f ( x n ) m n , whic h con tradicts Lemma 14. Pro of of Cor ollary 1. Let us denote by L the subs pace { (0 , 0) } ∪ { ( 1 n , 1 mn ) : n, m > 0 } of R 2 . According to [6, Lemma 8.3], a first coun table space contains a closed to polo gical cop y of L if and only if it fails to b e lo cally compact. Assume, contrary to our claim, that X is not scattered a nd fails to b e lo cally compact. Then X contains a to p olog ical copy of the space Q as well as a closed top ological copy of L . Since F X is g enerated by it s second-coun table subspace X , it has coun table pseudo c haracter. In addition, it is no r ma l b eing Lindel¨ of and Ty chono v. Applying Theorem 2 , w e conclude that F X con tains a to p olog ical cop y of S ω . It is w ell-kno wn [13] that a top olog ical group G con tains a (closed) top olog ical cop y of S ω if and only if it con tains a (closed) top ological copy of the Arens’ space S 2 , see [13] for its definition. It w as shown by Lin [12, Corollary 2.6] that a regular space Z with coun table pseudo c haracter contains a top ological cop y of S ω if and only if it contains a closed top ological cop y of S ω . It follow s fro m the a bov e that F X con tains a closed top ological cop y of S ω . In addition, F X con ta ins a closed copy of L b y L emma 18, whic h con tradicts [3, Theorem 1] asserting that a normal k -space con taining closed top ological copies of L and S ω is not homeomorphic to an y top ological group. ✷ References [1] A. V. Arkhangel’ski ˘ ı, F r e quency sp e ctrum of a top olo gic al sp ac e and classific ation of sp ac es , Dokl. Ak ad. Nauk SSSR 206 (1972), 265–26 8 . (In R uss ian) [2] A. V. Arkhangel’ski ˘ ı, O. G. Okunev, V. G. P esto v, F r e e top olo gic al gr oups over metrizable sp ac es, T op ology Appl. 33 (1989), 63–76. [3] T. O. Banakh, On top olo gic al gr oups c ontaining a F r´ echet-Urysohn fan , Matemat y- c hni Studii 9 (1 998), 145 –154. [4] T. O. 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Kechris , Classic al Descriptive Set The ory , GTM 156 , Springer, Berlin 1995. [11] J. Kulesza, Sp ac es wh ich do not emb e d in top olo gic al gr oups of the sam e dim ension , T op ology Appl. 5 0 (1993) , 139–14 5 . [12] S. Lin, A no te on Ar ens sp ac e and se quential fan, T op ology Appl. 81 (1997), 185– 197. [13] J. Nagata. Gener alize d Metric Sp ac es, I . In: K.Morita, J.Nag a ta (eds.), T opics in Gener al T op olo gy (Elsevie r Science Publishers B.V., 1989), 3 1 5–366. [14] P . Nic k olas, M. Tk ac henk o, T he char acter of fr e e top olo gic al gr oups, I , Appl. Gen. T op. 6 :1 (2005), 15–41. [15] P . Nick o las, M. Tk ac henk o, The c h ar acter of fr e e top olo gi c al gr oups, II , Appl. Gen. T op. 6 :1 (2005), 43–56. [16] D. B. Shakhmato v, A c ate gorial version of the L efschetz– N¨ ob eling-Pontryagin the- or em on emb e dding c omp acta in R n , T op ology Appl. 85 (1998 ), 345–349. [17] M. G . Tk aˇ cenk o, On top olo gies of fr e e gr oups, Czec hoslo v ak Mathematical Journal 34 :109 (1984), 541 –551. [18] J. E. V aughan, S m al l unc ountable c ar di n als a nd top o l o gy , In: Op en pr oblems in top olo gy (J. v an Mill, G .M. Reed, eds), Elsevier Sci. Publ., 1990, 197–216 . [19] M. M. Z a ric hn y ˘ ı, F r e e t op olo gic al gr oups of absolute neighb orho o d r etr acts and infinite-dimensio n al manifold s , D okl. Ak ad. Nauk SSSR 266 :3 (198 2 ), 541–54 4. ( In Russian) T aras Banakh, Inst ytut Matemat yki, Ak ademia ´ Swi¸ etokrzysk a, Kielce (P oland) and Departmen t o f Mec hanics and Mathematics, Iv an F rank o Lviv National Univ ersit y , Univ er- sytetsk a 1, Lviv, 79000, Ukraine. E-mail addr ess: tbanakh@franko.lviv. ua URL: http://www. franko.lviv.ua/faculty/mechmat/Departments/Topology/index.ht ml Du ˇ san Rep ov ˇ s, Institute of Mathematics, Ph ysics and Mec hanics, Jadransk a 19, P .O.B. 2964, Ljubljana, Slov enija 1001. E-mail addr ess: dusan.repovs@guest.a rnes.si URL: http://pef. pef.uni-lj.si/ ˜ dusanr/inde x.htm Lyub om yr Zdomskyy , Department of Mec hanics and Mathematics, Iv an F ranko Lviv Na- tional Univ ersit y , Univ ersytetsk a 1, Lviv, 79000, Ukraine. E-mail addr ess: lzdomsky@gmail.com 12