Cantor set, rationals, order topology, totally disconnected, zero dimensional. Pdf the cantor set is an interesting example of an uncountable set of. The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. Handwritten notes a handwritten notes of topology by mr. Introduction in this article, we recall the story of the discovery of set theory and point set topology by georg cantor 1845. In this paper, we extend our study to countably infinite products of topological spaces.
Thats because its length is 0, so it contains no continuous parts no intervals. Georg cantor at the dawn of point set topology number line vs. In addition, cantor sets are uncountable, may have 0 or positive lebesgue measures, and are nowhere dense. Several interesting remarks and theorems relating to cantor sets are then demonstrated, and their connection to real analysis and general topology explored. Georg cantor at the dawn of point set topology background. The idea is that if one geometric object can be continuously transformed into another, then the two objects are to be viewed as being topologically the same. Basic pointset topology 3 means that fx is not in o. Generically there is but one self homeomorphism of the cantor set. Cantor set plural cantor sets mathematical analysis, topology a subset of an interval formed by recursively removing an interval in the middle of every connected component of the set. Convergence georg cantor at the dawn of pointset topology by nicholas scoville ursinus college introduction abstract a first course in pointset topology can be challenging for the student because of the abstract level of the material. Syllabus textbooks essential topology introduction to topology course objectives. Our main reference for descriptive set theory is 30. Modern topology depends strongly on the ideas of set theory, developed by georg cantor in the later part of the 19th century. George cantor 18451918 was the originator of much of modern set theory.
Topologies on the group of homeomorphisms of a cantor set. These notes covers almost every topic which required to learn for msc mathematics. Jul 08, 20 an ideal text for courses in real analysis, topology, algebra, and set theory for undergraduate and graduatelevel courses within mathematics, computer science, engineering, and physics departments, the elements of cantor sets is also appropriate as a useful reference for researchers and secondary mathematics education majors. Algebraic topology for minimal cantor sets jeanmarc gambaudo and marco martens abstract. This article carefully analyzes this proof to clarify its logical reasoning. Hausdorff topological space is homeomorphic to 2 this is an well known fact already proved by brouwer in 1910. Geometric topology study of manifolds and their embeddings. Twotopologicaluniquenesstheoremsfor spacesofrealnumbers. A counterexample to this claim is the cantor set c. A systematic and integrated approach to cantor sets and their applications to various branches of mathematics. Topology has several di erent branches general topology also known as pointset topology, algebraic topology, di erential topology and topological algebra the rst, general topology, being the door to the study of the others. Cantor took this notion much further, showing that infinite sets come in an infinite number of sizes. Part ii is an introduction to algebraic topology, which associates algebraic structures such as groups to topological spaces. This will take us through several mathematical disciplines.
Fundamentals14 1 introduction 15 2 basic notions of pointset topology19 2. Network topologies are graphs consisting of nodes and edges. Mathdl georg cantor at the dawn of pointset topology. Since o was assumed to be open, there is an interval c,d about fx0 that is contained in o. We denote by c the standard middle thirds set of cantor, and by e the countable, dense subset of c consisting of endpoints of intervals deleted during the construction of c, along with 0 and 1. Georg cantor at the dawn of point set topology introduction. Cantors set has no interior points it is nowhere dense. Henry smith 18261883 wikimedia commonsthe cantor set, named after georg cantor 18451918, is no exception to this rule. The cantor set is an interesting example of an uncountable set of measure zero and has many interesting properties and consequences in the fields of set theory, topology, and fractal theory.
But the canonical example of a cantor space is the countably infinite topological product of the discrete 2point space 0, 1. We will follow munkres for the whole course, with some occassional added topics or di erent perspectives. This allows to study minimal cantor sets by algebraic topological means. In addition to establishing the basic ideas of set theory, cantor considered point sets in euclidean space as part of his study of fourier series. It will be shown that every minimal cantor set can be obtained as a projective limit of directed graphs. Connectedness pathconnected spaces cut points connected components and path components the cantor set exercises chapter 3. Next, we discuss a very special subset of 0, 1 called the cantor set. Cantor set and cantor function was neither motivated by geometry nor did it involve geometry, even though this is how these objects are often introduced see e. Introductory topics of pointset and algebraic topology are covered in a series of. The cantor set is created by deleting the open middle third from each of a set of line segments repeatedly. For further developments, see pointset topology and algebraic topology. Nevertheless, we are optimistic that some ideas of this paper may be used.
Definition and topological properties definition 1. In fact, cantor may have come upon them through a purely arithmetic program. By splitting the cantor set into halves depending on the value of this digit, one obtains a partition of the cantor set into two closed sets that separate the original two points. Also, notice the end points of the intervals at each step are always in the set however, we will see they are not the only points left in the set. Georg cantor at the dawn of pointset topology background. Solutions for the problems on the following page are due august 24, 2016. Cantor space is metrizable, and every compact metrizable space is a quotient space of cantor space see theorem below. With applications features a thorough introduction to cantor sets and applies these sets as a bridge between real analysis, probability, topology, and algebra the author fills a gap in the current literature by providing an introductory and integrated. Georg cantor at the dawn of pointset topology pointset. The points fx that are not in o are therefore not in c,d so they remain at least a. Basic pointset topology topological spaces interior, closure, and boundary basis for a topology metric spaces subspaces continuity and homeomorphisms product spaces exercises chapter 2. Basic point set topology one way to describe the subject of topology is to say that it is qualitative geometry. Georg cantor at the dawn of pointset topology pointset topology has a history, too.
Mathematics 490 introduction to topology winter 2007 what is this. Among his contributions to mathematics was the notion of the cantor set, which consists of points along a line segment, and possesses a number of fascinating properties. In this article, we narrate this fascinating story. Georg cantor at the dawn of point set topology the real numbers. The cantor set is a precisely selfsimilar fractal with hausdorff dimension log 3 2. August 2, 2016 this assignment is about the cantor set, a remarkable subset of 0. The following lemma summarizes how this cover x re.
Named for the mathematician georg cantor, this set is a fractal a type of selfsimilar object and possesses many strange properties. In the relative topology on the cantor set, the points have been separated by a clopen set. Georg cantor at the dawn of point set topology point set topology has a history, too. Foundations of a general theory of manifolds cantor, 1883, which i will refer to as the grundlagen,iscantors. This set of instructions, in conjunction with figure 1, gives us something of an idea of each stage, t n, of the cantor set. Convergence georg cantor at the dawn of point set topology by nicholas scoville ursinus college introduction abstract a first course in point set topology can be challenging for the student because of the abstract level of the material. The open sets there are the smallest collection of subsets that make all projections continuous. Di erential topology study of manifolds with smoothness at each point to allow calculus. It is useful to represent the cantor set in terms of real numbers written to basis 3, that is, ternaries.
Cantors grundlagen and the paradoxes of set theory w. Fundamentals14 1 introduction 15 2 basic notions of point set topology19 2. Consequently, the cantor set is totally disconnected. The cantor set is an example of a perfect nowhere dense set, where a perfect set is a closed set with no isolated points and nowhere dense set is a set whose closure has an empty interior. I aim in this book to provide a thorough grounding in. Types of topology general topology point set topology study of basic topological properties derived. Those examples often possess other structures in addition to topology and this pro vides the key link between topology and other branches of geometry.
Cantor sets are the only disconnected, perfect, compact metric space up to a. Pdf analysis of the proof of cantors theorem kuan peng. The systematic study of point set topology on the real line arose during the period. Georg cantor at the dawn of pointset topology introduction. Cantors theorem basic idea cantors theorem explained proof. It was a separate printing, with a preface and some footnotes added. Basic point set topology topological spaces interior, closure, and boundary basis for a topology metric spaces subspaces continuity and homeomorphisms product spaces exercises chapter 2. While cantor defined the set now bearing his name in an 1883 paper, a version of it was defined several years earlier in 1875 by henry john stephen smith in a paper that appeared in the proceedings of the london mathematical society entitled on the integration. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class. The cantor set c is the set of all points from 0,1 which have expansion 1.
For further developments, see point set topology and algebraic topology. General article how did cantor discover set theory and topology. X a cantor set if c is homeomorphic to the cantor space 2n. Georg cantor at the dawn of pointset topology the real numbers. General article how did cantor discover set theory and. In the middle of the project, students see smiths theorem concerning the construction of integrable functions with discontinuities at every point of a. A note on the history of the cantor set and cantor function. This is usually written as 2 n \displaystyle 2\mathbb n or 2.