ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM

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1 ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM A.P.M. KUPERS Abstract. These are otes for a Staford SUMO talk o space-fillig curves. We will look at a few examples ad the prove the Hah-Mazurkiewicz theorem. This theorem characterizes those subsets of Euclidea that are the image of the uit iterval. After a historical itroductio, we sped two sectios costructig space-fillig curves. I particular, we treat Peao s example because of its historical sigificace ad Lebesgue s example because of its relevace to the proof of the Hah-Mazurkiewicz theorem. This theorem is prove i the fourth sectio, with the exceptio of some poit-set topology lemma s that have bee delegated to the appedix. The first three sectios oly require basic kowledge of real aalysis, while the fourth sectio ad the appedix also require some kowledge of poit-set topology. The best referece for space-fillig curves is Saga s book [Sag94].. What are curves? The ed of the 9th cetury was a excitig time i mathematics. Aalysis had recetly bee put o rigorous footig by the combied effort of may great mathematicias, most importatly Cauchy ad Weierstrass. I particular, they defied cotiuity usig the well-kow ɛ,δ-defiitio: a fuctio f : R R is cotiuous if for every x R ad ɛ > 0, there exists a δ > 0, such that y x < δ implies f(y) f(x) < ɛ. We will be more iterested i curves: cotiuous fuctios f : [0, ] R. This defiitio was supposed to capture the ituitive idea of a ubroke curve, oe that ca be draw without liftig the pecil from the paper. Cotiuous curves should ot be the same as smooth curves, as they for example ca have corers. Though cotiuity might seem like the correct defiitio for this, it i fact admits mostrous curves: Weierstrass i 87 wrote dow a owhere differetiable cotiuous fuctio. By lookig at its graph i R, oe fids a curve havig corers everywhere! Shortly after this, i 878, Cator developed his theory of sets. Oe of the (the) shockig cosequeces was that R ad R had the same cardiality, i.e. the same umber of poits. This meas that there exists a bijectio betwee both sets, which Cator explicitely wrote dow. This result questioed ituitive ideas about the (rather elusive) cocept of dimesio ad a obvious questio after the previous discovery of badly-behaved curves was: Ca we fid a bijectio betwee R ad R which is cotiuous or maybe eve smooth? The aswer to this questio turs out to be o a 879 result by Netto somethig we will prove later this lecture. However, it remaied ope for more tha a decade whether it was also impossible to fid a surjective cotiuous map R R or, closely related, [0, ] [0, ]. Peao shocked the mathematical commuity i 890 by costructig a surjective cotiuous fuctio [0, ] [0, ], a space-fillig curve. It is pictured i figure. This result is historically importat for several reasos. Firstly, it made people realize how forgivig cotiuity is as a cocept ad that it does t always behave as oe ituitively would expect. For example, the space-fillig curves show it behaves badly with respect to dimesio or measure. Secodly, it made clear that result like the Jorda curve theorem ad ivariace of domai eed careful proofs, oe of the mai motivatios for the the ew subject of algebraic topology (back the called aalysis situs). I thik this makes space-fillig curves worthwhile to Date: February 8th 0. It is give by = a cos(b x) with a (0, ) ad b a odd iteger satisfyig ab > + π.

2 A.P.M. KUPERS Figure. The first, secod ad third iteratios f 0, f, f of the sequece defiig the Peao curve. kow about, eve though they are o loger a area of active research ad they hardly have ay applicatios i moder mathematics. If othig else, thikig about them will teach you some useful thigs about aalysis ad poit-set topology. I this lecture we ll costruct several examples, look at their properties ad ed with the Hah- Mazurkiewicz theorem which tells you exactly which subsets of R ca be image of a cotiuous map with domai [0, ].. Peao s curve We start by lookig at Peao s origial example. The mai tool used i its costructio is the theorem that a uiform limit of cotiuous fuctio is cotiuous: Theorem.. Let f be a sequece of cotiuous fuctios [0, ] [0, ] such that for η > 0 there is a N N havig the property that for all, m > N we have sup x [0,] f (x) f m (x) < η. The f(x) := lim f (x) exists for all x [0, ] ad is a cotiuous fuctio. Proof. Suppose we are give a x [0, ] ad ɛ > 0, the we wat to fid a δ > 0 such that y x < δ implies f(y) f(x) < ɛ. Note that for all N, f(y) f(x) f(y) f (y) + f (y) f (x) + f (x) f(x) By takig to be a sufficietly large umber N, we ca make two of the terms i the right had side be bouded by ɛ. By cotiuity of f N, we ca fid a δ > 0 such that y x < δ implies f N (y) f N (x) < ɛ. The if y x < δ we have f(y) f(x) f(y) f N (y) + f N (y) f N (x) + f N (x) f(x) ɛ + ɛ + ɛ = ɛ To get Peao s curve, we will defie a sequece of cotiuous curves f : [0, ] [0, ] that are uiformly coverget to a surjective fuctio. The previous theorem guaratees that this fuctio will i fact be cotiuous as well. We will get the f through a iterative procedure. The fuctios f are defied by dividig [0, ] ito 9 squares of equal size ad similarly dividig [0, ] ito 9 itervals of equal legth. It is clear what the i th iterval is, but the i th square is determied by the previous fuctio f by the order i which it reaches the squares. The i th iterval is the mapped ito the i th square i a piecewise liear maer. The basic shape of these piecewise liear segmets is that of f 0, give i figure. The ext iteratio is obtaied by replacig the piecewise liear segemets i each square with a smaller copy of f 0, matchig everythig up icely. Clearly each f is cotiuous, but also they become icreasigly close. Lemma.. For, m > N we have that sup x [0,] f (x) f m (x) < 9 mi(,m). Hece the sequece f is uiformly coverget. Proof. Without loss of geerality > m. To get from f m to f we oly modify f m withi each of the 9 m cubes. Withi such a cube, the poits that are furthest apart are i opposite corers ad their distace is 9 m.

3 ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM The theorem about uiform covergece of cotiuous fuctios ow tells us that the followig defiitio makes sese. Defiitio.. The Peao curve is the cotiuous fuctio f : [0, ] [0, ] give by Lemma.4. The Peao is surjective. f(x) = lim f (x) Proof. Note that th iterate f of the Peao curve goes through the ceters of all of 9 squares. Hece ay poit (x, y) [0, ] is o more tha 9 away from a poit o f. Usig the uiform covergece, we see that there is a poit o the image of f that is less tha 9 away from (x, y). Hece (x, y) lies i the closure of the image of f. We will see i propositio 4. that the image of the iterval uder ay cotiuous fuctio is compact ad hece closed, as (x, y) i fact lies i the image of f. Though each of the f is ijective, the limit f is ot. This is will prove later, as a cosequece of oe of the itermediate results i the proof of the Hah-Mazurkiewicz theorem.. Lebesgue s curve We will ow give a secod costructio of a space-fillig curve, due to Lebesgue i 904. The ideas i this costructio will play a importat role i the proof of the Hah-Mazurkiewicz theorem, ad alog the way we ll meet a iterestig space kow as a Cator space. Lebesgue s curve will be clearly surjective, because it will be give by coectig the poit i the image of a surjective map from a subset of [0, ] by liear segmets. This subset is the Cator set: Defiitio.. Let C be the subset [0, ] by deletig all elemets that have a i their terary decimal expasio. Alteratively, C is give by the repeatedly removig middle thirds of itervals. Oe starts with C 0 = [0, ], C = [0, /] [/, ], etc., ad sets C = C. Lebesgue the defied the followig fuctio from C to [0, ] : ( ) 0. a l(0. (a )(a )(a )...) = a a a a 4 a 6... This is easily see to be surjective. We will show it is cotiuous. Lemma.. The fuctio l is cotiuous. Proof. We will prove that if y x < (i.e. our δ), the l(y) l(x) < + (i.e. our ɛ). The idea is to ote that if y x <, the the first terary decimals are the same. Hece the + first biary decimals of the x ad y-coordiate of l(x) are the same. This meas that they are o further that apart. Now Lebesgue exteds this fuctio liearly over the gaps i [0, ] left by the Cator set. We deote such a gap by (a, b ). A example is (/, /). We obtai l : [0, ] [0, ] as follows l(x) = { l(x) if x C (l(b ) l(a )) x a b a + l(a ) if x (a, b ) ([0, ]\C) We claim that this fuctio is cotiuous. Clearly it is cotiuous o the gaps (a, b ) i the complemet of the Cator set because it is liear there. It is however ot clear it cotiuous at the poits of the Cator set. However, by cotiuity of l ad the fact that liear iterpolatios do t get far from their edpoits, we will also be able to prove cotiuity there. Lemma.. The fuctio l is cotiuous.

4 4 A.P.M. KUPERS Figure. The first, secod ad third iteratios of a sequece covergig to hte Lebesgue curve. Proof. By the remarks above it suffices to prove cotiuity at x C. Sice C has empty iterior or equivaletly every poit is a boudary poit, there are three situatios: (i) x is both o the left ad the right a limit of poits i C, (ii) x borders a gap (a, b ) o the left, (iii) x borders a gap (a, b ) o the right. We will do the last case ad skip the first two, because they are very similar to the last case. We do t eed to worry about cotiuity from the right, as l is liear there. We ca take a δ satisfyig > δ > 0 ad such that if y < x ad y x < δ, the y lies i C or i a gap + (a k, b k ) such that a k x <. The idea is just to take δ a bit smaller tha + such that + [x, x δ] cotais a elemet of C. + The, if y C, the we kow that l(y) l(x) = l(y) l(x) < by the previous lemma. If y / C, the y lies i aother gap (a k, b k ) with x a k ad a k x <. That l(y) l(x) < + ow follows by the covexity of balls i Euclidea space: if the edpoits of a lie segmet lie i a ball, the the etire lie segmet lies i the ball. Alteratively, we ca simply estimate l(y) l(x) = (l(b k ) l(a k )) y a k + l(a k ) l(y) b k a k = ( l(b k ) l(y) (y a k ) + l(a k ) l(y) (b k y)) b k a k < ( b k a k (y a k + b k y)) = where we have used that (a k, b k ) [x, x] implies that l(b + k ) l(y) < ad similarly for l(a k ) l(y). There is a iterative costructio of Lebesgue s curve similar to Peao s curve. It is give i figure. 4. The Hah-Mazurkiewicz theorem I this sectio we prove the Hah-Mazurkiewicz theorem for subsets of Euclidea space. Defiitio 4.. A subset A of R is said to be compact if every sequece has a coverget subsequece with limit i A. Equivaletly, by Bolzao-Weierstrass, it is bouded ad closed. The set A is said to be coected if we ca t write it A = (U V ) A, where U ad V are o-empty disjoit ope subsets of R. Fially, it is said to be weakly locally-coected if for all a A ad η > 0 we ca fid a smaller ɛ > 0 such that x B ɛ (a), the x ad a lie i the same coected compoet of A B η (a). The last defiitio, that of weakly locally-coectedess, is i fact equivalet to the more widely used otio of locally-coectedess. However, weakly locally-coectedess at a poit is ot equivalet to locally-coectedess at that poit! This is prove i the appedix. Theorem 4.. A subset A R is the image of some cotiuous map f : [0, ] R if ad oly if it is compact, coected ad weakly locally-coected.

5 ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM 5 We will split the proof ito two parts. The easier implicatio is that the image of a space-fillig curve is compact, coected ad locally-coected, i.e. that our coditios are ecessary. For the coverse we ll have to work a bit harder. The trick is to mimic Lebesgue s costructio i geeral, usig Hausdorff s theorem o the images of the Cator set. 4.. The coditios are ecessary. We will start by provig that the coditios of compactess, coectedess ad weakly locally-coectedess are ecessary. Propositio 4.. The image of a cotiuous map f : [0, ] R is compact, coected ad weakly locally-coected. Proof. Let s call the image A. Oe of the equivalet defiitios of beig compact is that every sequece i A has a coverget subsequece with limit i A. Let s prove this. Let a be a sequece i A, the by pickig a poit i each preimage of a a we get a sequece x i [0, ]. The iterval [0, ] is closed ad bouded, hece compact by Bolzao-Weierstrass. Thus there is a coverget subsequece x k i [0, ], with limit x. We claim that the a k coverge for f(x). But this is a direct cosequece of the cotiuity of f: it seds coverget sequeces to coverget sequeces. The image of a path-coected set is clearly path-coected. But path-coected implies coected, so A is i fact coected. For weakly locally-coectedess we have to a bit more careful. Suppose that for a A the coditio of weakly locally-coectedess fails. The we ca fid a ɛ > 0 ad η < ɛ with η 0 ad a B η (a) A such that a ad a do t lie i the same coected compoet of B ɛ (a). Note that by costructio the sequece {a } coverges to a. Pick a elemet x i the preimage of a. By compactess of [0, ] this has a coverget subsequece x k with limit x, which ecessarily is mapped to a. There is a δ > 0 such that y x < δ implies f(y) f(x) < ɛ. Take K sufficietly large such that x x K < δ. The the etire lie segmet [x K, x] (or [x, x K ], depedig o what makes sese), is mapped ito B ɛ (a) A ad hece a K ad a lie i the same coected compoet of B ɛ (a) A, cotradictig our assumptio o a K. This theorem is geeralized i poit-set topology, where oe proves that the image of ay compact, resp. coected, space is compact, resp. coected. 4.. Itermezzo: Netto s theorem. I the first part of the proof of the previous propositio, we actually proved the image of each closed subset B [0, ] uder a cotiuous map f : [0, ] R is closed. We will use this to prove that o space-fillig curve ca be ijective. This is kow as Netto s theorem. It is a special versio of the theorem that a cotiuous bijectio from a compact space to a Hausdorff space is homeomorphism. Propositio 4.4. A cotiuous surjective map f : [0, ] [0, ] ca ot be ijective. Proof. We claim that the iverse g := f : [0, ] [0, ] is cotiuous ad hece f is a homeomorphism (a cotiuous bijective fuctio with cotiuous iverse). Let C [0, ] be a closed subset. It is also bouded, so compact. We have that g (C) = f(c) ad every cotiuous fuctio seds compact sets to compact sets, as implicitly prove i propositio 4.. Every compact set i [0, ] is i particular closed. So g of closed sets are closed, hece by lookig at complemets uder g of ope sets we ca coclude that these are ope. Hece g = f is cotiuous. Coectedess is preserved by homeomorphisms, because if X ad Y are homeomorphic, a decompositio of oe of them ito a disjoit uio of o-empty ope sets ca be We ca fid a y it([0, ] ) whose preimage x is ot equal to 0 or. The the restrictio of f to [0, x) (x, ], gives a homeomorphism of the set [0, x) (x, ], havig two coected compoets, with the coected set [0, ] \{y}. This gives a cotradictio. 4.. Hausdorff s theorem ad a theorem o pathcoectedess. I this subsectio we will prove Hausdorff s theorem about the possible images i Euclidea space uder a cotiuous map of the Cator set. This was prove i 97 by Hausdorff ad idepedetly i 98 by Alexadroff. It says, i some sese, that the Cator set is the uiversal compact subset of Euclidea space of ay dimesio.

6 6 A.P.M. KUPERS Theorem 4.5. Every compact subset of R is a cotiuous image of the Cator set. Proof. Let A be our compact set. Cosider the cover of A give by B (a) for a A. By compactess there is a fiite subcover, which by duplicatig elemets of the cover if ecessary is of the form B (a i ) for i. Now cover A i = B (a i ) A by B / (a) for a B (a i ) A ad by duplicatig elemets of the fiitely may covers if ecessary we ca assume it has a fiite subcover B / (a i,j ) for i ad j with idepedet of i. We set A i,j = B / (a i,j ) B (a i ) A. Cotiue this procedure, makig the radius of the balls half as large each step. Notice that for a sequece k i {,..., i } we get a sequece of ested closed sets with radius goig to 0. This determies a uique elemet of A as follows: a is the uique elemet of i A k,k,...,k i. Sice we are dealig with icreasigly fie covers ad every elemet of A lies i some itersectio of elemets of the cover, we obtai each elemet of A this way. For coveiece, we defie a way of gettig from a sequece {a } N of 0, s to a sequece of umbers {K } N i {,..., i }. They are give by K = j= a j j, K = j= a j+ j ad i geeral by K i = i j= a j i j. We will defie a fuctio f : C A. It is give by f(0. (a )(a )(a )...) = uique elemet of i A K,K,...,K i This is surjective by the previous remark about every elemet lyig i a itersectio of some elemets of the cover. We also claim it is cotiuous. This is actually rather simple: if y x < + the the first digits i the terary expasio are equal. We ca fid a j such that j j+. Thus f(y) ad f(x) lie i the same A k,k,...,k j, which lies i a ball of radius. Hece f(y) f(x) < ad we coclude that f is cotiuous. k k We wat to costruct our space-fillig curve usig this map, by coectig usig paths i A over the gaps i the Cator set. However, to be able to do this we eed fid such paths ad show they ca be chose sufficietly short for poits i A that are close together. Let s make defiitios for these otios. Defiitio 4.6. We say that a subset A of R is path-coected if ay two poits ca be joied a cotiuous path with image i A. The subset A is said to be uiformly path-coected if for all ɛ > 0 there exists a η (0, ɛ) such that if a, a A satisfy a a < η, the there is a cotiuous path with image i B ɛ (a ) B ɛ (a) A coectig a ad a. Theorem 4.7. If A is compact, coected ad weakly locally-coected, it is path-coected ad uiformly path-coected. This is prove i the appedix The coditios are sufficiet. Now we ll combie Hausdorff s theorem ad our theorem o pathcoectedess to costruct a surjective map [0, ] A uder the coditios listed before. As said before, the idea is to copy the idea of Lebesgue s costructio of a space-fillig curve: use Hausdorff s theorem to get a surjective map from the Cator set ito A ad use our theorem o pathcoectedess to prove that we ca coect the image by paths our the gaps i the Cator set. Theorem 4.8. If A R is compact, coected ad weakly locally-coected, the there exists a surjective cotiuous fuctio f : [0, ] A. Proof. Sice A is compact, by Hausdorff s theorem there exists a cotiuous surjective fuctio g : C A. By uiformly path-coectedess, we ca fid a decreasig sequece of η such that a a < η implies that there is a cotiuous path with image i B / (a) B / (a ) A. By cotiuity of g ad compactess of C, we ca fid a decreasig sequece of δ > 0 such that y x < δ implies g(y) g(x) < η. If (a k, b k ) is a gap i C such that δ + b k a k < δ, the we ca fid a

7 ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM 7 cotiuous path γ k : [a k, b k ] A such that γ k lies i B / (g(a k )) B / (g(b k )) A coectig g(a k ) to g(b k ). We defie f as follows: { g(x) if x C f(x) = γ k (x) if x (a k, b k ) ([0, ]\C) This fuctio is clearly cotiuous at all poits of ([0, ]\C) ad as before the poits x C comes i three types: (i) x is both o the left ad the right a limit of poits i C, (ii) x borders a gap (a, b ) o the left, (iii) x borders a gap (a, b ) o the right. Agai we will do the last case ad skip the first two, because they are very similar to the last case. Cotiuity from the right is obvious, as f is just a cotiuous path there. For cotiuity from the left pick δ > 0 satisfyig δ > δ > 0 ad such that if y < x ad y x < δ, the y C or it lies i a gap (a k, b k ) such that a k x < δ. If y C, the y x < η ad hece f(y) f(x) = g(y) g(x) η. If y / C, the y lies o a path that does t get further tha from g(x) ad hece f(y) f(x). So give a ɛ > 0 just take a N so that mi({η, }) < ɛ ad set δ = δ. The y x < δ implies f(y) f(x) < ɛ. Appedix A. Differet otios of local coectedess I this appedix we prove several relatios betwee differet otio of coectedess. Let s recall their defiitios. We will use ope sets istead of ope balls for coveiece, except i the defiitio about uiform path-coectedess, which eeds explicit ɛ s. Defiitio A.. () A subset A R is said to be weakly locally-coected if for all a A ad ope eighborhoods U of a i A we ca fid ope eighborhood V U cotaiig a such that x V, the x ad a lie i the same coected compoet of U. () The subset A is said to be uiformly weakly locally-coected if for all ɛ > 0 there exists a η (0, ɛ) such that if a, a A satisfy a a < η the a, a lie i the same coected compoet of B ɛ (a) B ɛ (a ) A. () A R is said to be locally coected if for all a A ad all ope eighborhoods U of a i A we ca fid a eighborhood V A of a which is coected. (4) A is said to be locally path-coected if for all a A ad all ope eighborhoods U of a i A we ca fid a eighborhood V A of a which is path-coected. (5) The subset A is said to be uiformly locally path-coected if for all ɛ > 0 there exists a η (0, ɛ) such that if a, a A both satisfy a a < η the there is a cotiuous path with image i B ɛ (a) B ɛ (a ) coectig a ad a. Lemma A.. The subspace A is weakly locally-coected if ad oly if it is locally coected. Proof. The implicatio is clear. For the implicatio, we claim that the coected compoets of all ope subsets of A are ope. For if this is true, the take V to be the coected compoet cotaiig a i U. Let C be a coected compoet of U. We will cover C by ope sets V c A, the the expressio C = ( c V c ) A shows it s ope ad we are doe. Pick c C ad a ope eighborhood V c of it such that U c U. Usig weakly locally-coectedess, take a smaller ope eighborhood V c such that if x V c, the c ad x lie i the same coected compoet of U c. I particular c ad x must the both lie i C. This meas that V c C ad we are doe. Lemma A.. Coected ad locally path-coected implies path-coected. path-coected implies locally path-coected. Uiformly locally Proof. Let s start with the first statemet. Let a A ad P a be the subset of A of all poits that ca be coected by a path to A. It is ope by locally path-coectedess. To see it is closed as well, use that beig coected be a path is a equivalece relatio, hece A = P ai. Hece the complemet of P a is ope ad thus P a is closed. A ope ad closed set is a coected compoet ad sice A was assumed coected, we coclude A = P a. The secod statemet is trivially true.

8 8 A.P.M. KUPERS Lemma A.4. If A is compact ad weakly locally-coected it is uiformly weakly locally-coected. Proof. Suppose that it is ot uiformly weakly locally-coected. The there exists a ɛ > 0, η i 0 ad a i a i < η i such that a i ad a i lie i differet coected compoets of B ɛ(a) B ɛ (a ) A. By compactess, we ca assume that a i ad a i coverge, ecessarily to the same elemet a. Now we apply weakly locally-coectedess at a to B ɛ (a): we get a η > 0 such that x B η (a) implies that x, a lie i the same coected compoet of B ɛ (a). Take i large eough such η i < η. The a i ad a i lie i the same coected compoet of B ɛ(a) at a ad hece lie i the same coected compoet. This gives a cotradictio. Theorem A.5. If A is compact, coected ad weakly locally-coected, it is path-coected, locally path-coected ad i fact uiformly locally path-coected. Proof. By the previous lemma s, it suffices to prove uiform locally path-coectedess. Take ɛ > 0 ad set ɛ k = ɛ 4. Take η k k to be umber comig out of uiformly weakly locally-coectedess for ɛ k ad set η = η. Let a, a A be such that a a < η. We claim that by coectedess we ca fid a ad a sequece of poits x () i for 0 i such that x () 0 = a, x () = a ad x () i x () i+ < η. This is because the set of poits i A that ca be coected by a η -chai of poits is both ope ad closed, hece equal to A. Each of the x () i lie the same coected compoet of B ɛ (x () i ) B ɛ (x () i+ ) A ad hece we ca fid ad sequeces x () i,j i B ɛ (x () i ) B ɛ (x () i+ ) A for 0 i, 0 j with x () i,0 = x() i ad x () i, = x () i+. Cotiue this process, gettig k ad x (k). i Let J [0, ] be the set of poits with biary expasio that ca be writte as for k i some i ad k. This is dese. The we defie g : J A by sedig to the poit of the k x (k) that i correspods to. This ca be checked to be cotiuous, sice η k < ɛ k ad the ɛ k go to zero. Hece g exteds to a cotiuous fuctio γ : [0, ] A. The image lies i A sice A is compact, hece closed. The oly thig left to check is that γ stays withi B ɛ (a) B ɛ (a) A. But it stays withi k= ɛ k < ɛ of a ad a by costructio. Refereces [Sag94] Has Saga, Space-fillig curves, Uiversitext Series, Spriger-Verlag, 994.