Computing Space-Filling Curves

transcript

Theory of Computing Systems manuscript No.(will be inserted by the editor)

Computing space-filling curves

P.J. Couch · B.D. Daniel · Timothy H.McNicholl

Received: date / Accepted: date

Abstract We show that a continuous surjection of [0, 1] onto a EuclideanPeano continuum X can be computed uniformly from a name of X as a com-pact set and a local connectivity operator for X . We show by means of anexample that the second parameter is not superfluous. We then show that thisparameter is not necessary either in that there is a computable map of [0, 1]into R2 whose image is not e!ectively locally connected.

Keywords Computable analysis, constructive analysis, Peano continua,space-filling curves, locally connected spaces

Mathematics Subject Classification (2000) 03F60 · 30D55 · 54D05 ·54F15

1 Introduction

In 1887, Camille Jordan defined a continuous curve as a continuous func-tion whose domain is a closed segment of the line R. Interest in such spaces

P.J. CouchDepartment of MathematicsAuburn UniversityAuburn, Alabama 36849 USAE-mail: pjc0005@auburn.edu

B.D. DanielDepartment of MathematicsLamar UniversityBeaumont, Texas 77710 USAE-mail: dale.daniel@lamar.edu

Timothy H. McNichollDepartment of MathematicsLamar UniversityBeaumont, Texas 77710 USAE-mail: timothy.h.mcnicholl@gmail.com

arose due to Peano’s constructions [18] in 1890 of so-called space-filling curves(also known as Peano curves). In particular, Peano constructed an exampleof a continuous function of [0, 1] onto the square [0, 1]2. Peano’s constructionsmotivated e!orts to characterize continuous images of [0, 1] and to devise anappropriate definition of topological dimension. Such a characterization wasobtained independently by H. Hahn [9] and S. Mazurkiewicz [14]. The Hahn-Mazurkiewicz Theorem is a fundamental result on such curves and asserts thata topological space is a continuous image of the closed unit interval [0, 1] ifand only if it is metrizable, compact, connected, and locally connected. Dueto this characterization and Peano’s role in the study of such spaces, locallyconnected metric continua have come to be known as Peano continua. In 1916,R. L. Moore proved [16] that each Peano continuum is arcwise connected andis in fact locally arcwise connected. Sagan’s book [19] provides an excellentresource on the history of space-filling curves and many additional results.

More recently, the construction of space-filling curves has played a role inthe design of parallel computers. See, for example, [2], [8], [10], [13], and [20].In addition, these curves continue to be a topic of active research in topology.See, for example, [1], [3], [12], and [21].

Because of its prominence in the development of mathematics, its key rolein the definition of dimension, the use of space-filling curves in recent work inapplied computer science, and the work it continues to inspire in topology, it isinteresting, novel, and important to analyze the Hahn-Mazurkiewicz theoremfrom the viewpoint of e!ective mathematics. That is, what information is nec-essary and su"cient for the construction of a space filling curve onto a givenPeano continuum, X? We will restrict our attention to subspaces of Euclideanspace. Since a Peano continuum is compact, a natural first approximationto an answer to this question is the set of all finite covers of X by rationalrectangles; that is, rectangles whose vertexes have rational coordinates. For,in the two-dimensional case, this provides the precise amount of informationnecessary to draw the continuum on a computer screen with arbitrary preci-sion. While such information turns out to be necessary, we will demonstrateby means of an example that it is not su"cient. The question then turns towhat additional parameters are necessary and su"cient for the construction ofsuch a map. The natural candidate is information about the local connectivityof X . In fact, we will show that a local connectivity operator, a term whichwe will define below, provides su"cient additional information. However, itfollows from a recent example of Gu, Lutz, and Mayordomo that this addi-tional parameter is not necessary [7]. Namely, it is not possible to computeuniformly from a space-filling curve a local connectivity function for its image.The latter result indicates that the second direction of the most natural for-mulation of the Hahn-Mazurkiewicz theorem is not provable within a systemsuch as Bishop’s constructive mathematics or RCA0. This is surprising sinceclassically the second direction is by far the easiest to prove.

2 Background from computability and computable analysis

Our work is based on the Type Two E!ectivity foundation for computableanalysis which is described in great detail in [22]. We give an informal summaryhere of the points pertinent to this paper.

A rational rectangle is a set of the form (a1, b1) ! . . . ! (an, bn) such thataj , bj " Q and aj < bj .

A name of a point x " Rn is a list of all the rational rectangles to which xbelongs.

A name of an open U # Rn is a list of all the rational rectangles whoseclosure is contained in U .

A name of a closed C # Rn is a list of all the rational rectangles whichintersect C.

A name of a compact K # Rn is a list of all finite coverings of K by rationalrectangles each of which intersect K. Such a covering is called a minimalcovering. These names are called !mc-names in [22].

Once we establish a naming system for a space, an object of that space iscalled computable if it has a computable name.

Suppose f :# A $ B where A, B are spaces for which we have establishednaming systems. A Turing machine M with a one-way output tape and accessto an oracle X is said to compute f if whenever a name of a point x " dom(f)is written on the input tape, and M is allowed to run indefinitely, a name off(x) is written on the output tape. If X is computable, then f is said to becomputable. A name of such a function then consists of a set X # N and acode of a Turing machine with one-way output tape which computes f whengiven access to oracle X .

3 Background from topology

A path in a topological space X is the image of a continuous function from[0, 1] into X . If the function is one-to-one, then its image is referred to as anarc, the images of 0, 1 are called the endpoints of the arc, and the function isreferred to as a parametrization of the arc.

A metric space X is uniformly locally arcwise connected if for every " > 0there exists a # > 0 such that whenever x, y are points in X with d(x, y) < #,there is an arc in X from x to y whose diameter is at most ".

A continuum is a compact, connected metric space. A Peano continuum isa locally connected continuum. The Hahn-Mazurkiewicz Theorem states thata topological space is the image of a continuous function on [0, 1] if and onlyif it is a Peano continuum.

Finally, let B!(p) denote the open ball with center p and radius ". WhenX # Rn, let

B!(X) =!

B!(p).

4 Almost a counterexample

In this section, we show that computable compactness alone is not su"cientto guarantee the existence of a computable space-filling curve.

Theorem 1 There is a computable planar Peano continuum X that is not theimage of any computable function on [0, 1].

Proof Let A be a non-computable c.e. set. Let F : N $ N be a computablefunction such that A = ran(F ). Let At = {F (s) | s < t}.

For n " N, let:

an = 2"(n+1)

bn = 2"n

cn =bn + an

2Sn = (an, bn) ! (0, an)$n = cn % an(= bn % cn = 2"(n+2))

LetX0 = ([0, 1]2 % &nSn) &

({cn} ! [0, an]).

A picture of X0 is shown in Figure 1 in gray.For each n, let ln be the line segment from (an, an) to (cn, an/2). Then, let

LTn,t be the set of points in Sn that are above or on ln and whose x coordinateis in

[an, an +"

2"(j+1)$n].

Let dn,t be the right endpoint of this interval. ‘LT ’ is short for ‘left triangle’.LTn,t is shown in gray in the left half of Figure 2. Let LCn,t consist of theunion of all closed vertical line segments from the point

(an +dn,t % an

t + 1s, an)

to ln as s ranges from 1 to t + 1. ‘LC’ is short for ‘left comb’. LCn,4 is shownin the right half of Figure 2.

If n '" A, then letPn =

LTn,t.

If n " A, and if t is the least number such that n " At, then let

Pn = LCn,t.

Fig. 1 The set X0.

Fig. 2 LTn,t and LCn,3

LetX =

Pn & X0.

Lemma 1 X is closed.

Proof We first note that

(Sn ( X) & X0.

Clearly, X0 and Sn ( X are closed. Suppose p1, p2, . . . is a sequence of pointsin X that converges to p.

Suppose that for some n, pk " Sn ( X for infinitely many k. Since Sn ( Xis closed, it follows that p " X . Similarly, p " X if there are infinitely many kfor which pk " X0. In the only remaining case to consider, we are lead to theconclusion that

Sn ( X ( {p1, p2, . . .} '= )

for infinitely many n. Since

limn$%

d((0, 0), Sn) = 0

it follows that p = (0, 0) " X0. *+

Lemma 2 X is connected.

Proof Clearly, X0 and each Sn ( X is connected. SinceSn (X (X0 is non-empty for each n, it follows that X is connected. See, e.g.,Theorem 1-14 of [11]. *+

Lemma 3 X is locally connected.

Proof By examination of the cases n " A and n '" A, it follows that Sn (X islocally connected. Clearly, X0 is locally connected. It then follows that X islocally connected. *+

Lemma 4 The set of rational rectangles which intersect X is computably enu-merable.

Proof A rational rectangle intersects X is and only if it intersects X0 or Sn(Xfor some natural number n. The set of rational rectangles which intersect X0

is computable. So, it su"ces to show that the set of rational rectangles thatintersect !

n!NSn ( X

is computably enumerable. To this end, we note that for each n " N, the setof rational rectangles which intersect

LTn =df

n!NLTn,t

is computable. Moreover, it is computable uniformly in n. It follows that thereis a computable array of rational rectangles {Rn,t}n,t such that for each n,

LTn is the only closed set hit by all Rn,t, yet the diameter of each Rn,t is nosmaller than

dn,t+1 % an

t + 2and its right side is to the left of the line x = dn,t. It follows from elementarycalculus that (t + 2)"1(dn,t+1 % an) is decreasing as a function of t on [0,,).Hence, these additional requirements on the array {Rn,t}n,t ensure that foreach natural number t, every Rn,s with s - t intersects LCn,t+1.

Now, let {Sn,t,s}n,t,s!N be a computable array of rational rectangles suchthat for each n, t, {Sn,t,s}s!N enumerates precisely the rational rectangles thatintersect LCn,t. It now follows that a rational rectangle intersects Sn (X justin case it contains an Rn,t with n '" At or an Sn,t+1,s such that n " At+1%At.It then follows from Post’s Theorem that the set of all rational rectangles thatintersect an Sn ( X for some n is c.e.. *+

Lemma 5 The set of rational rectangles whose closure does not intersect Xis computably enumerable.

Proof Let Z be the set of all rational rectangles whose closures do not intersectX . Let U be the set of all rational rectangles whose closures do not intersect[0, 1]2. Hence, U is computable. Let Un be the set of all rational rectangleswhose closure is contained in (R2 % X) ( Sn. Hence,

Z = U &!

So, it su"ces to show that {Un}n is uniformly c.e..For all n, let mn be the line segment from (cn, 0) to (cn, an).If n '" At, then define Vn,t to be the set of all rational rectangles R such

that R . Sn, and R does not intersect mn nor#

s LTn,s. If n " At, then lets be the least number such that n " As, and define Vn,t to be the set of allrational rectangles R such that R # Sn and R does not intersect mn or LCn,s.

It then follows that {Vn,t}n,t is uniformly computable. At the same time,Un = &tVn,t. It then follows that {Un}n is uniformly computably enumerable.

It now follows that we can compute an enumeration of the minimal coversof X .

Lemma 6 Let f : [0, 1] $ X be continuous and surjective. Let y1 " X, andlet # > 0. Then, there exists " > 0 such that for all y2 " B!(y1) there existx1, x2 " [0, 1] such that |x1 % x2| < # and f(xi) = yi for i = 1, 2.

Proof By way of contradiction, suppose otherwise. For each n " N, let y2,n "B2!n(y1) be such that for all x1, x2 " [0, 1]

f(x1) = y1 / f(x2) = y2,n 0 |x1 % x2| 1 #.

Let x2,n " f"1({y2,n}) for all n " N. By the Bolzano-Weirstrass Theorem,there exist n1 < n2 < . . . such that {x2,ni}i converges to some x1 " [0, 1].Hence,

limi$%

f(x2,ni) = limn$%

f(x2,n) = limn$%

y2,n = y1.

Thus, since f is continuous, f(x1) = y1. But, there exists i such that |x2,ni %x1| < #, a contradiction. *+

The following is a consequence of Theorem 6.2.7 of [22]

Lemma 7 Let f :# R $ Rn be computable, and suppose dom(f) is com-pact and computable. Then, there exists a computable increasing h : N $ Nsuch that for all n " N and all x, y " dom(f), if |x % y| < 2"h(n), thend(f(x), f(y)) < 2"n.

The function h is called a modulus of continuity for f .

Lemma 8 X is not the image of any computable function from [0, 1] into R2.

Proof By way of contradiction, suppose otherwise. Let f : [0, 1] $ R2 be acomputable function such that X = ran(f). Let h : N $ N be a computablemodulus of continuity for f .

Now, fix n. Let %n,t be the upper side of LTn,t. We show how to determineif n " A. First, compute m such that 2"m < an/4. Set # = 2"h(m). Let&1(x, y) = x. Search for R0, R1, T0, T1, t such that n " At or n '" At and thefollowing hold.

Ri . [0, 1]d(R0, R1) < #

f [Ri] # Ti

(cn, an/2) " T1

T0 . LTn,t

max&1(T0) < min &1(T1)d(%n,t, T0) > an/4

It follows from the construction of X and Lemma 6 that this search mustsucceed. It then follows that if n '" At, then n '" A. For, f traces a path froma point in T0 to a point in T1. At the same time, the preimage of this path iscontained in a closed interval of diameter less than #. Hence, the diameter ofthis path must be less than 2"m < an/4. On the other hand, if n " A, thenthis path must have diameter at least as large as an/4. We are led to concludethat n '" A. However, this now yields a procedure for determining if a numbern is in A- a contradiction. Hence, such a function f can not exist. *+

*+Theorem 1 might appear to be a counterexample to an e!ective version of

the Hahn-Mazurkiewicz Theorem, but it is not. For, an e!ective version of theHahn-Mazurkiewicz Theorem ought to contain an e!ective rendition of localconnectivity. We explore such notions in the next section.

5 E!ective local connectivity

The definition of e!ective local connectivity we give here is essentially thatgiven in [4]. We first give some motivation and demonstrate that most attemptsat a simpler definition are likely to fail.

The classical definition of local connectivity reads in part

. . . for every point p in the space X , every neighborhood of p containsa neighborhood of p that is connected.

In the case of Euclidean space, this is equivalent to stipulating that for ev-ery point p in X , every rational rectangle R that contains p contains also aneighborhood of p, U , such that U ( X is connected. An e!ective version ofthis should require that the neighborhood U is in some way be computablefrom p and R. To this end, it is desirable, for the sake of simplicity, that weshould be able to choose the neighborhood U so that it is susceptible to afinite description; e.g. a rational rectangle or a sphere with rational center andradius. The following example shows that any such attempt at simplicity willfail spectacularly.

Theorem 2 There is a planar Peano continuum Z such that if x " [0, &] .Z then there is no local basis {Rn(x)} of rational rectangles at x such thatRn(x) ( X is connected for each n = 1, 2, 3, . . ..

Proof We construct a Peano continuum Z in the plane R2 such that Z con-tains the origin o = (0, 0) and no connected open set O containing o may bewritten as the union of finitely many rational rectangles each having connectedintersection with Z unless it contains all of Z.

Let I0 denote the closed interval [0, &]!{0} in R2. For each n = 0, 1, 2, . . .,let Vn denote the closed vertical interval Vn = {[ "

2n , y] : 0 - y - "2n }, Fn

denote the closed segment Fn = {(x, y) : y % "2n = 2

3 (x% "2n ), "

2n+2 - x - "2n },

and let Sn = Vn & Fn. Define Z0 to be the connected union I0 & (&%n=0Sn).

Figure 3 gives an indication of the construction of Z0. Note that if a rationalrectangle R contains the point o and has connected intersection with Z0 thenR must contain Z0.

For each k = 1, 2, . . ., we define Ik as ([0, "2k ] ! {0}) & (&%

n=kSn). For eachk = 1, 2, . . . and m = 1, 3, 5, . . ., we define Ik(m) to be a horizontal translationof Ik precisely m"

2k units.Define Z1 to be the connected union Z0 & I1(1). Define Z2 to be the con-

nected union Z1&(I2(1)&I2(3)). In general, we define Zj for each j = 1, 2, 3, . . .

as Zj = Zj"1 & (&ji=1Ij(2i % 1)). We then define Z by Z = &%

j=0Zj . It followsfrom the construction that Z is compact, connected, and locally connected.

Note that if b is a rational number and 0 < b < & then the intersectionof the line x = b and Z has infinitely many components. Moreover, since noendpoint of Z may lie on the line x = b then any open rational rectangle R hasthe property that R (Z has infinitely many components. It then follows thatif x " [0, &] . Z, then there is no local basis {Rn(x)} of rational rectangles atx such that Rn(x) ( X is connected for each n = 1, 2, 3, . . ..

Now assume that O is a connected open set containing o andR1, R2, . . . , Rh are rational rectangles such that O =

#Ri and Ri ( Z is

connected for each i = 1, .., h. Assume also that O does not contain Z. Foreach i = 1, .., h, let Ri = (ai

1, bi1) ! (ai

2, bi2). Assume without loss of generality

that bh1 = max{bi

1 : i = 1, .., h and bi1 ( [0, &] '= )}. By construction of Z, there

exists k and m such that bh1 " Ik(m), Ik(m)(O '= ), and Ik(m)((Z %O) '= ).

Then a translation of Rh contains the point o and has connected intersectionwith Z0 yet fails to contain all of Z0, a contradiction.

Fig. 3 The set Z0

Although we can not always choose U so that it has a finite description,the machinery of Type Two E!ectivity, because of the facility with which itdeals with objects capable only of infinite descriptions, points to an obvioussolution to the conundrum which now confronts us. We thus arrive at thefollowing definition.

Definition 1 Let X # Rn. A local connectivity operator for X is a continuous(though not necessarily computable) operator that, given a name of a pointp in X and a rational rectangle R that contains p, yields a name of an openU # Rn such that p " U ( X # R and U ( X is connected. We say that X ise!ectively locally connected if it has a computable local connectivity operator.

Again, the definition we have just given for e!ective local connectivity isessentially the same as that which appears in [4]. By considering local connec-tivity operators though, we have amplified the concept so that local connec-tivity is presented as additional computational information about the spacerather than its computability forming a restriction on the space.

A notion related to local connectivity is that of connected im kleinen. Inorder for a topological space to be connected im kleinen, it must be the casethat for each point p in the space, each neighborhood U of p must contain aconnected set C which in turn must contain a neighborhood of p. From thisdefinition, we extract the following.

Definition 2 A connected im kleinen function for a set X # Rn is a functionf : N $ N such that for all k " N and all x " X , there is a connected setC # X such that X ( B2!f(k)(x) # C # B2!k(x).

In [6], connected im kleinen functions are referred to as e!ective local con-nectivity functions. In [15], compact spaces for which there exists a computableconnected im kleinen function are called computably locally connected.

Let ' be a finite alphabet that contains 0, 1. By means of standard codingtechniques, a name of an object can be represented as a point in '#. See[22] for details. Again, by standard coding techniques, each rational rectanglecan be represented as a finite word in '&, and each pair in '# ! '& can berepresented as a single infinite word in '#. A local connectivity operator canthus be viewed as a partial continuous function from '# into '#. Moreover,by Theorem 2.3.8 of [22], we can assume the domain of such an operator isG$. The space of all such functions has a natural naming system. Informally,a name of such a function is an oracle Turing machine with one-way outputtape which computes the function. See Section 2.3 of [22]. From this, we makesense of the following theorem.

Theorem 3 It is possible to compute, uniformly from a name of a compactX # Rn and a local connectivity operator for X, ( , a name of a connected imkleinen function for X. Furthermore, from a name of a compact set X # Rn

and a connected im kleinen function for X, we can compute a name of a localconnectivity operator for X.

In order to prove the first direction of Theorem 3, suppose we are given aname of a compact set X # Rn and a name of a local connectivity operatorfor X , ( . We construct a connected im kleinen function for X , f . We will needthe following, which is proven in [6].

Lemma 9 From a name of a compact X # Rn, and a finite sequence ofrational rectangles R1, . . . , Rk such that

it is possible to compute a natural number k with the property that 2"k is aLebesgue number for this covering of X.

A name of a partial continuous function from '# into '# consists of anoracle Turing machine which computes the function. Accordingly, write R "((S, {Tj}j<k) if R is one of the rational rectangles the machine for ( listsafter it reads S, T0, . . . , Tk and nothing else as input. We then let

S = {(R, S) | R ( X '= ) / 2{Tj}j<k[R " ((S, {Tj}j<k)

/ X ( ($

Tj) '= )]}.

It follows that S is c.e. uniformly in the given data. Furthermore, for every(R, S) " S, there is a connected subset of X , C, such that R ( X # C # S.

We now claim that for every m " N and every p " X , there exists (R, S) "S such that p " R # S and diam(S) < 2"m. For, let p " S ( X where S is arational rectangle of diameter less than 2"m. Let {Tj}j!N be a name of p. LetU be the open set named by ((S, {Tj}j!N). Then, p " U (X # S. Therefore,there is a rational rectangle R1 # U such that p " R1 ( X # S. There isalso a rational rectangle R2 such that p " R2 and R2 # S. Let R = R1 ( R2.Therefore, R # U ( S. It follows that (R, S) " S.

We now define an oracle Turing machine which computes a function f :N $ N. Given m, compute (R1, S1), . . . , (Rt, St) " S such that X #

#j Rj ,

Rj ( X '= ), and diam(Sj) < 2"m. We then compute a Lebesgue number forthis covering, f(m).

We claim that f is a connected im kleinen function for X . For, let x " X ,and let m " N. If y is a point in X such that d(x, y) < 2"f(m), then there existsj such that x, y " Rj . There then exists a connected set Cy # X such thatX ( Rj # Cy # Sj . Define C to be the union of all the Cy’s as y ranges overall points in X whose distance from x is less than 2"f(m). Since x belongs toeach such Cy, it follows that C is a connected subset of X . Since the diameterof each Sj is less than 2"m, it follows that C is contained in the open ball withcenter x and radius 2"m. Finally, each point y " X such that d(x, y) < 2"f(m)

belongs to the corresponding set Cy. Hence,

X ( B2!f(m)(x) # C # B2!m(x).

It follows that f is a connected im kleinen function for X .In order to prove the second direction of Theorem 3, suppose we are given

in addition to a name of a compact set X , a name of a connected im kleinenfunction for X , f . We can assume f is increasing. It also follows that X islocally connected. We will use the following definition from [6].

Definition 3 A witnessing chain is a sequence (m, R1, . . . , Rk) such that

– Ri ( Ri+1 ( X '= ) whenever 1 - i < k, and– diam(Ri) < 2"f(m).

It is shown in [6] that from f and our name of X , we can compute anenumeration of the set of all witnessing chains.

Suppose ) = (m, R1, . . . , Rk) is a witnessing chain. It is shown in [6] thatwhenever x, y " Ri ( X , x, y belong to the same connected component ofB2!m(Ri) ( X . Let C#,i denote this connected component, and let

LetV# =

B2!m(Ri).

In [6], it is shown that C# is a connected subset of V# ( X .In order to complete our proof of the second direction of Theorem 3, we

prove the following.

Lemma 10 Let R be a rational rectangle. For every x, y " R(X, x and y arein the same connected component of R(X if and only if there is a witnessingchain ) = (m, R1, . . . , Rk) with x " R1 and y " Rk and such that V# # R.Furthermore, if x, y are in the same connected component of R ( X, then forevery " > 0, there is such a witnessing chain for which it is also true that2"m < ".

Proof Suppose x, y are in the same connected component of R ( X , and letC denote this component. At the same time, let X1 denote the connectedcomponent of x in X . Let C1 be the connected component of x in R ( X1.Hence, C1 is a connected subset of R(X . So, C1 # C. Since C is a connectedsubset of X that contains x, C # X1. Since C # R, C # R ( X1. Hence,C = C1.

Since X is locally connected, it follows that X1 is locally connected. SinceX is compact, X1 # X . However, the closure of a connected set is connected.So, X1 = X1. Hence, X1 is closed and so is compact. Thus, X1 is a Peanocontinuum.

It follows from Theorem 3-2 of [11] that C is open. It then follows fromTheorem 3-16 of [11] that C is arcwise connected. So, let A be an arc fromx to y in C. Since A is compact, there exists m such that 2"m < " andB2!m(A) # C. For each p " A, there is a rational rectangle Rp such thatp " Rp, B2!m(Rp) # R, and diam(Rp) < 2"f(m). By Theorem 3-4 of [11], wecan choose p1, . . . , pk so that (Rp1 ( A, . . . , Rpk ( A) is a simple chain from xto y in that Rpi ( Rpj ( A '= ) precisely when |i % j| - 1,

x " Rp1 %!

Rpi ( A, and

y " Rpk %!

Rpi ( A.

So, we let ) = (m, Rp1 , . . . , Rpk).Now, suppose ) = (m, Rp1 , . . . , Rpk) is a witnessing chain from x to y such

that V# # R. It follows that x " C#,1 and y " C#,k. Hence, x, y " C# . Asnoted above, C# is connected and is contained in V# . Hence, x, y belong tothe same connected component of R ( X . *+

We now define a local connectivity operator ( by showing how to compute( from the given data. Suppose we are given in addition to f and our name forX , a rational rectangle R and a name of a point x " X ( R. Read this namewhile cycling through all witnessing chains. When m, R1, . . . , Rk are foundsuch that ) = (m, R1, . . . , Rk) is a witnessing chain, x " R1, and V# # Renumerate Rk.

SupposeU =

where S ranges over all rational rectangles listed by this process. It follows thatU is open and is in fact the connected component of x in R(X . However, theset of rectangles listed by this process may not be a name of U ; some largerational rectangles whose closure is contained in U may be omitted. To mendthis flaw, we additionally list all rational rectangles T such that T is coveredby finitely many rational rectangles listed by the process we just described.The resulting list is a name of U . This completes the proof of Theorem 3.

Theorem 3 shows that the definition of e!ective local connectivity given inDefinition 1 and that given in [15] are equivalent.

We will need the following e!ective renditions of uniform local arcwiseconnectivity. The definitions are from [6].

Definition 4 A uniformly local arcwise connectivity witness for X is a func-tion f : N $ N such that for all k " N and all x, y " X , if d(x, y) < 2"f(k),then x, y are joined by an arc in X of diameter less than 2"k.

Definition 5 A pair (f, g) is called a strong witness of uniformly local arcwiseconnectivity for X if f witnesses the uniformly local arcwise connectivity ofX and if g is such that whenever k " N and p, q are names of points x, y " Xrespectively such that d(x, y) < 2"f(k), g(k, p, q) is a name of a parametrizationof an arc in X from x to y.

The following is proven in [6].

Theorem 4 From a name of a continuum X # Rn, and a connected imkleinen function for X, it is possible to compute uniformly a strong witnessof uniformly local arcwise connectivity for X. Furthermore, from a name of acontinuum X # Rn and a uniformly local arcwise connectivity witness for Xwe can compute a connected im kleinen function for X.

From Theorem 4 and Theorem 3, we now obtain the following.

Corollary 1 From a name of a Peano continuum X # Rn, names of distinctpoints x1, x2 " X, and a local connectivity operator for X, we can compute aname of a parametrization of an arc in X from x1 to x2.

6 Continuous images of the Cantor Set

Let C denote the Cantor middle-third set. Classically, the first step towards aproof of the Hahn-Mazurkiewicz Theorem is the Aleksandrov-Hausdor! The-orem. We give an e!ective version of this theorem here. A proof first appearedin [5].

Theorem 5 (E!ective Aleksandrov-Hausdor! Theorem)From a name of a compact set X # Rn we can uniformly compute a name ofa surjection of C onto X.

Proof Let #A denote the cardinality of A.For each i " N, let {Ji,j}j<2i enumerate the intervals which remain after

the i-th stage of the construction of the Cantor set. (See, e.g., Exercise 6,Section 3-6 of [17].) Hence, J0,0 = ). For each * " {0, 1}&, let J% = Ji,j if|*| = i and * is the (j + 1)-st element in the lexicographic ordering of {0, 1}i.It follows that

+ 3 * 4 J& . J%.

Given a name of a compact X # Rn, we compute a sequence of minimalcoverings of X , {Ut}t!N with the property that Ut+1 refines Ut for each t " N.Simultaneously, we compute a sequence of natural numbers {kt}t!N such that

2k0 1 #U0

and for all t " N and all X " Ut,

2kt+1 1 #{Y " Ut+1 | Y # X}.

We can then simultaneously compute for each t " N a sequence

{S%}%!{0,1}k0+...+kt

such that {S%}%!{0,1}k0 enumerates #U0 and for each * " {0, 1}k0+...+kt

{S%&}&!{0,1}kt+1

enumerates{Y " Ut+1 | Y # S%} .

It now follows that for every x " C there is a unique y in$

{S% | * " {0, 1}& / x " J%}.

So, for each x " C, we define f(x) to be the unique point in this set. It followsthat ran(f) = X . We claim that we can compute a name of f uniformly fromthe given data. To this end, it su"ces to show that if we are given in additiona name of an x " C, we can uniformly in the given data generate a nameof f(x). So, suppose we are additionally given a name of an x " C. We readthis name while reading the name of X and simultaneously generating theintermediate data. We concurrently cycle through {0, 1}&. Whenever we findR, R', * such that x " R # J% and R' 5 S%, we list R' as containing f(x).Since the diameter of S% tends to 0 as the length of * increases, it follows thata name of f(x) is generated by this process. *+

7 An e!ective version of the Hahn-Mazurkiewicz Theorem

Theorem 6 (E!ective Hahn-Mazurkiewicz Theorem) From a name ofa Peano continuum X # Rn and a local connectivity operator for X, we canuniformly compute a name of a surjection g : [0, 1] $ X.

Proof By Theorem 5 we can compute uniformly from these data a surjectionf : C $ X . Compute also a modulus of continuity for f , m.

Let I0, I1, . . . be the intervals of [0, 1]%C ordered by decreasing length andleft to right for intervals of equal length. Let Ij = (pj , qj). Note that pj , qj "Q ( C for all j. Furthermore, the sequences {pj}j and {qj}j are computable.It follows from Theorem 4 that we can compute uniformly in these data anincreasing h : N $ N such that for all n " N and all x, y " C, if d(x, y) <2"h(n), then f(x) and f(y) are joined by an arc in X of diameter less than2"n. Since {pj}j and {qj}j are computable, it now follows that we can computeuniformly in these data an h1 : N $ N such that I0, . . . , Ih1(0) have lengthat least 2"h(0) and Ih1(k)+1, . . . , Ih1(k+1) have length in [2"h(k+1), 2"h(k)). ByTheorem 4, for each k " N and i " {h1(k) + 1, . . . , h1(k + 1)} we can computea name of a function fi : Ii $ X which parametrizes an arc from f(pi) tof(qi) of diameter less than 2"k. By Corollary 1, if i - h1(0), we can computea name of such a function without bounding the diameter of its image.

Letg(x) =

%f(x) x " Cfj(x) x " Ij .

It follows that g maps [0, 1] onto X . We now show that a name of g can becomputed from the given data. So, we assume that we are given a name of anx " [0, 1] as additional input. We then read the given names and the namesgenerated so far in parallel. Relabel p0, q0, p1, . . . as p0, p1, . . ..

If we ever discover a rational interval R and an integer j such that x " R #Ij , then we begin running the computation of fj(x) and list as containing g(x)every rectangle it lists as containing fj(x). On the other hand, suppose thatat some point we have not found such R, j, but we have found R, R', j, k1, k2

such that the following hold.

1. pj, x " R.

2. 2"m(k1), 2"h(k2) > diam(R).3. max({diam(Ik) | Ik ( R '= ) / R '# Ik / Ik '# R} & {0}) < 2"h(k2).4. R' 5 B2!k1+2!k2 (f(pj)).

We then list R' as a rational rectangle which contains g(x).

Claim: g(x) belongs to every rational rectangle listed by this process.

If R' is listed by this process as a result of an R, j being discovered such thatx " R # Ij , then this follows immediately. So, suppose R' is listed as a resultof the discovery of R, R', j, k1, k2 such that (1) - (4) hold. We first consider thecase where x " C. We note that |pj%x| < 2"m(k). Hence, d(f(pj), f(x)) < 2"k

from which it follows that g(x) = f(x) " R'. Now, suppose x '" C. Let x " Ij" .Hence, diam(Ij" ) < 2"h(k2). One of the endpoints of Ij" lies in R. Let y besuch an endpoint. Thus, d(g(y), g(x)) < 2"k2 and d(g(y), g(pj)) < 2"k1 . Thus,g(x) " R'.

Claim: Every rational rectangle that g(x) belongs to is eventually listed .

Suppose g(x) " R'. If x '" C, then eventually we will discover a rationalrectangle R and an integer j such that x " R # Ij , and R' will be listedsometime after that. Suppose x " C. So, x is arbitrarily close to endpoints ofIj ’s. It then follows that there exist R,R',k1, k2, and j such that (1) - (4) hold.When these are discovered, R' will be listed. *+

8 A counterexample to the e!ective converse of theHahn-Mazurkiewicz Theorem

Theorem 7 There is a computable function from [0, 1] into R2 whose rangeis not e!ectively locally connected.

Proof It follows from Theorem 3.2 of [7] that there is a computable functionf : [0, 1] $ R2 whose image is an arc, A, with the property that any com-putable map of [0, 1] onto A must retrace itself infinitely often. Furthermore,the endpoints of A are computable. Hence, there is no computable homeomor-phism of [0, 1] with A. However, it then follows from Corollary 1 that A is note!ectively locally connected. *+

Acknowledgements We thank the referees for helpful comments. We also thank LawrenceJ. Osborne for helpful conversations and Jennifer Daniel for assistance with the figures. Thethird author thanks his wife Susan for support. He also thanks Jack Lutz and the computerscience department of Iowa State University for their hospitality. His visit to Iowa StateUniversity was generously supported by a developmental leave grant from Lamar University.

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