CHARACTERIZATIONS OF SPACES OF FINITE HAUSDORFF MEASURE: 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 Mathematics 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 dimension 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 point. Lastly, thank you to my family and friends for their reassurance and unwavering confidence in me. v Contents Page 1. PRELIMINARIES 1 1.1. HAUSDORFF MEASURE 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, Topology, and Fractal 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 dimensions 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 space with metric d. Consider a real number 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 metric space. 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 inductive dimension. 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.