Math 344-1: Introduction to Topology Northwestern University, Lecture Notes
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Generalized Inverse Limits and Topological Entropy
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of Faculty of Science, University of Zagreb FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Zagreb, 2016. FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Supervisors: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Mentori: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016. Acknowledgements First of all, i want to thank my supervisor professor Judy Kennedy for accept- ing a big responsibility of guiding a transatlantic student. Her enthusiasm and love for mathematics are contagious. I thank professor Vlasta Matijevi´c, not only my supervisor but also my role model as a professor of mathematics. It was privilege to be guided by her for master's and doctoral thesis. I want to thank all my math teachers, from elementary school onwards, who helped that my love for math rises more and more with each year. Special thanks to Jurica Cudina´ who showed me a glimpse of math theory already in the high school. I thank all members of the Topology seminar in Split who always knew to ask right questions at the right moment and to guide me in the right direction. I also thank Iztok Baniˇcand the rest of the Topology seminar in Maribor who welcomed me as their member and showed me the beauty of a teamwork. -
ELEMENTARY TOPOLOGY I 1. Introduction 3 2. Metric Spaces 4 2.1
ELEMENTARY TOPOLOGY I ALEX GONZALEZ 1. Introduction3 2. Metric spaces4 2.1. Continuous functions8 2.2. Limits 10 2.3. Open subsets and closed subsets 12 2.4. Continuity, convergence, and open/closed subsets 15 3. Topological spaces 18 3.1. Interior, closure and boundary 20 3.2. Continuous functions 23 3.3. Basis of a topology 24 3.4. The subspace topology 26 3.5. The product topology 27 3.6. The quotient topology 31 3.7. Other constructions 33 4. Connectedness 35 4.1. Path-connected spaces 40 4.2. Connected components 41 5. Compactness 43 5.1. Compact subspaces of Rn 46 5.2. Nets and Tychonoff’s Theorem 48 6. Countability and separation axioms 52 6.1. The countability axioms 52 6.2. The separation axioms 54 7. Urysohn metrization theorem 61 8. Topological manifolds 65 8.1. Compact manifolds 66 REFERENCES 67 Contents 1 2 ALEX GONZALEZ ELEMENTARY TOPOLOGY I 3 1. Introduction When we consider properties of a “reasonable” function, probably the first thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. What’s more, the composition of two continuous functions is also continuous. Usually, when we think of a continuous functions, the first examples that come to mind are maps f : R ! R: • the identity function, f(x) = x for all x 2 R; • a constant function f(x) = k; • polynomial functions, for instance f(x) = xn, for some n 2 N; • the exponential function g(x) = ex; • trigonometric functions, for instance h(x) = cos(x). -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Convergence of Functions: Equi-Semicontinuity
transactions of the american mathematical society Volume 276, Number 1, March 1983 CONVERGENCEOF FUNCTIONS: EQUI-SEMICONTINUITY by szymon dolecki, gabriella salinetti and roger j.-b. wets Abstract. We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem. Given a space X, by Rx we denote the space of all functions defined on X and with values in R, the extended reals. We are interested in the relationship between various notions of convergence in Rx, in particular, but not exclusively, between pointwise convergence and that induced by the convergence of the epigraphs. We extend and refine the results of De Giorgi and Franzoni [1975] (collection of "equi-Lipschitzian" functions with respect to pseudonorms) and of Salinetti and Wets [1977] (sequences of convex functions on reflexive Banach spaces). The range of applicability of the results is substantially enlarged, in particular the removal of the convexity, reflexivity and the norm dependence assumptions is significant in many applications; for illustrative purposes an example is worked out further on. The work in this area was motivated by: the search for " valid" approximations to extremal statistical problems, variational inequalities and difficult optimization problems, cf. the above mentioned articles. -
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). -
14. Arbitrary Products
14. Arbitrary products Contents 1 Motivation 2 2 Background and preliminary definitions2 N 3 The space Rbox 4 N 4 The space Rprod 6 5 The uniform topology on RN, and completeness 12 6 Summary of results about RN 14 7 Arbitrary products 14 8 A final note on completeness, and completions 19 c 2018{ Ivan Khatchatourian 1 14. Arbitrary products 14.2. Background and preliminary definitions 1 Motivation In section 8 of the lecture notes we discussed a way of defining a topology on a finite product of topological spaces. That definition more or less agreed with our intuition. We should expect products of open sets in each coordinate space to be open in the product, for example, and the only issue that arises is that these sets only form a basis rather than a topology. So we simply generate a topology from them and call it the product topology. This product topology \played nicely" with the topologies on each of the factors in a number of ways. In that earlier section, we also saw that every topological property we had studied up to that point other than ccc-ness was finitely productive. Since then we have developed some new topological properties like regularity, normality, and metrizability, and learned that except for normality these are also finitely productive. So, ultimately, most of the properties we have studied are finitely productive. More importantly, the proofs that they are finitely productive have been easy. Look back for example to your proof that separability is finitely productive, and you will see that there was almost nothing to do. -
Compactness of a Topological Space Via Subbase Covers
Compactness of a Topological Space Via Subbase Covers France Dacar, Joˇzef Stefan Institute [email protected] February 29, 2012 Exercise 1 for §9 of Chapter I in Bourbaki’s General Topology: Let X be a topological space and S a subbase of the topology of X. If every open cover of X by sets belonging to S contains a finite subcover, then X is compact. This result is known as the Alexander subbase theorem. The exercise provides a hint how to go about proving the theorem. Here we shall approach it from a different direction, using the following special case of Tychonoff’s theorem: The product of any family of finite topological spaces is a compact topological space. We will actually need only the compactness of the product of finite discrete topological spaces. It is easily seen that this even more special case of Tychonoff’s theorem implies X ι ∈ I O the special case above: given any finite topological spaces ι, , the topology of the product space ι∈I Xι is cruder than (i.e. is a subset of) the topology of the product of discrete topological spaces on the sets Xι; since (we are assuming that) the latter topology is compact, the topology O is also compact. For a start, observe that the exercise becomes trivial if “subbase” is replaced by “base”. Let B be a base of the topology of X, and suppose that every cover V⊆B of X has a finite subcover. Given any open cover U of X, assemble the set V of all sets B ∈Bfor which there exists an U ∈Uthat contains B.ThenV⊆Bcovers X, it has a finite subcover W, and we obtain a finite subset of U that covers X by selecting (and collecting) for each set B ∈W asetUB ∈Uthat contains B. -
CLASS NOTES MATH 551 (FALL 2014) 1. Wed, Aug. 27 Topology Is the Study of Shapes. (The Greek Meaning of the Word Is the Study Of
CLASS NOTES MATH 551 (FALL 2014) BERTRAND GUILLOU 1. Wed, Aug. 27 Topology is the study of shapes. (The Greek meaning of the word is the study of places.) What kind of shapes? Many are familiar objects: a circle or triangle or square. From the point of view of topology, these are indistinguishable. Going up in dimension, we might want to study a sphere or box or a torus. Here, the sphere is topologically distinct from the torus. And neither of these is considered to be equivalent to the circle. One standard way to distinguish the circle from the sphere is to see what happens when you remove two points. One case gives you two disjoint intervals, whereas the other gives you an (open) cylinder. The two intervals are disconnected, whereas the cylinder is not. This then implies that the circle and the sphere cannot be identified as topological spaces. This will be a standard line of approach for distinguishing two spaces: find a topological property that one space has and the other does not. In fact, all of the above examples arise as metric spaces, but topology is quite a bit more general. For starters, a circle of radius 1 is the same as a circle of radius 123978632 from the eyes of topology. We will also see that there are many interesting spaces that can be obtained by modifying familiar metric spaces, but the resulting spaces cannot always be given a nice metric. ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? As we said, many examples that we care about are metric spaces, so we'll start by reviewing the theory of metric spaces. -
3 Limits of Sequences and Filters
3 Limits of Sequences and Filters The Axiom of Choice is obviously true, the well-ordering theorem is obviously false; and who can tell about Zorn’s Lemma? —Jerry Bona (Schechter, 1996) Introduction. Chapter 2 featured various properties of topological spaces and explored their interactions with a few categorical constructions. In this chapter we’ll again discuss some topological properties, this time with an eye toward more fine-grained ideas. As introduced early in a study of analysis, properties of nice topological spaces X can be detected by sequences of points in X. We’ll be interested in some of these properties and the extent to which sequences suffice to detect them. But take note of the adjective “nice” here. What if X is any topological space, not just a nice one? Unfortunately, sequences are not well suited for characterizing properties in arbitrary spaces. But all is not lost. A sequence can be replaced with a more general construction—a filter—which is much better suited for the task. In this chapter we introduce filters and highlight some of their strengths. Our goal is to spend a little time inside of spaces to discuss ideas that may be familiar from analysis. For this reason, this chapter contains less category theory than others. On the other hand, we’ll see in section 3.3 that filters are a bit like functors and hence like generalizations of points. This perspective thus gives us a coarse-grained approach to investigating fine-grained ideas. We’ll go through some of these basic ideas—closure, limit points, sequences, and more—rather quickly in sections 3.1 and 3.2. -
Strongly Summable Ultrafilters: Some Properties and Generalizations
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by YorkSpace STRONGLY SUMMABLE ULTRAFILTERS: SOME PROPERTIES AND GENERALIZATIONS DAVID J. FERNANDEZ´ BRETON´ A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN DEPARTMENT OF MATHEMATICS AND STATISTICS YORK UNIVERSITY TORONTO, ONTARIO DECEMBER 2014 c David J. Fern´andezBret´on,2014 Abstract This dissertation focuses on strongly summable ultrafilters, which are ultrafilters that are related to Hindman's theorem in much the same way that Ramsey ul- trafilters are related to Ramsey's theorem. Recall that Hindman's theorem states that whenever we partition the set of natural numbers into two (or any finite num- ber of) cells, one of the cells must entirely contain a set of the form FS(X) for some infinite X ⊆ N (here FS(X) is the collection of all finite sums of the form P x2a x where a ⊆ X is finite and nonempty). A nonprincipal ultrafilter on N is said to be strongly summable if it has a base of sets of the form FS(X), this is, if (8A 2 p)(9X 2 [N]@0 )(FS(X) ⊆ A and FS(X) 2 p). These ultrafilters were first introduced by Hindman, and subsequently studied by people such as Blass, Eis- worth, Hindman, Krautzberger, Matet, Protasov and others. Now, from the view- point of the definitions, there is nothing special about N, and analogous definitions for FS(X) and strongly summable ultrafilter can be considered for any semigroup (in the non-abelian case, one must first fix an ordering for X on order-type !). -
HOMOTOPY THEORY for BEGINNERS Contents 1. Notation
HOMOTOPY THEORY FOR BEGINNERS JESPER M. MØLLER Abstract. This note contains comments to Chapter 0 in Allan Hatcher's book [5]. Contents 1. Notation and some standard spaces and constructions1 1.1. Standard topological spaces1 1.2. The quotient topology 2 1.3. The category of topological spaces and continuous maps3 2. Homotopy 4 2.1. Relative homotopy 5 2.2. Retracts and deformation retracts5 3. Constructions on topological spaces6 4. CW-complexes 9 4.1. Topological properties of CW-complexes 11 4.2. Subcomplexes 12 4.3. Products of CW-complexes 12 5. The Homotopy Extension Property 14 5.1. What is the HEP good for? 14 5.2. Are there any pairs of spaces that have the HEP? 16 References 21 1. Notation and some standard spaces and constructions In this section we fix some notation and recollect some standard facts from general topology. 1.1. Standard topological spaces. We will often refer to these standard spaces: • R is the real line and Rn = R × · · · × R is the n-dimensional real vector space • C is the field of complex numbers and Cn = C × · · · × C is the n-dimensional complex vector space • H is the (skew-)field of quaternions and Hn = H × · · · × H is the n-dimensional quaternion vector space • Sn = fx 2 Rn+1 j jxj = 1g is the unit n-sphere in Rn+1 • Dn = fx 2 Rn j jxj ≤ 1g is the unit n-disc in Rn • I = [0; 1] ⊂ R is the unit interval • RP n, CP n, HP n is the topological space of 1-dimensional linear subspaces of Rn+1, Cn+1, Hn+1. -
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.