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Tomáš Ye Rectangles Inscribed in Jordan Curves BACHELOR THESIS Tom´aˇsYe Rectangles inscribed in Jordan curves Mathematical Institute of Charles University Supervisor of the bachelor thesis: doc. RNDr. Zbynˇek S´ır,Ph.D.ˇ Study programme: Mathematics Study branch: General mathematics Prague 2018 I declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act. In ........ date ............ signature of the author i Title: Rectangles inscribed in Jordan curves Author: Tom´aˇsYe Institute: Mathematical Institute of Charles University Supervisor: doc. RNDr. Zbynˇek S´ır,Ph.D.,ˇ department Abstract: We will introduce quotients, which are very special kinds of continuous maps. We are going to study their nice universal properties and use them to formalize the notion of topological gluing. This concept will allow us to define interesting topological structures and analyze them. Finally, the developed theory will be used for writing down a precise proof of the existence of an inscribed rectangle in any Jordan curve. Keywords: Jordan curve, rectangle, topology ii I would like to thank my supervisor doc. RNDr. Zbynˇek S´ır,Ph.D.ˇ for helping me willingly with writing of this bachelor thesis. It was a pleasure working under his guidance. iii Contents Introduction 2 1 Topological preliminaries 3 1.1 General topology . .3 1.2 Topological gluing . .6 1.3 Special quotient spaces . 14 2 Rectangles and curves 21 2.1 Jordan curves . 21 2.2 Rectangles inscribed in curves . 25 2.3 Remarks . 31 Conclusion 33 Bibliography 34 List of Figures 35 1 Introduction The theory of curves has been a classical mathematical discipline for centuries. One of the reasons for that is that curves naturally show up in many areas of mathematics, for instance in differential geometry, algebraic geometry, complex analysis, and so on. What makes the study of curves both fulfilling and depressing is the fact that most of the problems involving them are very hard. No exception to this is the inscribed rectangle problem, on which we will take a close look in this thesis. In 1977, mathematician Vaughan proved that every Jordan curve in the plane contains an inscribed rectangle. To do this, he used a gluing argument combined with the fact that projective plane cannot be embedded into three dimensional Euclidean space. The essential trick of the proof is that every simple closed curve has a hidden topological structure homeomorphic to the Mobius strip. To make this notion precise, one has to formalize the concept of topological gluing. This is best done via quotients, which are special kinds of surjective topological maps that naturally preserve continuity of other functions. However, even with the machinery of quotient maps, there are still some technical subtleties left that need to be resolved to make Vaughan’s argument bulletproof. The purposes of the thesis is to prepare all the theoretical arsenal needed for making Vaughan’s argument rigorous and then precisely proving the inscribed rectangle theorem. The entire content of the paper is divided into two chapters. The first chapter is purely topological whereas the second one is focused on curves and on the final proof. Starting off as elementary as possible, assuming onlythe basics of general topology, we are going to define quotients and study how they interact with other maps. We will see that any projection induced by an equiv- alence relation is a quotient which will be the key for understanding topological gluing. In that regard, we are going to discover some general truths about the re- lationship between quotients and equivalences. These results will then enable us to comfortably work with concrete topological spaces and prove useful identities. The second chapter will begin with some basic properties of curves followed by intuitive, yet extremely hard to prove, results like the theorems of Jordan and Schoenflies. In the last section, there can be found the desired proof ofthe inscribed rectangle theorem together with reasoning why the use of topology in the proof was necessary. 2 1. Topological preliminaries The following section will serve as an intro to the topological notions we will use in our proofs. Although later we will be working exclusively with metric spaces and quotients of metric spaces, there is no reason to limit ourselves to these special cases right now since most of the theory we will use works in general and is actually easier to get a grasp on when done generally. The basic knowledge of general topology including metric spaces is assumed. Throughout the paper, we will use standard mathematical notation. 1.1 General topology All the proofs of theorems in this section and much more can be found in a remarkable topological textbook Munkres [2000]. The entire section is contained within the chapters 2 and 3. We will only mention results that are going to directly impact the proofs done in latter sections. Definition 1.1. Let X be an abstract set. By topology on X we mean a collection τ ⊂ 2X that is closed under finite intersections and arbitrary unions and contains ? and X. Remark. Throughout the paper, we are going to follow this standard topological terminology: The pair (X, τ) is called a topological space. The sets in the col- lection τ are called open, their complements are called closed. If the topology τ is known from context, we will omit it and just talk about open subsets of X. Set U ⊂ X is called a neighborhood of a point x ∈ X if there exists an open set G ⊂ U such that x is in G. Subcollection B ⊂ τ is called a base of topology τ if any open set can be written as the union of some subcollection B˜ ⊂ B. If Y is a subset of X, then the collection τ˜ = {G ∩ Y : G ∈ τ} is a topology on Y and (Y, τ˜) is called a topological subspace of X. In the following pages, when we say that a subset Y of X has some topological property we mean that Y has such property as a space equipped with the subspace topology. Definition 1.2. Topological space is called Hausdorff if every two distinct points can be separated by disjoint open sets. Hausdorffness is also known as the T2 separation axiom. There are other separation axioms with different separation conditions but for our purpose, the Hausdorff axiom will be enough. The reason for this is that continuous functions into Hausdorff spaces, in some sense, behave well. We will discuss precisely how ”well” in the following pages. Figure 1.1: Visualization of the Hausdorff property 3 Lemma 1.3. Subspace of a Hausdorff space is Hausdorff. Theorem 1.4. Let (X, d) be a metric space and let B(x, δ) be the open ball centered at x with radius δ > 0, then the system τ defined by ∀G ⊂ XG ∈ τ ⇐⇒ ∀x ∈ G ∃δ > 0 : B(x, δ) ⊂ G is a Haussdorff topology on X and the collection {B(x, δ): x ∈ X, δ > 0} is a base of the induced topology. Topology τ will be referred to as the topology induced by metric d. Definition 1.5. Map f : X −→ Y between topological spaces X and Y is called continuous if one of the following equivalent conditions holds 1. f −1(G) is open for every open G ⊂ Y 2. f −1(F ) is closed for every closed F ⊂ Y Moreover, if f is a bijection and f, f −1 are both continuous then f is called a homeomorphism. Remark. Equivalence of 1. and 2. follows from the identity f −1(Y \ G) = f −1(Y ) \ f −1(G) = X \ f −1(G) Theorem 1.6. For continuous maps the following statements are true: 1. Composition of continuous maps is continuous 2. Restriction of a continuous map is again a continuous map with respect to the subspace topology Theorem 1.7. Let f : X −→ Y be a map between two metric spaces X and Y. Then f is continuous in the metric sense if and only if f is continuous in the topological sense with respect to topologies induced by the metrics. The motivation for creating topological spaces was to find a way to formalize the notions of closeness and continuity which is broader then the ϵ − δ approach with metrics, as these concepts also make sense in non-metrizable structures. According to the previous theorem, the given definition of topological spaces suc- ceeds in this regard. Two topological spaces X and Y are said to be homeomorphic, if there exists a homeomorphism between them. This fact will be denoted X ≃ Y . The relation ≃ is an equivalence on the class of all topological spaces. If X ≃ Y and Y is a subspace of Z, then we say that X embeds into Z and the corresponding homeomorphism is called an embedding. Homeomorphism transforms open sets into open sets and since topological properties are defined through open sets, in a purely topological sense, homeomorphic spaces are the same. Example. The mapping X idX :(X, 2 ) −→ (X, {?,X}) is a continuous bijection which is never a homeomorphism (unless X is a one-point set). 4 We see, that a continuous bijection between two general topological spaces need not be a homeomorphism. However, if we include some additional properties we get a sufficient condition for homeomorphism.
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