Connected, not separably connected complete metric spaces

CONNECTED, NOT SEP ARABL Y CONNECTED COMPLETE METRIC SP A CES T. BANAKH, M. VO VK, M. R. W ´ OJCIK Abstract. In a separably connected s pace an y tw o p oints are con tained in a sepa rable conn ected subset. W e show a mec hanism that tak es a connected bounded metric space and pr o duces a complete connect ed metric s pace whose separablewise componen ts form a quotien t space i sometric to the or iginal space. W e r ep eatedly apply this mec hanism to construct, as an i nv erse limit, a complete co nnected met ric space whose eac h separable subset is zero-dimensi onal. A topo logical space is separably connected iff any tw o of its po ints are co n- tained in a s eparable connected subset. A separablewise comp onen t is the unio n of all separable connected subsets containing a given p oint. Since the closure of a separable connected set is a gain separ able a nd connected, it follows that separa- blewise comp onents are closed. A connected space who se all connected s eparable subsets a re s ingletons will b e called nonseparably connected . The first example of a nonsepa rably connected metric s pace was g iven by Pol in 1975, [4]. Another example w as given by Simon in 20 01, [5]. In 2003, Aron and Maestre c onstructed a connected, not sepa rably connected metr ic spa ce that contains many a rcs, [6]. In 2 008, Morayne and W´ ojcik obta ined a nonseparably connected metric space as a gr aph of a function from the real line sa tisfying Cauch y’s equation and thus forming a top o logical g roup, [7] or [8]. None of these spaces ar e completely metr izable. Our first result is that for any connected metric space X we pro duce a co mplete connected metric s pace whose s eparablewise comp onents form a q uotient space homeomorphic to X , Corolla ry 15. (This yields a lo t of top ologica lly different completely metr izable co nnected, not sepa rably connec ted s paces.) Our second result is the fir st known example of a nonsepar ably connected metric space that is co mplete, Corolla ry 19. In fact, o ur space satisfies a stronger prop erty . Namely , the metric is ec onomical in the following sense. Economically m etrizable spaces. Given a metric space ( X, d ), we say that the metric d is e c o nomic al iff | d ( A × A ) | ≤ dens( A ) for a ny infinite s ubset A ⊂ X , where d ( A × A ) = { d ( a, b ) : a, b ∈ A } , dens( A ) = min {| D | : D ⊂ A ⊂ D } . Prop ositi on 1. Eve ry sep ar a ble subset of an e c o nomic al ly metrizable sp ac e is zer o- dimensional. 1991 Mathematics Subje ct Classific ation. 54D05; 54C30. 1 2 T. BANAKH, M. VO VK, M . R. W ´ OJCIK Prop ositi on 2. Every c onne cte d e c onomic al ly metrizable sp ac e is nonsep ar ably c onne cte d. Prop ositi on 3 . The Cantor set is e c onomic al ly metrizable. Pr o of. Let N = { 1 , 2 , 3 , . . . } . Consider the following metric d on the space P ( N ) of all subse ts of N , d ( A, B ) =    0 if A = B , 1 min(( A \ B ) ∪ ( B \ A )) if A 6 = B . Now, the eco nomically metriza ble spac e ( P ( N ) , d ) is ho meomorphic to the terna ry Cantor s et with the euclidean metric, which is not eco nomical.  Prop ositi on 4 . Every met r izable sp ac e c ontaining a c opy of t he Cantor set admits a metric that is not e c onomic al. Pr o of. Recall that given a metriza ble space X and a closed subset M ⊂ X , every admissable metric on M can b e extended to an admissable metr ic o n X , E ngelking 4.5.21(c). So if X co nt ains a co py of the Can tor set, M ⊂ X , w e may choose a metric for M that is not eco nomical and extend it to the who le X .  1. P reliminaries: connected inverse limits In this section we prepare the too ls that we use to show that the spaces we construct are co nnected. Lemma 5 a nd Lemma 6 a re us ed in Theo rem 14 to obtain connected, no t separably connected s paces. The r emaining lemmas and Theo rem 10 ar e used in Theorem 18 to show that our inv erse limit is connected. W e nee d to make a n umber of s imple observ a tions — Lemma s 7, 8, 9 — b efore we a re ready to apply E. Puzio’s Theorem 11 from [2]. Let X and Y be to po logical s paces. Then a function f : X → Y is • m onotone iff f − 1 ( y ) is connected for ea ch y ∈ Y , • op en at x ∈ X iff f ( x ) ∈ I nt ( f ( U )) for every neighbo rho o d U of x , • her e ditarily quotient iff f − 1 ( y ) ⊂ U = ⇒ y ∈ I nt ( f ( U )) for every op en set U ⊂ X and every po int y ∈ Y . Lemma 5. If every fib er of f : X → Y c ontains a p oint at which f is op en, t hen f is her e ditarily quotient. Pr o of. This is immediate fro m the definitions.  Lemma 6 (E ngelking 6.1.I) . Supp ose that f : X → Y is monotone and her e ditarily quotient. Then X is c onn e cte d if f(X) is c onn e cte d. Pr o of. Let U b e a clop en subset of X . T o sho w that f ( U ) is op en tak e any y ∈ f ( U ). Since the fib er f − 1 ( y ) is connected and intersects the clop en set U , it follows that f − 1 ( y ) ⊂ U . Since f is hereditarily quotient, y ∈ I n t ( f ( U )), showing that f ( U ) is op en. Since X \ U is a lso clop en, it follows by ana logy that f ( X \ U ) is op en, to o. Now, the sets f ( U ), f ( X \ U ) are disjo int , bec ause the clop en se ts U , X \ U contain who le fib ers. The connected s pace f ( X ) = f ( U ) ∪ f ( X \ U ) is a union of t wo dis joint op en subsets. Th us U = X o r U = ∅ , showing that X is connected.  CONNECTED, NOT SEP ARABL Y CONNECTED COMP LETE METRIC SP ACES 3 Lemma 7. If f : X → Y is monotone and her e ditarily qu otient then f − 1 ( E ) is c onne cte d whenever E ⊂ Y is c onne cte d. Pr o of. A s traightforward subspace topo logy a rgument shows that the res triction f | f − 1 ( E ) : f − 1 ( E ) → E is heredita rily quo tient . It is also e vidently monotone. T herefore, by Lemma 6, f − 1 ( E ) is co nnected if E is connected.  Lemma 8. If f : X → Y and g : Y → Z ar e monotone and f is her e ditarily quotient, then g ◦ f is monotone. Pr o of. Since g − 1 ( z ) is connected a nd f is monotone a nd her editarily quotient, b y Lemma 7, ( g ◦ f ) − 1 ( z ) = f − 1 ( g − 1 ( z )) is connected.  Lemma 9. If f : X → Y and g : Y → Z ar e her e ditarily quotient then their c om- p osition g ◦ f : X → Z is her e ditarily quotient. Pr o of. T ake any o pe n set U ⊂ X and any p o in t z ∈ Z such that ( g ◦ f ) − 1 ( z ) ⊂ U . Then ( g ◦ f ) − 1 ( z ) = f − 1 ( g − 1 ( z ))) = [ y ∈ g − 1 ( z ) f − 1 ( y ) ⊂ U. Since f is hereditarily quotient, we hav e y ∈ I nt ( f ( U )) for each y ∈ g − 1 ( z ). Thu s g − 1 ( z ) ⊂ I nt ( f ( U )). No w, since g is hereditarily quotient, z ∈ I nt ( g ( I n t ( f ( U )))) ⊂ I nt (( g ◦ f )( U )) .  Theorem 10 (E. Puzio , 1972) . L et X 1 , X 2 , . . . b e a se qu en c e of c onne cte d sp ac es. Supp ose that e ach function f n : X n +1 → X n is a c ontinu ous monotone her e ditarily quotient surje ction. Then the inverse limit of this system X =  ( x n ) ∞ n =1 ∈ ∞ Y n =1 X n : ( ∀ n ∈ N )  x n = f ( x n +1 )  is c onne cte d. Pr o of. Theorem 11 in [2] requires the additional assumption that the functions f n ◦ f n +1 ◦ . . . ◦ f m : X m +1 → X n are monotone hereditarily quotient sur jections for all n < m ∈ N . But this follows from o ur a ssumptions tha nks to Lemma 8 and Lemma 9. So X is connec ted.  The little example [0 , 1) ∪ { 2 } → [0 , 1] shows that the assumption that f is hereditarily q uotient is needed in all these lemmas. 4 T. BANAKH, M. VO VK, M. R. W ´ OJCIK 2. P reliminaries: first count able sp aces Our m echanism (Theorem 14) that returns an appropriate complete metric space for a g iven metric space actually works fo r any first countable space, so we decided to write it more g enerally at the cost of in tro ducing some technical details fo r dealing with fir st countable space s. Prop ositi on 1 1 . A top olo gic al sp ac e X is first c ountable if and only if to e ach p air of p oints x, u ∈ X we may assign a numb er d x ( u ) ∈ [0 , 1] so that inf { d x ( u ) : u ∈ E } = 0 ⇐ ⇒ x ∈ E for al l x ∈ X and E ⊂ X . Pr o of. If X is first countable, there are op en sets { U n ( x ) : x ∈ X , n ∈ N } such that x ∈ U n +1 ⊂ U n ( x ) and the sequence U n ( x ) is a lo cal basis at x . Then d x ( u ) = inf { 1 /n : u ∈ U n ( x ) } is eas ily see n to b e as desired. On the other ha nd, if X a dmits s uch a function d x ( u ), let B ( x, r ) = { u ∈ X : d x ( u ) < r } for a ll x ∈ X , r > 0. Then inf  d x ( u ) : u ∈ X \ B ( x, r )  ≥ r > 0 and co nsequently , x 6∈ X \ B ( x, r ) . In other words, x ∈ I nt ( B ( x, r )). If G is an op en set c ontaining x , then inf { d x ( u ) : u ∈ X \ G } ≥ r > 0 , for s ome r > 0 and consequently B ( x, r ) ⊂ G . So, U n ( x ) = I nt ( B ( x, 1 /n )) are the op en sets which show X to b e first countable.  3. Com plete connected, not sep arabl y connected sp a ces Lo cally constan t functions. Let X be a top ologica l spa ce. W e say that a func- tion f : X → Y is lo c al ly c onstant iff ea ch point x ∈ X ha s a neighbor ho o d U x such that f | U x is constant. Naturally , the car dinality of the ima ge f ( X ) canno t e xceed the dens it y of X , | f ( X ) | ≤ dens( X ) = min {| D | : D = X } . Metrically discrete subsets. Let ( X , d ) b e a metric space. W e s ay that A ⊂ X is a met ric al ly discr ete subset o f X iff ther e is an ε > 0 such that d ( a, b ) ≥ ε for a ny distinct a, b ∈ A . Naturally , the cardinality of a metr ically discrete subset cannot exceed the density of the metric space X , | A | ≤ dens ( X ). Lemma 12. If X is a metric sp ac e and f : X → Y is lo c al ly c onstant exc ept on a metric al ly discr ete subset, then | f ( X ) | ≤ dens( X ) . Pr o of. Let K , D b e dis joint subsets of X suc h that f | K is loc ally constant and D is metrica lly discr ete. Then | f ( X ) | ≤ | f ( K ) | + | f ( D ) | ≤ dens( X ) + | D | ≤ dens( X ) + dens( X ) .  CONNECTED, NOT SEP ARABL Y CONNECTED COMP LETE METRIC SP ACES 5 F unctionally Hausd o rff s paces. W e say that a top o logical space X is fun ction- al ly Hausdorff iff for a ny t wo distinct p oints a, b ∈ X there is a co ntin uous function f : X → R with f ( a ) 6 = f ( b ). Each connected subset E of such a spa ce is either a singleton o r contains a set of cardinality c , as the following easy argument shows. Let a, b ∈ E with f ( a ) < f ( b ). Then the co nnected set f ( E ) m ust contain the int erv al [ f ( a ) , f ( b )]. Lemma 13. L et X b e a m et ric sp ac e. L et Y b e a functional ly Hausdorff sp ac e. L et f : X → Y b e a Darb oux function that is lo c al ly c onstant exc ept on a met ric al ly discr ete subset. T hen f is c onstant on every c onne cte d sep ar able subset of X . Mor e over, if f is not c onst ant, then X is not sep ar ably c onne cte d. Pr o of. Let E ⊂ X be a connected separable subset. Since f is Darb oux and E is connected, the set f ( E ) is a connected subset of a functionally Hausdorff s pace. Thu s f ( E ) is either a sing leton or | f ( E ) | ≥ c . By Lemma 12, | f ( E ) | ≤ dens( E ) = ℵ 0 , so the set f ( E ) is countable and th us a single ton.  T o arg ue that the s paces constructed in Theo rem 14 are completely metrizable, we o btain them as closed subsets of a certain ca nonical space (which we call the cobw eb) tha t is e asily seen to b e complete. The hed g ehog. Le t κ b e a cardinal num b er. A hedgehog with κ s pikes, each of length ε > 0, is the space H = { (0 , 0) } ∪ ( κ × (0 , ε ]) equipp ed with the metric ρ given by ρ (( x, t ) , ( u, s )) = ( | t − s | if x = u, t + s if x 6 = u. It is ea sy to see that it is a complete one-dimensional arcwise connected metric space. The cob web. Let ε > 0 be a fixed num b er. Let V b e a subset of a no rmed spa ce with k u − v k = ε f or all distinct u, v ∈ V . Let W = S { [ u, v ] : u, v ∈ V } , where [ u, v ] denotes the line segment betw een vectors u, v . Let ρ ( a, b ) b e the infim um ov er finite s ums n X i =1 k x i − x i − 1 k where a = x 0 and x n = b a nd ( ∀ i )( ∃ u, v ∈ V )  { x i , x i − 1 } ⊂ [ u , v ]  . Then ρ is clearly a metric on W , the distance b eing measur ed alo ng the thr e ads , so tha t a t each vortex the space ( W, ρ ) is lo cally iso metric to a hedg ehog with a p- propriately shortened spikes, and at the re maining p oints it is lo cally is ometric to appropria tely sho rt euclidean interv als. The metr ic space ( W, ρ ) will b e called the c obweb sp un ove r V . It is clear ly zero- dimensional. The cobw eb is complete . Indeed, if x n is a Cauch y sequence , w e hav e a k ∈ N such that ρ ( x n , x k ) ≤ ε / 4 fo r all n ≥ k . It is evident that there is a vortex u ∈ V with ρ ( x k , u ) ≤ ε/ 2. Then ρ ( x n , u ) ≤ ρ ( x n , x k ) + ρ ( x k , u ) ≤ 3 ε/ 4 for all n ≥ k . So the Cauch y subse quence x k , x k +1 , . . . is cont ained in the hedgeho g with vortex u and spikes of leng th 3 ε/ 4, so it co n verges bec ause the hedge hog is complete. The fo llowing theorem is written so as to rev eal a ll the interesting prop erties of 6 T. BANAKH, M. VO VK, M. R. W ´ OJCIK the s paces constructed and to allow fir st countable functionally Hausdo rff spac es in the plac e of m etric spaces. How ever, so me of the conclus ions are str onger for metric spa ces a nd these ar e summarized in Corollar y 15. Theorem 14. L et X b e a first c ountable functional ly Hausdorff sp ac e. Then ther e ex ists a c omplete one-dimensional metric sp ac e Z and a c ont inuous monotone her e ditarily quotient surje ction f : Z → X such t hat (1) e ach fi b er of f is home omorphic to a he dgeho g with | X | -many spikes, thus | Z | = c | X | (2) e ach fi b er Z a = f − 1 ( a ) has exactly one p oint a ∗ ∈ Z a at which f is op en; these p oints form a metric al ly discr ete subset (3) Z is c onne cte d ⇐ ⇒ X is c onne ct e d (4) Z a \ { a ∗ } is op en in Z fo r e ach a ∈ X , thus dens( Z ) ≥ | X | and f is lo c al ly c onstant ex c ept at those discr etely sp ac e d p oints a ∗ (5) | f ( A ) | ≤ dens( A ) for any subset A ⊂ Z (6) e ach c onne cte d sep ar able s u bset of Z lies in one of the fib ers (7) Z is n ot sep ar ably c onne cte d (8) its ar cwise c omp onents c oincide with the fib ers (9) the sp ac e Z | X = { Z a : a ∈ X } with E ⊂ Z | X de clar e d op en iff S E is op en in Z is home omorphic to X via Z | X ∋ Z a 7→ a ∈ X . (10) Z is lo c al ly c onne cte d at z ∈ Z ⇐ ⇒ z / ∈ { a ∗ : a ∈ X } Pr o of. Since X is first co un table, let d b e as in Pro p osition 11. Let X = X × { 0 } and let X ∗ = X × { 1 } . Let Ω = X ∪ X ∗ be consider ed with the counting meas ure. Let Y b e the Hilb ert space L 2 (Ω), that is Y =  h : Ω → R : X ω ∈ Ω | h ( ω ) | 2 < ∞  . F or each x ∈ X , let x ∈ Y b e the characteristic function o f the singleton { ( x, 0) } , and let x ∗ ∈ Y b e the characteristic function of the singleton { ( x, 1) } . Let V = { x : x ∈ X } ∪ { x ∗ : x ∈ X } . Then k u − v k = √ 2 for distinct u , v ∈ V . Let [ u, v ] denote the line se gment joining tw o vectors u, v ∈ Y . F or any distinct a, b ∈ X , le t a b denote the po int lying in [ a, b ∗ ] with k a b − b ∗ k = d b ( a ) ≤ 1. Since X is Hausdorff, d b ( a ) > 0, and so a b 6 = b ∗ . Let Z a = [ a, a ∗ ] ∪ [ { [ a, a b ] : b ∈ X, b 6 = a } for each a ∈ X . Le t Z = S { Z a : a ∈ X } . Let Z b e equipped with the induced metric fro m the cobw eb spun over V . Notice that Z is obtained by taking aw ay selected op en interv als from s ome of the threads of the cob web spun o ver V . So Z is a clo sed subse t of this cob web. Thu s Z is a c omplete zer o-dimensional metric space. Let f : Z → X be given by f ( Z a ) = { a } for a ll a ∈ X . Clearly , f − 1 ( a ) is connected for each a ∈ X , s o f is monoto ne. W e claim that f is o pen a t each a ∗ ∈ Z b ecause f ( a ∗ ) = a ∈ I nt ( B ( a, r )) ⊂ f ( B ( a ∗ , r )) for all a ∈ X , r > 0 . Indeed, for any b ∈ B ( a, r ), w e hav e k b a − a ∗ k = d a ( b ) < r , and so b = f ( b a ) ∈ f ( B ( a ∗ , r )). By Lemma 5, f is hereditarily quotient. CONNECTED, NOT SEP ARABL Y CONNECTED COMP LETE METRIC SP ACES 7 By Lemma 6, Z is connected if X is co nnected. Later we show that f is contin- uous, so X is connected if Z is connected. F or each a ∈ X , the set Z a \ { a ∗ } is op en in Z . Thus f is lo cally constant except on the metrically dis crete s et { a ∗ : a ∈ X } ⊂ V . Therefore, for any set A ⊂ Z , the restriction f | A is lo cally constant except po ssibly on a metrica lly discrete set, and by Lemma 12, | f ( A ) | ≤ dens( A ). T o show tha t f is co nt inuous, it is sufficient to analyz e those p oints where f is not lo cally constant. T ake any a ∈ X and an y s ∈ (0 , √ 2). Then, by Pr op osition 11, f ( a ∗ ) ∈ I nt ( B ( a, s )), and there is an r ∈ (0 , s ) with B ( a, r ) ⊂ I nt ( B ( a, s )). W e claim that f is contin uous at a ∗ bec ause f ( B ( a ∗ , r )) ⊂ B ( a, r ) ⊂ I nt ( B ( f ( a ∗ ) , s ) . Indeed, ta ke any z ∈ B ( a ∗ , r ). If z ∈ Z a , then f ( z ) = a ∈ B ( a, r ). If z ∈ Z b with b 6 = a , then z ∈ [ b, b a ] because r is sufficiently small. Consequently , k z − b a k + k b a − a ∗ k = k z − a ∗ k < r . Thus d a ( b ) = k b a − a ∗ k < r , and s o f ( z ) = b ∈ B ( a, r ). Hence f is contin uous. Now, f is a Dar b oux function that is lo cally constan t except on a discrete s ubset. Since X is functionally Hausdorff, by Lemma 13, w e c onclude that each co nnected separable s ubset of Z lies in o ne o f the fib ers. This means t hat the fib ers coin- cide with the separa blewise comp onents of Z , each of which is homeomor phic to a hedgehog a nd thus ar cwise connected.  Corollary 15 . Le t X b e a metric sp ac e b ounde d by one. Then ther e ex ists a c omplete metric sp ac e Z (c onne cte d if and only if X is c onne cte d) and a Lipschitz monotone her e ditarily quotient su rje ction f : Z → X whose fib ers c oincide with the sep ar ablewise c omp onent s of Z and form a quotient s p ac e isometric to X . Pr o of. Notice that in the course of the pr o of o f The orem 14 we hav e min { ρ ( x, u ) : x ∈ Z a , u ∈ Z b } = min { d a ( b ) , d b ( a ) } . Then the metric z ( Z a , Z b ) = d ( a, b ) is a n isometr y b ecaus e f ( B ( a ∗ , r )) = B ( a, r ) for a ll a ∈ X , r > 0. W e a lso hav e d ( f ( x ) , f ( u )) ≤ ρ ( x, u ) for a ll x, u ∈ Z .  In the next section we obtain a complete nonsepa rably connected spa ce as an inv er se limit o f a s equence o f spa ces gener ated by rep eatedly applying Theo rem 14. The follo wing Co rollar y 16 lists only tho se prop erties that a re essen tial for that purp ose. Corollary 16. T o every metric sp ac e X t her e c orr esp onds a un iquely determine d c omplete c onne ct e d metric sp ac e Z and a c ontinuous monotone her e ditarily qu otient surje ction f : Z → X such that | f ( A ) | ≤ dens( A ) for any subset A ⊂ Z . Recall that a connected, lo ca lly c onnected complete metric space m ust b e a rcwise connected, Engelking 6.3 .11. The spac e Z obtained in Theorem 1 4 is no t lo ca lly connected, although it is lo cally co nnected e xcept o n a metrically discrete subs et. This illustrates how imp ortant it is to assume that the s pace is lo cally co nnected at each p oint if we wan t to conclude that it is arcw ise connected. 8 T. BANAKH, M. VO VK, M. R. W ´ OJCIK 4. A complete nonsep arabl y conn ected sp ace W e mak e a n appro priate choice o f the pro duct metric for our inv erse limit to ensure that it is economica lly and completely metrizable a t the s ame time. Lemma 17. L et ( X n , d n ) b e a se quenc e of u niformly b ounde d m et ric sp ac es. L et X = Q n ∈ N X n and let π n : X → X n b e given by π n ( x 1 , . . . , x n , . . . ) = x n . Then d ( x, u ) = max n d n ( π n ( x ) , π n ( u )) 2 n : n ∈ N o for e ach x, u ∈ X defines a pr o duct metric that is c omplete if al l factor metrics ar e c omplete. Mor e over, for any infi nite s u bset A ⊂ X we have | d ( A × A ) | ≤ sup n ∈ N | π n ( A ) | . Pr o of. F or any a, b ∈ X w e have d ( a, b ) ∈ [ { 2 − n d n ( π n ( a ) , π n ( b )) : n ∈ N } . Therefore, if A ⊂ X is infinite, we hav e | d ( A × A ) | ≤ X n ∈ N | d n ( π n ( A ) × π n ( A )) | ≤ X n ∈ N | π n ( A ) | 2 ≤ sup n ∈ N | π n ( A ) | .  Theorem 18. Ther e exists a c omplete c onne cte d e c onomic al metric sp ac e. Pr o of. Let X 1 be a co nnected complete metric spac e b o unded by o ne. By Cor ol- lary 1 6, w e have a s equence ( X n , d n ) of co mplete c onnected metr ic spa ces b o unded by one and contin uous monotone hereditar ily quo tient surjections f n : X n +1 → X n such that | f n ( A ) | ≤ dens( A ) for any subset A ⊂ X n +1 . Let X b e the inv erse limit of this system. By Theo rem 1 0, X is connected. Equipping the product Q ∞ n =1 X n with the product metr ic from Lemma 17 in- duces a complete metric d on X that satisfies | d ( A × A ) | ≤ sup n ∈ N | π n ( A ) | for any infinite subset A ⊂ X . It remains to show that ( X , d ) is ec onomical. Let A ⊂ X be a ny infinite subs et. W e hav e | π n ( A ) | = | f n ( π n +1 ( A )) | ≤ dens( π n +1 ( A )) ≤ de ns( A ) for a ll n ∈ N . Therefore, | d ( A × A ) | ≤ sup n ∈ N | π n ( A ) | ≤ dens( A ) .  Corollary 19. Ther e exists a c omplete c onne cte d metric sp ac e whose e ach sep ar a- ble subset iz zer o-dimensional. In p articular, ther e exists a c omplete nonsep ar ably c onne cte d metric sp ac e. Pr o of. By Theo rem 1 8 a nd Pr op osition 1.  Connected punctiform spaces. Recall tha t a top ologica l spa ce is punctiform if all of its connected co mpact subsets are singletons. F or example, an y Bernstein subset of the euclidean plane is a connected punctiform metric space. A sepa - rable complete connected punctiform space was constructed by Kuratowski and Sierpi ´ nski in 1922 , [1]. A v ariation of their idea was presented in [7]. Our complete nonsepara bly co nnected space is a new example of a connec ted punctiform space. CONNECTED, NOT SEP ARABL Y CONNECTED COMP LETE METRIC SP ACES 9 5. A ppendix: a non-const ant continuo us locall y extremal function Alessandro F edeli and A ttilio Le Donne c onstructed a connected metric space X and a no n-constant con tin uous function f : X → [0 , 1] that has a lo cal max im um or a lo cal minimum at every p oint, without claiming that the domain X is c omplete, [9 ] and [1 0]. By modifying the proof o f our Theo rem 14, we make a similar construction that is less technically bur densome, and the doma in of the function is ea sily seen to b e complete. Theorem 20. Ther e is a c omplete c onne cte d metric sp ac e ( X , ρ ) and a Lipschitz monotone her e ditarily quotient surje ction f : X → (0 , 1) that has a lo c al extr emum at every p oint. Pr o of. Let a, a ↑ , a ↓ for a ∈ (0 , 1) b e identified a s distinct po int s forming a discrete set V in some nonsepar able normed vector space, e .g. k u − v k = 1 for u 6 = v . F or a ∈ (0 , 1) and b ∈ ( a, 1), let b ↑ a be the point lying in [ a, b ↓ ] wit h k b ↑ a − b ↓ b k = b − a . F or a ∈ (0 , 1) and b ∈ (0 , a ), let b ↓ a be the point lying in [ a, b ↑ ] wit h k b ↓ a − b ↑ b k = a − b . Let H a = [ a, a ↑ ] ∪ [ a, a ↓ ] ∪  [ a, b ↑ a ] : b ∈ ( a, 1)  ∪  [ a, b ↓ a ] : b ∈ (0 , a )  for ea ch a ∈ (0 , 1). Le t X = S { H a : a ∈ (0 , 1) } . Notice that X is a clos ed subset of the cobweb spun ov er V . Equipp ed with the metric ρ induced from the cobw eb, X is a co mplete metric spa ce. Let f : X → (0 , 1) be given by f ( H a ) = { a } f or each a ∈ (0 , 1 ). Notice that | f ( x ) − f ( u ) | ≤ ρ ( x, u ) for all x, u ∈ X , so f is contin uous. F or each a ∈ (0 , 1 ), f has a lo ca l minimum at a ↑ and a lo cal maximum at a ↓ . It is lo cally consta nt at every o ther p oint. Now, f is hereditar ily quo tient b eca use f ( B ( a ↑ , r )) = [ a, a + r ) f ( B ( a ↓ , r )) = ( a − r , a ] for all a ∈ (0 , 1) and a ll r ∈ (0 , a ) ∩ (0 , 1 − a ). Since f is a monotone hereditarily quotient s urjection o nt o (0 , 1 ), by Lemma 6, X is connected.  F urther dev elopments. W e a re preparing a sequel pap er in which we construct a co mplete nonsepa rably co nnected s pace as a metric group, [11]. Moreov er, w e plan to devote a separate pap er to the cons truction of the cobw eb ⊛ ( X ) ov er a first countable space X , having the prop erties listed in Theorem 1 4. W e feel that this op era tion of obtaining the spac e Z out of X is interesting in itself a nd ne eds to b e inv estigated mor e clos ely , outside the context of co nstructing connected, not se parably connected s paces. W e ha v e a num be r of different wa ys of desc ribing this cobw eb op eration, e ach having its adv antages and disadv a ntages, and we feel that this topic deser ves separa te treatment. Ac kno wledgements. W e would like to thank Pa we l Krupski for his top olo gical seminar at which we had a chance to present our work and improv e it greatly a long the lines sugg ested by Krz ysztof Omiljanowski. W e r ely on [2] in o ur Theorem 18 to arg ue that the inv erse limit is connected. Besides that, the pro ofs of our main r esults ar e se lf-contained. 10 T. BANAKH, M. VO VK, M. R. W ´ OJCIK References [1] K. Kur atowski and W. Sierpi´ nski, Les f onctions de classe 1 et les e nsembles co nnexes punc- tiformes, F undamen ta Mathematicae 3 (1922), 303-313. [2] E. Puzio, Limit mappings and pr oje ctions of inve rse systems , F undameta M athematicae 80 (1973), 57-73. [3] R. Engelking, General T op ology , PWN, 1977. [4] R. Po l, Two e x amples of non-sep ar able metrizable sp ac es, Collo quium Mathematicum 33 (1975), 209-211. [5] P . Simon, A c onne c te d, not sep ar ably co nne cte d metric sp ac e, Rend. Istit. M at. Univ. T rieste Suppl. 2 V ol. XXXII (2001), 127-133. [6] R. M. Aron and M. M aestre, A co nne cte d m etric sp ac e that is not sep ar ably c onne cte d, Con tempor ar y Mathematics 328 (2003), 39-42. [7] M. R. W´ ojcik, Close d and c onne cted gr aphs of functi ons; ex amples of c onne c t e d punctiform sp ac es, Ph. D. T hesi s , Kato wice 2008 (h ttp://www.apron us.com/math/MR W o jcikPhD.htm ). [8] M. Morayne , M. R. W´ ojcik, A nonsep ar ably c onne cte d metric sp ac e as a dense c onne cte d gr aph , prepri nt (htt p://arxiv.org/abs/0811.2808 ). [9] A. Le Donne, A. F edeli, On metric sp ac es and lo c al ex t r e ma , VI I Ib eroamerican Conference on T opology and its Applications, 25-28 June 2008, V alencia, Spain (h ttp://cita.w ebs.up v.es/abstracts/files/p160.pdf ). [10] A. Le Donne, A. F edeli, On metric sp ac e s and lo c al ext rema , T op ology and its A ppl i cations (to appear). [11] T. Banakh, M. R. W´ ojcik, A c omplete nonsep ar ably c onne c t e d metri c gr oup , in preparation. T aras Banakh: Iv an Franko L viv Na tional University, L viv, Ukraine, and Unwersytet Hu manistyczno-Przyrodnicz y im. Jana K ochanowskiego, Kielce, Poland E-mail addr ess : tbanakh@yah oo.com M. Vovk: Na tional University “L vivska Politechnika”, L viv, Ukraine Micha l R yszard W ´ ojcik E-mail addr ess : michal.rysz ard.wojcik@gmail.com