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DIMENSION THEORY BY DANIELLE WALSH A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for the Degree of MASTER OF ARTS Mathematics May 2014 Winston-Salem, North Carolina Approved By: Sarah Raynor, Ph.D., Advisor Jason Parsley, Ph.D., Chair Brian Pigott, Ph.D. Acknowledgments There are many people who helped to make this thesis possible. First I would like to thank my family and friends, specifically my parents, for always being there for me and supporting me throughout my academic career. Next I would like to thank Dr. Raynor for all the time and direction she has given me throughout this process. I greatly appreciate how she has put up with me during her year on sabbatical. Moreover, I would also like to thank Dr. Parsley for tolerating me through all the classes we have had together, and even adding an extra class to his already extremely busy schedule. Additionally, I would like to thank him for serving as one of my thesis committee members. Furthermore, I would like to express my gratitude for Dr. Pigott for letting me bully him into teaching measure theory. I also really appreciate him serving as one of my thesis committee members. I deeply appreciate all of the professors at Wake Forest for being an important part of my mathematical journey. ii Table of Contents Acknowledgments . ii Abstract . iv Chapter 1 Introduction . 1 1.1 Basic Definitions and Notation . .3 1.1.1 Topology . .3 1.1.2 Measure Theory . .5 Chapter 2 Dimension . 8 2.1 Topological Dimension . .8 2.1.1 Small Inductive Dimension . .8 2.1.2 Large Induction Dimension . 12 2.2 Hausdorff Dimension . 20 Chapter 3 Embedding Theorems . 33 3.1 The Whitney Embedding Theorem . 33 3.2 The Van Kampen-Flores Theorem . 37 Chapter 4 Conclusion . 51 4.1 Further Study . 51 Bibliography . 52 Curriculum Vitae . 54 iii Abstract Danielle Walsh We explore the concept of dimension using mathematical tools. We start by providing definitions, examples, and basic properties for two types of topological dimension: small inductive dimension and large inductive dimension. We then repeat this process to investigate a type of fractal dimension called Hausdorff dimension. We then focus on embedding n-dimensional topological objects into Euclidean space. This discussion brings up three main theorems. The first theorem states that any n-dimensional topological object can be embedded into R2n+1. This is a very well known result and will be used to spark the conversation for the next two theorems. The next theorem was made famous by Hassler Whitney in 1944 and states that any smooth n-dimensional manifold may be embedded into R2n. The last theorem is called the Van Kampen-Flores theorem, and it states that in general, the best bound for an embedding of an n-dimensional object into Euclidean space is 2n + 1. We compare these theorems and why particular hypotheses are necessary to ensure these embeddings exist. iv Chapter 1: Introduction Dimension is a very broad topic that most mathematicians use without thinking about the details. Dimension can lead to the study of fractals, embeddings, and many other theories in math. One simple question can lead you down a path of combinatorial topology that you think will never end. Other ideas lead to the basics of measure theory or metric space topology. A. R. Pears states that dimension theory \reveals the properties of dimension functions", with a dimension function defined as a function that takes the class of topological spaces to the nonnegative extended integers and function values are topo- logical invariants [10]. Henri Poincar´efirst proposed defining dimension inductively in 1912 [10]. The next year Luitzen Brouwer acted on Poincar´e'sproposition and rigorously defined inductive dimension [10]. He defined what is now known as small inductive dimension. This is the definition of topological dimension that is the most intuitive, and we encounter basic properties and examples of this topological dimension. We then define large inductive dimension and come across similar properties and examples. The biggest result of this section is that for separable metric spaces, the small inductive dimension and large inductive dimension coincide. This section is guided by [2]. Although useful, dimension functions with a codomain of the nonnegative ex- tended integers are limited. Theorem 2.1.1 states that the Cantor set has small inductive dimension 0, while Lemma 2.1.1 states that finite point sets also have small inductive dimension 0. The Cantor set has much more complex structure than any finite point set, but we are unable to deduce this fact from a topological dimension function. 1 Broadening the range of a dimension function to the positive extended real num- bers allows for further distinction between sets. In section 2.2 we define a new dimen- sion function called Hausdorff dimension that is based on Hausdorff measure. Finite point sets still have dimension zero, while the Cantor set has Hausdorff dimension log 2= log 3 [2]. This section is also guided by [2]. The ideas behind topological dimension follow us to the next chapter on embedding theorems. Although this is a slight detour from defining types of dimensions, the transition into embedding theorems is a smooth one. We can compute the dimension of simple geometric objects, but an interesting question to note is how can these objects be realized in Euclidean space. This question leads us to embeddings. The first embedding theorem encountered comes from Hurewicz and Wallman. This theorem states that any arbitrary n-dimensional object (in the topological sense) can be embedded into R2n+1 [5]. This is a very well known result, and is proved many different ways (see [7], [5], [8], [4], [1]). We try to make this bound on embedding objects into Euclidean spaces more exact by confronting the Whitney embedding theorem. This theorem states that n-dimensional smooth manifolds can be embedded into R2n. We use Hassler Whitney's original paper along with [1] to direct us through this result. Even though the Whitney embedding theorem gives us a better bound, it excludes certain topological objects. The Van Kampen-Flores theorem states that n-skeletons of (2n + 2)-dimensional simplices, which are n-dimensional, cannot be embedded into R2n. These n-skeletons are not manifolds and need the extra dimension described by the first embedding theorem to be realized in Euclidean space. This section is guided by [7] through the means of topological combinatorics and is supplemented by [6]. 2 1.1 Basic Definitions and Notation Dimension theory is a subject that is dense with ideas from point-set topology and measure theory. The main background information from these areas are listed here for reference. All of these basic definitions and notation come from [2], [8], and [12]. 1.1.1 Topology Definition 1. A topology on a set X is a collection τ of subsets of X such that (i) both X and Ø are in τ, (ii) the finite intersection of elements in τ is in τ, and (iii) the arbitrary union of elements in τ is in τ. Definition 2. If X is a set, then a basis for a topology on X is a collection, B, of subsets of X such that (i) for all x 2 X, there exists B 2 B such that x 2 B, and (ii) if x 2 B1 \ B2 for B1;B2 2 B, then there exists B3 2 B such that x 2 B3 and B3 ⊂ B1 \ B2. Definition 3. A metric space is a pair (X; d) where X is a set and d : X ×X ! R+ is a function with the following properties: (ia) d(x; y) ≥ 0 for all x; y 2 X (ib) d(x; y) = 0 if and only if x = y (ii) d(x; y) = d(y; x) for all x; y 2 X (iii) d(x; z) ≤ d(x; y) + d(y; z) for all x; y; z 2 X The function d is called a metric. 3 Let (X; d) be a metric space. Then the topology generated by the metric d is the topology whose basis is the collection of the open balls Br(x) := fy 2 X j d(x; y) < rg: Definition 4. A topological space X is called a Hausdorff space if for each pair x; y of distinct points of X, there exist neighborhoods U and Y of x and y respectively such that U \ V = Ø. Lemma 1.1.1. All metric spaces are Hausdorff spaces. 1 Proof. Let (X; d) be a metric space. Let x; y 2 X with x 6= y. Let r = 3 d(x; y). Then x 2 Br(x), y 2 Br(y) and Br(x) \ Br(y) = Ø. Definition 5. Let X and Y be topological spaces. A function f : X ! Y is contin- uous if for each open set V ⊂ Y , the set f −1(V ) is open in X. Definition 6. A function h : X ! Y is a homeomorphism if h is bijective, contin- uous, and has a continuous inverse. A property preserved by every homeomorphism is called a topological property. Definition 7. A function f : X ! Y is regular if every non-zero element is mapped to a non-zero element. Definition 8. An embedding is a continuous function f : X ! Y such that f^ : X ! f(X) is a homeomorphism. We say that X embeds into Y . Definition 9. A subset A of a space X is dense in X if the closure of A equals the set X.
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