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A Guide to Topology
i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
MATH 3210 Metric Spaces
MATH 3210 Metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1. Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences. Continuity of mappings. (6) 2. Real inner-product spaces, orthonormal sequences, perpendicular distance to a subspace, applications in approximation theory. (7) 3. Cauchy sequences, completeness of R with the standard metric; uniform convergence and completeness of C[a; b] with the uniform metric. (3) 4. The contraction mapping theorem, with applications in the solution of equations and differential equations. (5) 5. Connectedness and path-connectedness. Introduction to compactness and sequential compactness, including subsets of Rn. (6) LECTURE 1 Books: Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985. M. O.´ Searc´oid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006. D. Kreider, An introduction to linear analysis, Addison-Wesley, 1966. 1 Metrics, open and closed sets We want to generalise the idea of distance between two points in the real line, given by d(x; y) = jx − yj; and the distance between two points in the plane, given by p 2 2 d(x; y) = d((x1; x2); (y1; y2)) = (x1 − y1) + (x2 − y2) : to other settings. [DIAGRAM] This will include the ideas of distances between functions, for example. 1 1.1 Definition Let X be a non-empty set. A metric on X, or distance function, associates to each pair of elements x, y 2 X a real number d(x; y) such that (i) d(x; y) ≥ 0; and d(x; y) = 0 () x = y (positive definite); (ii) d(x; y) = d(y; x) (symmetric); (iii) d(x; z) ≤ d(x; y) + d(y; z) (triangle inequality). -
Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. -
Chapter P Prerequisites Course Number
Chapter P Prerequisites Course Number Section P.1 Review of Real Numbers and Their Properties Instructor Objective: In this lesson you learned how to represent, classify, and Date order real numbers, and to evaluate algebraic expressions. Important Vocabulary Define each term or concept. Real numbers The set of numbers formed by joining the set of rational numbers and the set of irrational numbers. Inequality A statement that represents an order relationship. Absolute value The magnitude or distance between the origin and the point representing a real number on the real number line. I. Real Numbers (Page 2) What you should learn How to represent and A real number is rational if it can be written as . the ratio classify real numbers p/q of two integers, where q ¹ 0. A real number that cannot be written as the ratio of two integers is called irrational. The point 0 on the real number line is the origin . On the number line shown below, the numbers to the left of 0 are negative . The numbers to the right of 0 are positive . 0 Every point on the real number line corresponds to exactly one real number. Example 1: Give an example of (a) a rational number (b) an irrational number Answers will vary. For example, (a) 1/8 = 0.125 (b) p » 3.1415927 . II. Ordering Real Numbers (Pages 3-4) What you should learn How to order real If a and b are real numbers, a is less than b if b - a is numbers and use positive. inequalities Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. -
Topology - Wikipedia, the Free Encyclopedia Page 1 of 7
Topology - Wikipedia, the free encyclopedia Page 1 of 7 Topology From Wikipedia, the free encyclopedia Topology (from the Greek τόπος , “place”, and λόγος , “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct A Möbius strip, an object with only one discipline developed, which was referred to in Latin as the surface and one edge. Such shapes are an geometria situs (“geometry of place”) or analysis situs object of study in topology. (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20 th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms , which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line. Topology includes many subfields. The most basic and traditional division within topology is point-set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology , which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology , which primarily studies manifolds and their embeddings (placements) in other manifolds. -
Section 27. Compact Subspaces of the Real Line
27. Compact Subspaces of the Real Line 1 Section 27. Compact Subspaces of the Real Line Note. As mentioned in the previous section, the Heine-Borel Theorem states that a set of real numbers is compact if and only if it is closed and bounded. When reading Munkres, beware that when he uses the term “interval” he means a bounded interval. In these notes we follow the more traditional analysis terminology and we explicitly state “closed and bounded” interval when dealing with compact intervals of real numbers. Theorem 27.1. Let X be a simply ordered set having the least upper bound property. In the order topology, each closed and bounded interval X is compact. Corollary 27.2. Every closed and bounded interval [a,b] R (where R has the ⊂ standard topology) is compact. Note. We can quickly get half of the Heine-Borel Theorem with the following. Corollary 27.A. Every closed and bounded set in R (where R has the standard topology) is compact. Note. The following theorem classifies compact subspaces of Rn under the Eu- clidean metric (and under the square metric). Recall that the standard topology 27. Compact Subspaces of the Real Line 2 on R is the same as the metric topology under the Euclidean metric (see Exam- ple 1 of Section 14), so this theorem includes and extends the previous corollary. Therefore we (unlike Munkres) call this the Heine-Borel Theorem. Theorem 27.3. The Heine-Borel Theorem. A subspace A of Rn is compact if and only if it is closed and is bounded in the Euclidean metric d or the square metric ρ. -
Chapter 2 Metric Spaces and Topology
2.1. METRIC SPACES 29 Definition 2.1.29. The function f is called uniformly continuous if it is continu- ous and, for all > 0, the δ > 0 can be chosen independently of x0. In precise mathematical notation, one has ( > 0)( δ > 0)( x X) ∀ ∃ ∀ 0 ∈ ( x x0 X d (x , x0) < δ ), d (f(x ), f(x)) < . ∀ ∈ { ∈ | X 0 } Y 0 Definition 2.1.30. A function f : X Y is called Lipschitz continuous on A X → ⊆ if there is a constant L R such that dY (f(x), f(y)) LdX (x, y) for all x, y A. ∈ ≤ ∈ Let fA denote the restriction of f to A X defined by fA : A Y with ⊆ → f (x) = f(x) for all x A. It is easy to verify that, if f is Lipschitz continuous on A ∈ A, then fA is uniformly continuous. Problem 2.1.31. Let (X, d) be a metric space and define f : X R by f(x) = → d(x, x ) for some fixed x X. Show that f is Lipschitz continuous with L = 1. 0 0 ∈ 2.1.3 Completeness Suppose (X, d) is a metric space. From Definition 2.1.8, we know that a sequence x , x ,... of points in X converges to x X if, for every δ > 0, there exists an 1 2 ∈ integer N such that d(x , x) < δ for all i N. i ≥ 1 n = 2 n = 4 0.8 n = 8 ) 0.6 t ( n f 0.4 0.2 0 1 0.5 0 0.5 1 − − t Figure 2.1: The sequence of continuous functions in Example 2.1.32 satisfies the Cauchy criterion. -
Mariya!Tsyglakova!
! ! ! ! ! ! ! ! Mariya!Tsyglakova! ! ! Transcendental+Numbers+and+Infinities+of+ Different+Sizes+ ! ! Capstone!Advisor:! Professor!Joshua!Lansky,!CAS:!Mathematics!and!Statistics! ! ! ! University!Honors! ! Spring!2013! Transcendental Numbers and Infinities of Di↵erent Sizes Mariya Tsyglakova Abstract There are many di↵erent number systems used in mathematics, including natural, whole, algebraic, and real numbers. Transcendental numbers (that is, non-algebraic real numbers) comprise a relatively new number system. Examples of transcendental numbers include e and ⇡. Joseph Liouville first proved the existence of transcendental numbers in 1844. Although only a few transcendental numbers are well known, the set of these numbers is extremely large. In fact, there exist more transcendental than algebraic numbers. This paper proves that there are infinitely many transcendental numbers by showing that the set of real numbers is much larger than the set of algebraic numbers. Since both of these sets are infinite, it means that one infinity can be larger than another infinity. 1 Background Set Theory In order to show that infinities can be of di↵erent sizes and to prove the existence of tran- scendental numbers, we first need to introduce some basic concepts from the set theory. A set is a collection of elements, which is viewed as a single object. Every element of a set is a set itself, and further each of its elements is a set, and so forth. A set A is a subset of a set B if each element of A is an element of B, which is written A B.Thesmallestsetisthe ⇢ empty set, which has no elements and denoted as . -
Chapter 3. Topology of the Real Numbers. 3.1
3.1. Topology of the Real Numbers 1 Chapter 3. Topology of the Real Numbers. 3.1. Topology of the Real Numbers. Note. In this section we “topological” properties of sets of real numbers such as open, closed, and compact. In particular, we will classify open sets of real numbers in terms of open intervals. Definition. A set U of real numbers is said to be open if for all x ∈ U there exists δ(x) > 0 such that (x − δ(x), x + δ(x)) ⊂ U. Note. It is trivial that R is open. It is vacuously true that ∅ is open. Theorem 3-1. The intervals (a,b), (a, ∞), and (−∞,a) are open sets. (Notice that the choice of δ depends on the value of x in showing that these are open.) Definition. A set A is closed if Ac is open. Note. The sets R and ∅ are both closed. Some sets are neither open nor closed. Corollary 3-1. The intervals (−∞,a], [a,b], and [b, ∞) are closed sets. 3.1. Topology of the Real Numbers 2 Theorem 3-2. The open sets satisfy: n (a) If {U1,U2,...,Un} is a finite collection of open sets, then ∩k=1Uk is an open set. (b) If {Uα} is any collection (finite, infinite, countable, or uncountable) of open sets, then ∪αUα is an open set. Note. An infinite intersection of open sets can be closed. Consider, for example, ∞ ∩i=1(−1/i, 1 + 1/i) = [0, 1]. Theorem 3-3. The closed sets satisfy: (a) ∅ and R are closed. (b) If {Aα} is any collection of closed sets, then ∩αAα is closed. -
18 | Compactification
18 | Compactification We have seen that compact Hausdorff spaces have several interesting properties that make this class of spaces especially important in topology. If we are working with a space X which is not compact we can ask if X can be embedded into some compact Hausdorff space Y . If such embedding exists we can identify X with a subspace of Y , and some arguments that work for compact Hausdorff spaces will still apply to X. This approach leads to the notion of a compactification of a space. Our goal in this chapter is to determine which spaces have compactifications. We will also show that compactifications of a given space X can be ordered, and we will look for the largest and smallest compactifications of X. 18.1 Proposition. Let X be a topological space. If there exists an embedding j : X → Y such that Y is a compact Hausdorff space then there exists an embedding j1 : X → Z such that Z is compact Hausdorff and j1(X) = Z. Proof. Assume that we have an embedding j : X → Y where Y is a compact Hausdorff space. Let j(X) be the closure of j(X) in Y . The space j(X) is compact (by Proposition 14.13) and Hausdorff, so we can take Z = j(X) and define j1 : X → Z by j1(x) = j(x) for all x ∈ X. 18.2 Definition. A space Z is a compactification of X if Z is compact Hausdorff and there exists an embedding j : X → Z such that j(X) = Z. 18.3 Corollary. -
Math 131: Introduction to Topology 1
Math 131: Introduction to Topology 1 Professor Denis Auroux Fall, 2019 Contents 9/4/2019 - Introduction, Metric Spaces, Basic Notions3 9/9/2019 - Topological Spaces, Bases9 9/11/2019 - Subspaces, Products, Continuity 15 9/16/2019 - Continuity, Homeomorphisms, Limit Points 21 9/18/2019 - Sequences, Limits, Products 26 9/23/2019 - More Product Topologies, Connectedness 32 9/25/2019 - Connectedness, Path Connectedness 37 9/30/2019 - Compactness 42 10/2/2019 - Compactness, Uncountability, Metric Spaces 45 10/7/2019 - Compactness, Limit Points, Sequences 49 10/9/2019 - Compactifications and Local Compactness 53 10/16/2019 - Countability, Separability, and Normal Spaces 57 10/21/2019 - Urysohn's Lemma and the Metrization Theorem 61 1 Please email Beckham Myers at [email protected] with any corrections, questions, or comments. Any mistakes or errors are mine. 10/23/2019 - Category Theory, Paths, Homotopy 64 10/28/2019 - The Fundamental Group(oid) 70 10/30/2019 - Covering Spaces, Path Lifting 75 11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80 11/6/2019 - The Brouwer Fixed Point Theorem 85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88 11/13/2019 - Deformation Retracts and Homotopy Equivalence 91 11/18/2019 - Computing the Fundamental Group 95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99 11/25/2019 - Universal Covering Spaces, Free Groups 104 12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109 2 9/4/2019 - Introduction, Metric Spaces, Basic Notions The instructor for this course is Professor Denis Auroux. His email is [email protected] and his office is SC539.