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CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF : ONE-DIMENSIONAL CASE

JORDAN DOUCETTE

Supervisor: Dr. Murat Tuncali

MAJOR RESEARCH PAPER

Submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in

Department of Computer Science and Mathematics Nipissing University North Bay, Ontario

c Jordan Doucette December 2012 !

Abstract. An intriguing problem that was proposed by Samuel Eilenberg and O.G. Harrold in 1941 involves the characterization of separable metric spaces, whose topological is n, for which there is a metrization so that the n- dimensional Hausdorff measure is finite. The focus of this paper will be to examine several conditions in order to characterize metric continua of finite one-dimensional Hausdorff measure. Our main theorem will consist of showing that for a metric continuum being of finite one-dimensional Hausdorff measure is equivalent to being totally regular, as well as being an inverse limit of graphs with monotone bonding maps.

iv ACKNOWLEDGEMENTS

First and foremost, I am grateful to my supervisor Dr. Murat Tuncali, for his invaluable support, instruction and encouragement throughout the preparation of this paper. Without Murat’s continual guidance this paper may never have come to fruition. Thanks are also due to my second reader, Dr. Alexandre Karassev, external examiner, Dr. Ed Tymchatyn and examination committee chair, Dr. Vesko Valov for their helpful feedback. I would also like to extend thanks the entire mathematics department at Nipissing University and visiting professors throughout the years, whose teaching has lead me to this . Lastly, thank you to my family and friends for their reassurance and unwavering confidence in me.

v Contents Page 1. PRELIMINARIES 1 1.1. and HAUSDORFF DIMENSION 1 1.2. COVERING DIMENSION 3 1.3. SOME CONTINUUM THEORY 5 2. PROBLEM and EXAMPLES 13 3. ONE-DIMENSIONAL CASE 21 3.1. EILENBERG AND HARROLD THEOREM 21 3.2. THEOREMS ON TOTALLY REGULAR CONTINUA 23 3.3. THE MAIN THEOREM 35 References 39

vi CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 1

1. PRELIMINARIES

In this chapter we provide some of the basic definitions and concepts that are needed throughout the paper. For any other definitions one may be interested in or not able to recall immediately, please refer to Edgar’s “Measure, , and Geometry” [2], Engelking’s “General Topology” [4], and Nadler’s “Contin- uum Theory: an Introduction” [9]. The main focus of this paper will be the one- dimensional case of the problem posed by Samuel Eilenberg and O.G. Harrold, Jr in [3]. Though we will not prove all of the theorems presented in this paper, those which are significant to this problem with be given in detail. All spaces considered in this paper are separable and metric.

1.1. HAUSDORFF MEASURE and HAUSDORFF DIMENSION.

Some notions of measure and dimension for certain spaces are quite intuitive, while others are not necessarily so. We will consider both the topological and Hausdorff as they relate to continua. The Eilenberg and Harrold problem involves the Hausdorff dimension of separable metric spaces. We use Edgar’s book [2] for the following definitions. 2 JORDAN DOUCETTE

DEFINITION 1.1.1 (Hausdorff Measure): Let X be a metric with metric d. Consider a s>0 a candidate for the dimension. For each !> 0, define

∞ s s H! (X, d) = inf [diam(Xi)] , i=1 ! where the infimum is over all decompositions X = X X ..., such that diam(X ) < 1 ∪ 2 ∪ i s ! for all i =1, 2, .... Then H! is a non-decreasing function of !, and we may define

s s s H (X, d) = lim H! (X, d) = sup H! (X, d). ! 0 → !>0 This number (which may be ) is called the s-dimensional Hausdorff measure ∞ of X. Since this value depends strictly on the choice of the metric d on X, we have made sure to include it in the definition, but we will often refer to this measure simply as Hs(X) when our choice of d is clear. When we have s = 1, if H1(X, d) < , we ∞ say that (X, d) is of finite linear measure.

This paper will be focusing on the one-dimensional case, so in the above definition we only need to replace s with 1 in order to understand what we will be dealing with.

DEFINITION 1.1.2 (Hausdorff Dimension): For a given set X, there is a unique ‘critical value’ s [0, ] such that: 0 ∈ ∞ Hs(X)= for all s < s ; ∞ 0

s H (X) = 0 for all s > s0.

This value s0 is called the Hausdorff dimension of the set X. We will write

s s0 = dimH X. It is possible that H (X) = 0 for all s>0, in that case dimH X = 0. Similarly, if Hs(X)= for all s>0, then dim X = . ∞ H ∞ CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 3

The concept of Hausdorff dimension can be difficult to grasp, so it may be more manageable to consider how we come to this dimension through a series of steps. Hausdorff dimension involves first covering our space, X, with arbitrarily small !- covers and taking the infimum of the sum of the diameters of the members of these

s !-covers raised to the power s (H! (X)). Then, taking the limit as ! approaches zero, s in other words, the supremum of our H! (X), we get the s-dimensional Hausdorff measure of X, and denote it Hs(X). Finally, the Hausdorff dimension is achieved by finding the infimum of the set of s values (s can be any real number, not necessarily an integer) for which the s-dimensional Hausdorff measure of X, Hs(X), is zero. That is, the smallest value of s we can take while still being able to cover X. We will give a variety of examples in Chapter 2 of spaces with different Hausdorff dimensions.

1.2. COVERING DIMENSION.

In addition to understanding Hausdorff dimension, it will also be important to know how to define dimension for an arbitrary separable . What we will define as topological dimension is the“covering dimension” originally defined by Lebesgue. First, recall that given a collection of subsets of a topological space A X, a collection is said to refine if for each element B there is an element B A ∈B A such that B A. We will also need to know that in the following definition ∈A ⊂ order m + 1 means that some point of X lies in m + 1 elements of , but no point A of X lies in more than m + 1 elements of . A 4 JORDAN DOUCETTE

DEFINITION 1.2.1: A space X is said to be finite dimensional if there is some integer m such that for every open covering of X, there is an open covering of A B X that refines and has order m + 1. The topological dimension of X is defined A to be the smallest value m for which this statement holds; we denote it by dim X.

Maybe a more intuitive way to think about topological dimension is that dim X ≤ n, where n 0, means that every point x X can be separated from any closed set ≥ ∈ F which does not contain x, by a set of dimension n 1, where the empty set is ≤ − given dimension 1. This is the same as saying that dim X n (n 0) if and only − ≤ ≥ if there exists a base for X such that dim bd(B) n 1 for every B . This is B ≤ − ∈B also known as the small . For further details on these ideas, one can refer to Engelking [4].

One theorem about the relationship between the topological dimension and the Hausdorff dimension of a space that was given by Edgar in [2] is the following.

Theorem 1.2.2 ([2], Theorem 6.2.9, p. 155) Let X be a metric space. Then dim X dim X. ≤ H

We will be referring to this fact in Chapter 2. In [3], Eilenberg and Harrold mention work done by Szpilrajn who studied the connection between the two dimensions we are considering. Among other results, Szpilrajn proved that a separable metric space X has dimension less than or equal to n if and only if there is a metrization d of X such that Hn+1(X, d) = 0. This theorem implies that if dim X = n, then for every metrization d of X, we have Hn(X, d) > 0. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 5

1.3. SOME CONTINUUM THEORY.

The spaces we are interested in will be continua, X, with the property that H1(X, d) < , where d is the metric on our space X. Since we will be talking ∞ about continua frequently, we provide some basic definitions and facts about such spaces below.

DEFINITION 1.3.1:Acontinuum is a nonempty, compact, connected, metric space. A nonempty, compact, connected subset of a continuum is called a subcon- tinuum.

A very simple example of a continuum is an arc. An arc is a continuum which is homeomorphic to the closed interval [0, 1].

Let us recall that components of a space are maximal connected subsets of that space. The symbol U denotes the closure of the set U. In [9], one can find three theorems, referred to as the ‘boundary bumping theorems’ that give some facts about how, under certain moderate conditions, a component of a set must intersect (or “bump into”) the boundary of that set. We will only provide one of these theorems, though the other two are similar in nature.

Theorem 1.3.2 (Boundary Bumping Theorem) ([9], Theorem 5.4, p. 73) Let X be a continuum, and let U be a nonempty, proper open subset of X. If K is a component of U, then K bd(U) = (equivalently, since K U and U is open, ∩ * ∅ ⊂ K (X U) = ). ∩ − * ∅ 6 JORDAN DOUCETTE

The proof of this theorem, as well as the two other boundary bumping theorems can be found in [9].

Our focus in this paper will be on one-dimensional locally connected continua with some special properties, which we will discuss next. The following concepts can be found in Whyburn [16], and Nadler [9].

DEFINITION 1.3.3: Let X be a continuum. A point x X is said to be of finite ∈ degree provided that for all !> 0 there is an uncountable family of neigbourhoods U of x such that: { α}

(i) diam(Uα)

Notice that this definition does not depend on the metric on X. Another way to state this definition is by saying that for each neighbourhood U containing our point x, there is an uncountable family of neighbourhoods of x, all lying inside of U, each with finite boundary, and for any two sets in the family the closure of one is contained in the interior of the other.

As we know, continua of finite degree are our main interest in this paper. It turns out that continua of finite degree have been studied under the name totally regular continua. This will be the term we use more so in this paper, so it is important that we give some background and details about this notion. We will start with what it means for a continuum to simply be called regular, and expand from there. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 7

DEFINITION 1.3.4: If X is a continuum and p X, then X is said to be regular at ∈ p provided there is a local base at p such that each B has finite boundary. Bp p ∈Bp If X is regular at each of its points, then we say that X is a regular continuum. i.e. there exists a basis of X so that each B has finite boundary. (Please note B ∈B that regular continua are also known under the name rim-finite.)

Every regular continuum is locally connected, and it is clear from the definition that every subcontinuum of a regular continuum is regular as well. We also have that each regular continuum is hereditarily locally connected, meaning that every subcontinuum is locally connected. One may find these facts in [9]. Some other interesting facts about hereditarily locally connected (hlc) continuum are given by Whyburn in [16]. Two in particular are that a continuum X is hlc if and only if it has no continuum of convergence ([16], (2.1), p. 89). This means that there is no non-degenerate subcontinuum A of X such that there is a sequence A ∞ of { i}i=1 subcontinua A of X such that A = lim A , A A = for each i. Secondly, X is hlc i i ∩ i ∅ if and only if the components of every subset of X forms a null family ([16], (2.6), p. 92). Recall that a family of sets is called a null family provided that for any !> 0, at most finitely many of its elements have diameter greater than !. Next we will define totally regular continuum; these are the spaces we will see examined most heavily throughout Chapter 3.

DEFINITION 1.3.5: A continuum X is said to be totally regular provided that, for each P X, there is a basis of open sets for X such that for ⊂ B each B ,P bd(B)= and B has finite boundary. ∈B ∩ ∅ 8 JORDAN DOUCETTE

Again, every subcontinuum of a totally regular continuum is totally regular, and clearly totally regular implies that a continuum is also regular. Any graph (i.e. a compact, connected 1-dimensional polyhedron) is also totally regular. Nikiel proved some interesting results about totally regular continua in [10]. These are provided in the next theorem and the corollary that follows, but first we need a couple of definitions.

DEFINITION 1.3.6: A continuum X is said to be completely regular provided X contains no nowhere dense subcontinuum.

DEFINITION 1.3.7: A continuous map f : X Y is said to be monotone provided → 1 f − (y) is connected for every y Y . ∈

DEFINITION 1.3.8: An inverse sequence is a “double sequence” X ,f ∞ of { i i}i=1 spaces X , called coordinate spaces, and continuous functions f : X X i i i+1 → i called bonding maps. If X ,f ∞ is an inverse sequence, sometimes written { i i}i=1

f1 f2 fi 1 fi fi+1 X X − X X 1 ← 2 ← ··· ← i ← i+1 ← ··· then the inverse limit of X ,f ∞ , denoted by lim(X ,f), is the subspace of the { i i}i=1 i i ←− cartesian product space i∞=1 Xi defined by

" ∞ lim(X ,f)= (x )∞ X : f (x )=x for all i . i i i i=1 ∈ i i i+1 i ←− # i=1 % $

Though we will not be proving the following theorem until Chapter 3, we mention it here along with a corollary, as they both pertain to the topic at hand. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 9

Theorem 1.3.9 ([10], Theorem 3.6, p. 131)

If (Xn,fn) is an inverse sequence such that all the spaces Xn are totally regular continua and all the bonding maps f : X X are monotone surjections, then n n+1 → n X = lim(Xn,fn) is a totally regular continuum. ←−

Corollary 1.3.10 ([10], Corollary 3.7, p. 132) If X is the inverse limit of an inverse sequence of connected graphs (completely reg- ular continua) with monotone bonding maps, then X is a totally regular continuum.

DEFINITION 1.3.11: A continuum X is said to be rectifiable if there exists a continuous mapping f([0, 1]) = X such that the path length S(f) is finite. That is,

n

S(f) = sup d(f(ti),f(ti 1)) < , − ∞ i=1 ! where d is the metric on X and the supremum is taken over all finite subsets t n { i}i=1 of [0, 1] such that 0 = t0

Throughout several of the proofs that will be given in this paper, we will deal with covers of the spaces which are under consideration. Two special types of covers that were outlined by Eilenberg and Harrold in [3] are defined below.

DEFINITION 1.3.12: A cover X = F F ... F will be called regular1 if 1 ∪ 2 ∪ ∪ k

(i) Fi is a continuum for i =1, 2, , ..., k, and (ii) F F is finite for i = j. i ∩ j * 1Note that the use of the term regular here is not to be confused with how we have used regular in our definitions regarding special types of continua. 10 JORDAN DOUCETTE

DEFINITION 1.3.13: A cover X = F F ... F will be said to be of mesh ! if 1 ∪ 2 ∪ ∪ k the diameter of each Fi is less than !.

The last little bit of key information given here is important in understanding some of the framework of various proofs given by Buskirk, Nikiel and Tymchatyn in [1], which we will have reason to look at closely.

DEFINITION 1.3.14: Let d be a metric on a space X. We say that d is convex if for every x, y X, there is a z X such that ∈ ∈ 1 d(x, z)=d(y, z)= d(x, y). 2

DEFINITION 1.3.15: A point x of a connected space X is said to be a separating point of X if X x is not connected. Such an x may sometimes be referred to as \{ } a cut point.

DEFINITION 1.3.16: A point x is said to be a local separating point of a con- nected space X if there is a connected, open set U containing x such that U x is \{ } not connected.

In [17] (Corollary 4.1, p. 15), Whyburn proved that a continuum X is of finite degree if and only if every subcontinuum of X contains uncountably many local separating points of X . This was one of the most important characterizations used by O.G. Harrold in [7] to prove two more characterizations of continua of finite degree; namely CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 11

Theorem 1.3.17 ([7], Theorem A, p. 952) A continuum X is of finite degree if and only if X is locally connected and for every pair of closed, disjoint subsets X0,X1 in X, there is a finite collection of disjoint, perfect sets (i.e. closed sets that contain no isolated points) N1,N2, ..., Nk such that any continuum K in X intersecting both X0 and X1 contains some Ni.

Theorem 1.3.18 ([7], Theorem B, p. 953) A continuum X is of finite degree if and only if for nondegenerate continua (i.e. containing more than one point) K and Ki, (i =1, 2, ...), in X, with limi Ki = K, →∞ there exists an integer n such that n∞ Ki is an . & We will end this section with a few fundamental remarks on some of the essentials of cyclic element theory. In the definitions that follow, X will be a locally connected continuum.

DEFINITION 1.3.19: A connected subset D X is said to be a cyclic element of ⊂ X provided D is a maximal subset of X that cannot be separated by a point.

If we have two elements p, q X, then the set of all points of X which separate p ∈ and q is denoted E(p, q).

DEFINITION 1.3.20: The cyclic chain from p to q is defined as the set

C(p, q)=E(p, q) p, q D , ∪{ }∪ j j ' where D is the set of all cyclic elements of X such that D (E(p, q) p, q ) { j} j ∩ ∪{ } consists of exactly two points. 12 JORDAN DOUCETTE

If Z is a connected subset of a totally regular continuum X, and p, q Z, then we ∈ define the cyclic chain from p to q with respect to Z as the cyclic chain from p to q in the subspace Z, and denote it C(p, q, Z). Analogously, E(p, q, Z) is the set of all points in the subspace Z that separate p and q, and finally C(p, q, X)=C(p, q). CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 13

2. PROBLEM and EXAMPLES

Eilenberg and Harrold [3] proposed the following general problem. In this paper though, we will be focusing on the one-dimesional case only.

The Problem: To characterize separable metric spaces X of dimension n for which there is a metrization d such that Hn(X, d) < . ∞

This problem dates back to the early 1940s, and is interesting because being able to recognize such spaces would be quite valuable. In this paper, several properties will be presented, which allow us to determine whether or not a space is of finite degree. We will be focusing on some results in papers by Eilenberg and Harrold, as well as Buskirk, Nikiel and Tymchatyn that give us various ways to check that the space in question is a continuum of finite degree, that is, a totally regular continuum. There has been a lot of research done by the authors we mention most frequently in this paper, and by others such as Whyburn, who have found several different properties of continua that are equivalent to being of finite degree. The ability to pinpoint any one of the characteristics we will discuss allows one to immediately distinguish a space as being a totally regular continuum. In [3], Eilenberg and Harrold completely solve the problem for the case where X is a continuum and n = 1. We will state their main theorem, though the proof will not be given in its entirety. Instead we will be considering some other interesting equivalent conditions for continua of finite one-dimenisional Hausdorff measure that have come about by the work of other authors. Eilenberg and Harrold do set out 14 JORDAN DOUCETTE the proof of their main theorem in [3] for those who may be interested in studying it further. Through research done since this problem was posed, there have been numerous results proven that allow us, under certain conditions for instance the case when − X is a continuum and n =1 to characterize separable metric spaces of dimension − n with finite Hausdorff measure. However, a general characterization is still an open problem. This problem is still of interest to many mathematicians, and work continues on this now seventy-year-old question.

Let us go through some examples that will give us some insight as to why this problem is an interesting one, and at the same time so complex.

Example 2.1

We will start out with simply looking at the real numbers. In the metric space R, the one-dimensional Hausdorff measure H1 coincides with the L.

To see this, first we will start with a subset A R whose diameter, r, is finite. Then ⊂ sup A inf A = r, so A is contained in a closed interval of length r, call it I . We − r obtain the Lebesgue , L(A), on A by covering A with countably many half-open intervals whose total length is small. That is,

∞ L(A) = inf (b a ) j − j j=1 ! where the infimum is taken over all countable families [aj,bj):j N of half open { ∈ } intervals with A j N[aj,bj). By one of the basic properties of Lebesgue outer ⊂ ∈ measure, since A (Ir, we have L(A) L(Ir)=r. But by Theorem 5.2.2 of [2], ⊂ ≤ 1 H is the largest outer measure satisfying (A) diam(A) for all sets A with ! M M ≤ 1 1 diam(A)

Now, if we have a half-open interval [a, b), and !> 0, we can find points a = x0 < x1 < ... < xn = b such that (xj xj 1)

diam([xj 1,xj]) = (xj xj 1)=b a − − − − j=1 j=1 ! ! 1 Therefore H ([a, b)) b a. But, again by Theorem 5.2.2 of [2], L is the largest ! ≤ − outer measure satisfying L([a, b)) b a for all half-open intervals [a, b). Therefore, ≤ − L(F ) H(F ) for all F . ≥ So, we have shown that the two outer measures coincide. Recall that a set A R ⊂ is Carath´eodory measurable if and only if L(E)=L(E A)+L(E A) for all sets ∩ \ E R ([2], p. 130), and a set A R is Carath´eodory measurable if and only if it is ⊂ ⊂ Lebesgue measurable ([2], Proposition 5.1.16, p. 130). In both cases the measurable sets are given by the criterion of Carath´eodory, so in fact, the measures L and H1 also coincide. In Proposition 6.2.3 of [2] (p. 152), Edgar proves that the Hausdorff dimension of the real R is 1, which we know is the same as its topological dimension. He also proves that the Hausdorff dimension of the two-dimensional R2 is 2. We know that this is the same as the topological dimension of the plane ([2], p. 93). In fact, it is also the case that for any n>0 the Hausdorff dimension and the topological dimension of Rn correspond.

It turns out that for many common spaces the Hausdorff and the Lebesgue mea- sures are actually identical. This is not the case in general, but it is interesting that there are certain kinds of spaces where this does occur. As a rule, the topological dimension of a space must take on an integer value, however this is not the case 16 JORDAN DOUCETTE for the Hausdorff dimension. As we mentioned earlier, the Hausdorff dimension of a space can be any non-negative real number.

The two examples that we look at next will both be spaces where the two di- mensions we are considering are indeed quite distinct. They will, however, give us an idea of the relationship between Hausdorff dimension and topological dimension. The will be an example where we exhibit the rule that the topological dimension will always be an integer, while the Hausdorff dimension can be any real number, not necessarily integer-valued. Recall from Theorem 1.2.2 that for met- ric spaces, the topological dimension is always less than or equal to its Hausdorff dimension.

Example 2.2 The Cantor set,

C = An n Z+ )∈ 1 2 where A0 = [0, 1] in R; A1 is obtained by deleting the “middle third” ( 3 , 3 ) from A0; 1 2 7 8 A2 is obtained from deleting the “middles thirds” ( 9 , 9 ) and ( 9 , 9 ) from A1, and in general ∞ 1+3k 2+3k An = An 1 , . − − 3n 3n k'=0 * + This is a familiar set that has been studied in much detail. We know that the topological dimension of C is zero because every point in C has arbitrarily small neighbourhoods whose boundaries do not intersect the set (also see [2], Proposition 3.1.3, p.81). The Hausdorff dimension of the Cantor set, however, is a little bit more interesting, and probably the most familiar set of real numbers with a non-integer Hausdorff dimension. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 17

The Cantor set is a special kind of set called self-similar. This notion is fairly intuitive and means that sets are built by pieces that are similar to the entire set at a smaller scale. Basically, zooming in on smaller and smaller regions will reveal scaled copies of the original set. In [5], the author gives a method for calculating the Hausdorff measure of a self-similar set, and we find that for the Cantor set, this

log 2 dimension is log 3 . So as we can see, the topological dimension is indeed less than the Hausdorff dimension of the Cantor set, but this does not really give us much information in regards to to the problem of Eilenberg and Harrold.

Example 2.3 The Sierpinski gasket is a slightly more difficult example, though the set is still quite a familiar one. We start with a solid of side length one, call it S0. It is then subdivided into four smaller triangles by joining the midpoints of all three sides by lines. We then remove this middle open triangle, which is rotated 180 degrees compared to the others, the set that remains we will call S1. This process is repeated by removing the open middle triangles from each of the remaining three triangles with side length 1/2. The result is S2. We continue in the same way, each time removing the middle open triangle from each of the triangles that remain from the step before. In this way we obtain a sequence Sk of sets, where each Sk+1 is a subset of Sk. 18 JORDAN DOUCETTE

The Sierpinski gasket is the limit of this sequence, and we denote it by S. Since the sequence is decreasing, this means

S = Sk. k N )∈ This is another self-similar set, and its topological dimension is one because each point of S has arbitrarily small neighbourhoods whose boundaries meet S in a set of points, which is homeomorphic to a subset of the Cantor set.

k 1 Now, it is clear that each Sk has 3 equilateral triangles with side length 2k , so the total area of each S is 3k 1 √3 . As k , this total area converges to k · (2k)2 · 4 →∞ zero. This means that the total “area” of S is zero, so area doesn’t seem to be very useful for measuring the size of the Sierpinski gasket. Area is used to measure the size of sets of dimension two. From this, it can be said that S must have dimension less than two. We encounter another issue if, instead of looking at area, we consider the length of the line segments making up the boundary of S. At each stage the total length from the step before remains, so the “total length” of S is at least 3k 3 1 , which · · 2k CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 19 goes to as k . This would indicate that the total length of S is infinite, so ∞ →∞ “length” also seems like it is not very useful to measure the size of our set. Length is used to measure the size of a set whose Hausdorff dimension is one. This seems to lead to the idea that the Sierpinski gasket has Hausdorff dimension greater than one. So now we have a seemingly reasonable supposition that the Hausdorff dimension of S must be greater than one, but also less than two. Obviously there is no integer between 1 and 2, so it is with the Hausdorff dimension that we are given a different way to consider this dilemma. It is proven in [2] that the Sierpinski gasket has Hausdorff dimension log 3 1.58. We refer the reader to Edgar’s proof that uses an log 2 ≈ . This example again shows us a space that has very different topological and Haus- dorff dimensions, and inspires one to investigate further some way to characterize the type of spaces mentioned in the Eilenberg and Harrold problem.

Example 2.4 One final example we will consider is described by Eilenberg and Harrold in [3], and shows us why one has to assume connectedness to solve the problem they posed.

If we have a compact metric space X0 that consists of a countably infinite set of points, and X is the cartesian product In X , where In is the n-dimensional , n × 0 then we can see that dim X = n but for every metrization d, Hn(X ,d)= . It is n n ∞ this example that Eilenberg and Harrold have as the one that leads to the suggestion of their problem of characterizing separable metric spaces of dimension n such that there is a metrization d such that Hn(X, d) < . ∞ 20 JORDAN DOUCETTE

We will consider only the one-dimensional case, and some work that has been done on this problem which has moved us closer to being able to characterize such spaces. In the one-dimensional case, the example noted above would involve something like a sequence limiting to a point, and the one-dimensional cube, that is, the [0, 1], i.e. X = 1 : n =1, 2, 3, ... 0 [0, 1]. 0 { n } ∪ { } × , - Through these examples, it seems as though the topological and Hausdorff di- mensions coincide only when we have “nice”, relatively simple spaces. Even a slight change can make a substantial difference and cause something like the Hausdorff mea- sure to become infinite. What kind of spaces fall into which category, what small changes in parameter can cause a measure to go from finite to infinite, and how we can easily identify these spaces is the motivation behind the interesting problem we are dealing with. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 21

3. ONE-DIMENSIONAL CASE

3.1. EILENBERG AND HARROLD THEOREM.

Let I denote the unit interval, and 0 t 1. If f(X)=Y , let φ(f) denote the ≤ ≤ 1 set of all points y Y such that f − (y) is a finite set. Now we will state the main ∈ theorem of Eilenberg and Harrold [3].

Theorem 3.1.1 ([3], Theorem 3, p. 139) For every continuum X, the following conditions are equivalent:

(A) X can be imbedded in the Hilbert cube Qω so as to have a finite linear measure. (B) X has a homeomorphic image of finite linear measure. (C) X is of finite degree.

(D) Given any two disjoint, closed subsets X0,X1 in X, there is a continuous mapping f(X)=I such that (d ) f(x)=i, for x X ,i=0, 1; 1 ∈ i (d2) φ(f) is uncountable.

(E) Given any two closed, disjoint subsets X0,X1 in X, there is a continuous mapping f(X)=I such that (e ) f(x)=i, for x X ,i=0, 1; 1 ∈ 1 (e ) for every irrational t [0, 1],t φ(f). 2 ∈ ∈ 22 JORDAN DOUCETTE

(F) Every subcontinuum of X contains uncountably many local separating points of X.

(G) X is locally connected and for every pair of closed, disjoint subsets X0,X1 in

X, there is a finite collection of disjoint, perfect sets N1,N2, ..., Nk such that

any continuum in X intersecting both X0 and X1 contains some Ni.

(H) Given a sequence X0,X1,X2, ... of subcontinua such that

lim Xi = X0 i →∞

there is an integer n for which

∞ Xi i=n ) is uncountable.

This theorem was proved by Eilenberg and Harrold in [3] and has inspired much research since they published this work in 1941. Though we will not give the proof of this theorem, we will use parts of it to format and prove a smaller theorem that will allow us to provide the characterization of continua X with metric d such that their one-dimensional Hausdorff measure is finite, i.e. H1(X, d) < . ∞ CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 23

3.2. THEOREMS ON TOTALLY REGULAR CONTINUA.

We will begin with a lemma that is used in the proof of the main theorem of [1] and that will be essential in understanding several of the results presented in this paper.

Lemma 3.2.1 ([1], Lemma 2, p. 321) If X is a totally regular continuum, and !> 0, then there exists a monotone decom- position of X such that diam(K)

Proof: First off, we will note that if is a decomposition of a totally regular continuum G X into subcontinua. Since X is totally regular, it is rim-finite ([11], Proposition 7.4, p. 47), and thus hereditarily locally connected ([16], Chapter V, 5, p. 99). Then § for every open cover of X, the set G : G ! U is finite for any U (i.e. U { ∈G } ∈U G is a null-family) ([16], (2.6), p. 92). Hence is upper semi-continuous. That is, the G collection G : G U is open for each open set U in X. { ∈G ⊂ } Let be a countable basis for X consisting of connected sets. For every V , P ∈P let DV be a countable dense subset of V ; i.e. DV = V . Take any connected subset V X and any two points a, b V . Let ⊂ ∈

E&(a, b, V )= a, b E(a, b, V ). { }∪

0 This set E&(a, b, V ) is compact ([16], (4.2), p. 51). Let F (a, b, V ) denote the set of all condensation points of E&(a, b, V ). Condensation points are those points for which 24 JORDAN DOUCETTE

0 every one of their neighbourhoods is uncountable. Then E&(a, b, V ) F (a, b, V ), i.e. \ the non-condensation separating points, will be a countable set; and if there exists at least one condensation point, that is, F 0(a, b, V ) = , then F 0(a, b, V ) is compact * ∅ and has no isolated points. Obviously condensation points cannot be isolated points by nature of their definition; neighbourhoods of condensation points consist of an uncountable number of points. Put

F (a, b, V )=F 0(a, b, V ) a, b . \{ }

That is, F (a, b, V ) is the set of condensation points of E&(a, b, V ) not including a or b. Let

A = E&(a, b, V ) F (a, b, V ):V and a, b D , 1 { \ ∈P ∈ V } ' and note that A is countable because E&(a, b, V ) F (a, b, V ) and are both countable 1 \ P sets. Put

A = x X : x is a local separating point of X and ord X>2 . 2 { ∈ x }

Recall that ordxX>2 means that in a small neighbourhood around x, the mini- mum number of points of the set X that intersect the boundary of this neighbourhood is 2. Because all but a countable number of the local separating points of X are points of order 2 ([16], Theorem 9.2, p. 61), we know that A2 is also countable. Now, let A = A A . 1 ∪ 2 Since X is a totally regular continuum, we can find a basis for X such that if Q U , then U is a connected open set, U = int(U), bd(U) is finite, and bd(U) A = ∈Q ∩ . Note that because the boundary of U is finite for U , if we have a point ∅ ∈Q CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 25 x bd(U), then we can find a neighbourhood around x so small that it contains no ∈ other boundary points of U, hence x is a local separating point of X. Let T , ..., T be a finite covering of X such that each T and diam(T ) < ! { 1 n} i ∈Q i 3 for i =1, ..., n. Put B = bd(T ) ... bd(T ). Since T , bd(T ) is finite for 1 ∪ ∪ n i ∈Q i every 1 i n, clearly B itself is finite. Additionally, if x B, then because ≤ ≤ ∈ bd(T ) A = for i =1, ..., n, we know that bd(B) A = ; we also have that x/A i ∩ ∅ ∩ ∅ ∈ 1 and x/A , so x is a local separating point of X with ord X = 2. ∈ 2 x For all x B, let U be connected open sets with the following properties: ∈ x

(i) x separates Ux; ! (ii) diam(Ux) < 3 ; (iii) U U = provided x = y. x ∩ y ∅ * Now, take any point x B and let V be such that x V V U . Next, ∈ x ∈P ∈ x ⊂ x ⊂ x we let Wx, Wx& be two connected, open subsets of X such that

x W W W & W V ∈ x ⊂ x ⊂ x ⊂ x& ⊂ x

with the boundaries of W , W & as follows: bd(W )= a ,b and bd(W & )= a& ,b& ; x x x { x x} x { x x} as we know the boundary sets are only two points since ordxX = 2. Notice that it must be the case that in Wx, x separates ax and bx, as well as separating ax& and bx& in Wx& because both sets are connected and have only two-point boundary sets, so we must pass through x on the way between these boundary points. Thus, we know that x E&(a ,b ,W ). ∈ x x x Recall that D is a countable, dense subset of V . There exist points a∗,b∗ Vx x x x ∈ D W such that C(a ,b ,W ) C(a∗,b∗,V ). It is clear that all the points that Vx \ x x x x ⊂ x x x separate ax from bx in Wx also separate ax∗ from bx∗ in Vx, as well as ax and bx 26 JORDAN DOUCETTE themselves; that is E&(a ,b ,W ) E&(a∗,b∗,V ). Hence x x x ⊂ x x x

0 0 F (a ,b ,W ) a ,b = E&(a ,b ,W ) F (a∗,b∗,V ) a ,b . x x x \{ x x} x x x ∩ x x x \{ x x}

We have already noted that x/A and that x E&(a ,b ,W ) E&(a∗,b∗,V ), ∈ 1 ∈ x x x ⊂ x x x 0 so it follows that x F (a∗,b∗,V ). Therefore, with x E&(a ,b ,W ) and x ∈ x x x ∈ x x x ∈ 0 0 0 F (a∗,b∗,V ), we have that x F (a ,b ,W ), ergo F (a ,b ,W ) is not empty. x x x ∈ x x x x x x 0 This tells us that F (ax,bx,Wx) is compact, non empty and has no isolated points.

0 On that account, F (ax,bx,Wx) is uncountable, as is E&(ax,bx,Wx). By [1] (Lemma 1, p.321), there is a monotone decomposition of W such that Gx x W / is homeomorphic to [0, 1] and there are distinct elements M ,N of such x Gx x x Gx that a M and b N . Moreover, since is a decomposition of W and x ∈ x x ∈ x Gx x W U , by recalling property (ii) of U , we see that if K , then x ⊂ x x ∈Gx ! diam(K) diam(W ) diam(U ) < . ≤ x ≤ x 3

Let

& = K : K is a component of T T W , 1 i n G i\ m ∪ x ≤ ≤ # .m

Note that for those components K &, bd(K) x B ax,bx because of the ∈G ⊂ ∈ { } choice of the finite covering T1, ..., Tn . Since B is finite,( it follows that & is finite { } G as well.

For each K &, let ∈G

S = K M : a bd(K),x B N : b bd(K),x B . K ∪ { x x ∈ ∈ }∪ { x x ∈ ∈ } ' ' Then SK is a continuum and because each of the three pieces making up SK have ! diameter < 3 , we get diam(SK )

Finally, put

= S : K & ( M ,N ). G { K ∈G }∪ Gx\{ x x} x B '∈ This is our desired decomposition of X. G !

In [1], Buskirk, Nikiel, and Tymchatyn gave the following theorem as their main result.

Theorem 3.2.2 ([1], Theorem 3, p. 323)

If X is a totally regular continuum, then there exists an inverse sequence (Xn,fn) such that:

(i) each Xn is a connected graph; (ii) each f : X X is a monotone surjection; and n n+1 → n (iii) X is homeomorphic to lim(Xn,fn). ←−

Proof: This proof makes use of Lemma 3.2.1. According to this lemma, there is a mono- tone decomposition of X such that diam(K) < 1 for each K , and the G1 ∈G1 quotient space X = X/ is a graph. The family = K : diam(K) 1 , 1 G1 F1 { ∈G1 ≥ 2 } that is the set of components of whose diameter is 1 , is finite. We know this G1 ≥ 2 because is a null family, so there can only be finitely many sets greater than a G1 certain size. By the lemma, if K , then there exists monotone decomposition ∈F1 of K such that diam(L) < 1 for each component L and the quotient space GK 2 ∈GK K/ is a graph. Let GK =( ) . G2 G1\F1 ∪ GK K 1 '∈F 28 JORDAN DOUCETTE

So by removing the components of with diameters 1 and adding decom- G1 ≥ 2 positions of these components into small pieces, we have that is a monotone G2 decomposition of X such that diam(L) < 1 for each component L . Since is 2 ∈G2 F1 finite, only finitely many points of are replaced by our little decomposition graphs, G1 so the quotient space X = X/ is again a graph. 2 G2 In the analogous way, we construct monotone decompositions ,n=1, 2, ..., of X Gn such that refines , diam(L) < 1 for each component L and X = X/ Gn+1 Gn n ∈Gn n Gn is a graph. i.e. = K : diam(K) 1 , then let =( ) and 2 2 3 3 2 2 K 2 K F { ∈G ≥ } G G \F ∪ ∈F G X3 = X/ 3 is a graph. ( G = K : diam(K) 1 , then let =( ) and X = X/ 3 3 4 4 3 3 K 3 K 4 4 F { ∈G ≥ } G G \F ∪ ∈F G G is a graph. ( . . = K : diam(K) 1 , then let =( ) and n n n n+1 n n K n K F { ∈G ≥ } G G \F ∪ ∈F G Xn+1 = X/ n+1 is a graph. ( G Let g : X X denote the quotient map, n =1, 2, .... Since refines , there n → n Gn+1 Gn is a unique map f : X X such that g = f g . This f is a monotone n n+1 → n n n ◦ n+1 n surjection. Let Y = lim(X ,f ) and let g : X Y denote the map induced by n n → ←− the quotient maps gn,n=1, 2, .... Then this g is a continuous surjection. Since diam(L) < 1 for each L , it follows that the maps g ,n=1, 2, ... separate n ∈Gn n points of X. Hence, g is one-to-one, and thereby a because Y is compact. So, X is homeomorphic to Y and thus is an inverse limit of graphs with monotone bonding maps. ! CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 29

In [1], the authors use this result to provide a proof of a theorem, which, given a totally regular , allows us to determine a convex metric such that we satisfy condition (C) of Theorem 3.1.1. This theorem is as follows, and it will be proved in detail as it is important to the problem we wish to solve in this paper.

Theorem 3.2.3 ([1], Theorem 4, p. 324) If X is a totally regular curve, then there exists a convex metric d on X such that(X, d) is of finite linear measure.

In the proof of this theorem, the main result from [1], Theorem 3.2.2 above, is applied to give a proof that is simpler than one previously given by O.G. Harrold. Other proofs of this theorem have been also been obtained by different authors.

Before the proof is given, we will note that a polygonal arc is an arc whose end- points are connected by finitely many straight line segments. For each polygonal arc

ζ in R3, we let l(ζ) denote the length of ζ.

Proof of Theorem 3.2.3:

We let X = lim(Xn,fn), where each Xn is a connected graph and every fn : ←− X X is a monotone surjection, according to Theorem 3.2.2. At each stage n+1 → n when constructing our decomposition, we dealt with only finitely many points, so instead of dealing with all of them at once, we can do them one at a time. In doing it this way, we may assume that for each n, there is exactly one point x = xn X 0 ∈ n 1 such that more than one point in Xn+1 maps to it. i.e. fn− (x) is nondegenerate for exactly one point x = xn X for each n. The case where X itself is a graph is 0 ∈ n trivial because we would not need to worry about doing the decomposition if our 30 JORDAN DOUCETTE space was a graph. Furthermore, we may assume that X = x1 , and, for n = 1, 2, 1 { 0} ..., the following conditions are satisfied:

3 3 (1) Xn R and each subarc of Xn is a polygonal arc of R , ⊂ (2) if ζ X is an arc such that xn / f (ζ), l(ζ)=l(f (ζ)), and ⊂ n+1 0 ∈ n n 1 n (3) for each triangulation of f − (x ), we have K n 0 mn 1 l(ζ ) < i 2n i=1 ! 1 n where ζ , ..., ζ is an enumeration of all edges of (f − (x ), ). { 1 mn } n 0 K For u, v X , let ∈ n

d (u, v) = min l(ζ):ζ X is an arc with end points u and v . n { ∈ n }

We can do this because between each pair of points there are at most finitely many arcs, so we are able to find the minimum. A metric d on a compact space X is convex if and only if, for any pair of points x, y X, x = y, there is an arc ζ X with endpoints x and y such that ζ is isometric ∈ * ∈ to the interval [0,d(x, y)] in the reals [1]. So clearly dn is a convex metric on Xn. Moreover, for each triangulation of X , we have H1(X ,d )= kn l(ζ ), where L n n n i=1 i ζ1, ..., ζk is an enumeration of all edges of (Xn, ). Because of the0 way each Xn is { n } L constructed, an inductive argument shows that, by (2) and (3),

1 1 1 H (Xn,dn) < + ... + n 1 < 1 2 2 − for each n>1.

Let s =(s1,s2, ...) and t =(t1,t2, ...) be any pair of points of X. Note that

1 d (s ,t ) d (s ,t )

Let

d(s, t) = sup d (s ,t ):n =1, 2, ... . { n n n } Then d(s, t) 1 + 1 + ... = 1. Hence, d is a metric on the set X. Moreover, the ≤ 2 4 space (X, d) is compact (being the inverse limit of compact spaces). Notice that each projection g :(X, d) (X ,d ) is continuous (because d (s ,t ) d(s, t)). It n → n n n n n ≤ follows that d induces the original topology on X. Further, we can easily verify that H1(X, d) = sup H1(X ,d ) 1. { n n }≤ Since Hausdorff measure depends on what metric we chose, how we defined d is essential to consider. We must use the metric to find the measure H1(X, d), and because d is defined in such a way that it is the supremum of all the dn’s and the measure of each dn is defined as the summation of the lengths of the edges in the

1 kn triangulation of Xn (H (Xn,dn)= i=1 l(ζi)), we must take the supremum of all 1 1 these summations. And we already0 know that H (Xn,dn) < 1, and so H (X, d)= sup H1(X ,d ) 1. { n n }≤ Now we only need to show that our metric d is convex.

Let x =(x ,x , ...),y =(y ,y , ...) X. For each positive integer k, there is a 1 2 1 2 ∈ point zk =(zk,zk, ...) X such that 1 2 ∈ 1 d (zk,x )=d (zk,y )= d (x ,y ) k k k k k k 2 k k k since we know that each dk is a convex metric. Because each dk is convex, we can

find a midpoint of each coordinate pair (xk,yk) of (x, y). (Note that this midpoint is not necessarily unique since zk X = lim(X ,f )) ∈ n n ←− i.e.

1 1 1 1 1 z =(z1,z2,z3, ...) where z1 is a midpoint of (x1,y1) in d1 32 JORDAN DOUCETTE

2 2 2 2 2 z =(z1, z2,z3, ...) where z2 is a midpoint of (x2,y2) in d2 3 3 3 3 3 z =(z1,z2, z3, ...) where z3 is a midpoint of (x3,y3) in d3 . .

k k k k k z =(z1 ,z2 , ..., zk, ...) where zk is a midpoint of (xk,yk) in dk. k k Since X is compact, there exists a convergent subsequence z i ∞ of z ∞ . Put { }i=1 { }k=1 ki limi z = z =(z1,z2, ...). →∞ Let !> 0. There is a positive integer i such that for each i i , we have d(zki ,z) < ! ≥ ! ki !, and since d is the supremum of all the dn’s, we have dn(zn ,zn)

1 distance between points is not greater than k 1 . So, provided i i!, 2 i− ≥ 1 d(z, x) d(z, zki )+d(zki ,x) ! + d (zki ,x )+ . ki ki ki k 1 ≤ ≤ 2 i−

Hence,

ki 1 d(z, x) ! + lim dki (zk ,xki ) + lim i i i ki 1 ≤ →∞ →∞ 2 − and since d (zki ,x )= 1 d (x ,y ) ki ki ki 2 ki ki ki 1 d(z, x) ! + lim dk (xk ,yk ) i i i i ≤ 2 →∞ So, 1 d(z, x) ! + d(x, y) ≤ 2 Since ! was chosen arbitrarily, we can say that d(z, x) 1 d(x, y), and analogously, ≤ 2 we can show that d(z, y) 1 d(x, y). Since d(x, y) d(x, z)+d(z, y) by the trian- ≤ 2 ≤ gle inequality, and d(x, z)+d(z, y) 1 d(x, y)+ 1 d(x, y)=d(x, y), it follows that ≤ 2 2 CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 33

1 d(x, y)=d(x, z)+d(z, y) and so we must have that d(z, y)=d(z, x)= 2 d(x, y). Thus d is a convex metric. !

Another theorem that will be important when proving our main theorem is the following, which was stated in Chapter 1 as Theorem 1.3.8, but we restate here along with the proof.

Theorem 3.2.4 ([10], Theorem 3.6, p. 131)

If (Xn,fn) is an inverse sequence such that each Xn is a totally regular continuum and each bonding map f : X X is a monotone surjection, then X = lim(X ,f ) n n+1 → n n n ←− is a totally regular continuum.

Proof:

We know that each Xn is compact, connected and locally connected. Theorem 6.1.20 of [4] (p. 355) tells us that the limit of an inverse system of continua is a continuum and the inverse limit of locally connected continua with monotone bonding maps is locally connected ([6], Corollary 2, p. 414). Hence, X is a locally connected continuum. Let Q be a countable subset of X. For n =1, 2, ..., let g : X X n → n denote the projection map. Then for each n =1, 2, ..., gn is a continuous monotone

1 surjection, and g = f g . Let P = y X : g− (y) is nondegenerate for each n n ◦ n+1 n { ∈ n n } n.

Suppose that for some positive integer n, Pn is uncountable. Hence, there is

1 an !> 0 such that A = y P : diam(g− (y)) ! is an uncountable set. { ∈ n n ≥ } 1 There exists an integer m>nsuch that diam(g− (z))

3.3. THE MAIN THEOREM.

To form our main theorem, we combine different parts of the theorems presented in [3] and [1] to give some conditions as they apply to the one-dimensional case of continua of finite Hausdorff measure.

Theorem 3.3.1 Suppose X is a continuum, then the following are equivalent:

(i) X admits a metric whose one-dimensional Hausdorff measure is finite; (ii) X is totally regular; (iii) X is an inverse limit of graphs with monotone bonding maps.

We set our attention on these three items in particular, because for the one- dimensional case, graphs play an important role in construction of the spaces we are interested in.

Proof: (i) (ii) → Let X be a continuum which admits a metric d with finite one-dimensional Haus- dorff measure. The fact that X is totally regular is a consequence of the following.

Claim 1 ([3], Theorem 2, p. 139): If H1(X) < there is a continuous mapping ∞ f([0, 1]) = X such that the path-length S(f) 2 H1(X) diam(X). ≤ · −

First we will prove this more general statement 36 JORDAN DOUCETTE

Claim 2 ([3], Theorem 2a, p. 141): Let X be a continuum such that its one- dimensional Hausdorff measure is finite, and let A X be an arc with endpoints ⊂ a0,a1. There is a mapping f([0, 1]) = X such that f(0) = a0,f(1) = a1 and S(f) 2 H1(X) H1(A). ≤ · −

In the case that our continuum X is a tree (i.e. a connected graph containing no closed curve) we can establish this by induction with respect to the number of ramification points of X. That is, the number of vertices in X with degree greater than 1. For details on how such a continuous surjection can be formed in this case, please refer to Ward ([15], Lemma 5, p. 371). To show this for X in general, we first start by realizing that since H1(X) < , ∞ X is a locally connected continuum and therefore there is a sequence of trees T { i} such that

A T X; lim T = X. ⊂ i ⊂ i

Hence we get a sequence of mappings fi([0, 1]) = Ti such that fi(0) = a0,fi(1) = a1, and

S(f ) 2 H1(T ) H1(A) 2 H1(X) H1(A). i ≤ · i − ≤ · − Since the functions f have a uniformly bounded variation there is a subsequence { i} f that converges uniformly to a mapping f([0, 1]) X which will satisfy all the { ik } ⊂ requirements of Claim 2. Again, one may wish to refer to Ward [15] for details. Then, to prove Claim 1, notice that because H1(X) < , X is arcwise connected ∞ and we can choose an arc A so that diam(A)=diam(X). Since diam(A) H1(A), ≤ we have that 2 H1(X) H1(A) 2 H1(X) diam(A), and so we have shown · − ≤ · − Claim 1. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 37

So we have shown that X is rectifiable, and since the inequality H1(X) S(f) ≤ is readily established from the definitions, it follows that if X is rectifiable, then H1(X) < . Then, by the Corollary of Eilenberg and Harrold’s Theorem 1 ([3], p. ∞ 138) if we fix a point x X and f(x)=d(x, x ), then because H1(X) < and 0 ∈ 0 ∞ x X, the set of points whose distance from x is exactly t, 0 ∈ 0 S(t)= x : d(x, x )=t { 0 } is finite for almost all t. Please note that B (x ) has finite boundary and the set t :0 t ! is uncount- t 0 { ≤ ≤ } able. Hence, Bt(x0) 0 t ! forms an uncountable collection of nested neighbourhoods { } ≤ ≤ with finite boundaries. Then for any !> 0 and any p X, it is not difficult to es- ∈ tablish the following: there exists an uncountable family of neighbourhoods U of { α} p such that

(i) for each α, diam(Uα)

Since this is true for all points in our set X, we have, by the definition given by Eilenberg and Harrold [3], that X is totally regular.

(ii) (iii) → This step is a consequence of Theorem 3.2.2, which we have already proven, so we will not give it here.

(iii) (i) → In the first part of the proof of Theorem 3.2.3, we began by letting X be the inverse limit of connected graphs whose bonding maps were monotone surjections. 38 JORDAN DOUCETTE

From here, we went on to show that our space had finite one-dimensional Hausdorff measure. Not only that, but our space X also admits a convex metric. Hence, we have already proven this step within the proof of Theorem 3.2.3, so we are done.

So we have shown (i) (ii) (iii) (i), hence all three statements are equiva- → → → lent, and we have completed the proof of our main theorem. !

We have now seen several ways to characterize totally regular continua. Interest- ingly enough, in the one-dimensional case this means the space has finite Hausdorff measure, and that it is the inverse limit of graphs with monotone bonding maps. Moreover, in the one-dimensional case, such spaces admit a convex metric so that the measure remains finite. CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE 39

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