DOCSLIB.ORG

Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 5-1973 Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies James A. Capps Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/gradreports Part of the Geometry and Topology Commons Recommended Citation Capps, James A., "Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies" (1973). All Graduate Plan B and other Reports. 1137. https://digitalcommons.usu.edu/gradreports/1137 This Report is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Plan B and other Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. INFINITE PRODUCT SPACES UNDER THE TYCHONOFF AND GOOFYNOFF TOPOLOGIES James A.by Capps A report submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Plan B UTAH STATE UNIVERSITY Logan, Utah 1973 ACKNOWLEDGEMENTS I would like to express my thanks to all chose professors who have influenced the preparation of this report. I especially wish to thank my major professor, Dr. L. D. Loveland for his many valuable suggestions and criticisms and for the encouragement he offered me. I would like to thank Dr. Robert Heal for his suggestions and guidance. I wish to express my gratitude to my wife for her great patience which she has shown me during the writing of this report. James A. Capps TABLE OF CONTENTS INTRODUCTION ........ Page I. DEFINITIONS AND NOTATION 1 II. THE GOOFY CUBE .... 2 III. PRODUCTIVE PROPERTIES. 146 LITERATURE CITED 23 VITA ... 24 LIST OF TABLES Table Page w w 1. A Comparison of I and G 13 2. Productivity of Topological Properties 15-17 INTRODUCTION The only topology considered for the infinite product of topolo­ gical spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topo­ logy of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not re­ fer to any person of that name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of the popular topology texts provide much indication of why the Tychonoff topology is generally chosen instead of the more natural Goofynoff topology. This paper provides the reasons from a topologist's point of view. In Section III of this paper we compare the productivity of various properties under these two topologies. This paper was motivated by a question asked in the notes for a topology class at Utah State University in the fall of 1972. This question, concerned with the connectedness of an infinite product of connected spaces (under the Goofynoff topology), went unanswered through two graduate classes. An example showing that such a space may fail to be connected is given in Section II. In some respects the w example resembles the Hilbert Cube I and, for that reason, is called w the Goofy Cube and is denoted by G . In order to emphasize the difference, we include a table at the end of Section II which compares w w the topological properties of I with those of G . 2 I. DEFINITIONS AND NOTATION Given two sets A and B, the product, denoted AXB, is defined to be the set of all ordered pairs (a,b) where aEA and bEB. If X and Y are topological spaces, then the family of all sets of the form DXV, where U is open in X and Vis open in Y is a basis for a topology for XX Y. The topology generated by this basis is called the product topology for XXY. The definitions and remarks given thus far easily extend to products of any finite collection of topological spaces. An equivalent basis for the product topology on a finite product X1 X X2 X · • • X Xn is the set of all products of the form . U 1 X U2 X • · · X U0 where each Ui is restricted to a fixed basis for Xi This fact allows us to picture a basis element for the product topology as a n-sided box and thus many people refer to the product topology as the "box" topology. The mappings from XXY onto X and Y defined by TI ( (x,y) ) = x X and IT ( (x,y) ) = y, respectively, are called projections or Y projections onto the coordinate spaces. As can be easily shown, these projections are continuous and open. The box topology is the smallest topology on XXY which makes these projections continuous. We now extend these ideas on product spaces to infinite products. Let A be an indexing set (possibly uncountably infinite) and for each aEA, let A a be a topological space. The cartesian product a X {Aa: EA} (abbreviated X Aa where only one indexing set is involved) is the set of all functions f: A-UAa such that f (a) E Aa for each aEA. This product could also be thought of as a set of "infinite-tuples" 3 th where the a coordinate would be f(a), an element of Aa. For each 13 E: A, the map IT : X Aa....._ A , defined by I1 (f) = f (8), is called the 13 8 8 l3th th projection mapping of X Aa on to A . A is called the 8 factor 8 B th space or 8 coordinate space. We denote an element f of X Aa by th <xa> or more frequen tly by <f(a)> and call f(a) the a coordinate of f. There are many ways to describe a topology on the cartesian pro­ duct X Aa. Of course we shall want a topology which agrees with the box topology in the case of finite products and we would hope that the projection mappings would be continuous. The most well-known topology for X Aa, called the Tychonoff topology, is defined as the smallest topology for which the projection mappings are all continuous. The family of all subse ts of X Aa of the form X Ua, where each Ua is open in Aa and Ua = Aa for all but a finite number of indices a, is the usual basis taken for the Tychonoff topology [l; p. 98]. Perhaps a more natural way of defining a topology for X Aa would be to merely extend the definition of the box topology for finite products. The Goofynoff topology for X Aa is defined to be the topology generated by taking as a basis the set D of all subsets of X Aa of the form X Ua, where each Ua is a basis element of Aa. It is important to note that the sets generated in this manner are independent of the selection of bases for the factor spaces. We need to prove at this point that the set Dis indeed a basis for some topology on X Aa. Clearly the set D covers the space X Aa. Suppose B1, B2 E: D and g E: (Bl n B2). Then Bl = X ua and B2 = X Va for some collection of a Ua's and Va's where Ua and Va are basis elements in Aa for each E: A. E: t a) E: (U V ) for each a E: A. Thus Since g (B1 (\B2), we know tha g( a n a for each a E: A, there exists a basis elemen t Ya of Aa such that 4 g(a) E Ya C (Ua n Va). Hence X Ya is an element of Dsuch that g E X ya C (Bl n B2). Therefore, the set Dis a basis for some topology on X Aa [2; Th. 1.5, p. 22]. Given a product space S = X {Aa: aEA}, the Tychonoff topology on S will be denoted by T or just T (when no confusion arises) while S the Goofynoff topology will be denoted by G or just G. We will assume S that bases for the Aa are fixed and refer to the elements of the basis D for G as basic open sets. S Since XUa, where each Ua is open in Aa and Ua = Aa for all but a finite number of indices a, is clearly the union of basic open sets, we see that the Tychonoff topology is a subset of the Goofynoff topology for XAa. Consequently, any continuous function on (XAa,T) will also be continuous when viewed as a function on (XAa,G). Thus, for instance, the projection mapping TI : (XAa,G)�A is continuous S S th for each SEA. Furthermore, rr (XUa) = u and thus the s projection s s of a basic open set is a basis element of A implying that the S projection mappings on (XAa,G) are also open functions. There are several concepts in topology whose definitions vary slightly from text to text. We now give the version of each of these concepts which we will be using in this paper. A To-space is a space X such that, if p,q E Xand p # q, there exists an open set U with either p EU and qi U or pi U and q EU. A T1-space is a space X such that, if p,q EXand p # q, there exists an open set U with p EU and qi U.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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..........................
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Math 344-1: Introduction to Topology Northwestern University, Lecture Notes
    Math 344-1: Introduction to Topology Northwestern University, Lecture Notes Written by Santiago Can˜ez These are notes which provide a basic summary of each lecture for Math 344-1, the first quarter of “Introduction to Topology”, taught by the author at Northwestern University. The book used as a reference is the 2nd edition of Topology by Munkres. Watch out for typos! Comments and suggestions are welcome. Contents Lecture 1: Topological Spaces 2 Lecture 2: More on Topologies 3 Lecture 3: Bases 4 Lecture 4: Metric Spaces 5 Lecture 5: Product Topology 6 Lecture 6: More on Products 8 Lecture 7: Arbitrary Products, Closed Sets 11 Lecture 8: Hausdorff Spaces 15 Lecture 9: Continuous Functions 16 Lecture 10: More on Continuity 18 Lecture 11: Quotient Spaces 19 Lecture 12: More on Quotients 20 Lecture 13: Connected Spaces 21 Lecture 14: More on Connectedness 22 Lecture 15: Local Connectedness 23 Lecture 16: Compact spaces 24 Lecture 17: More on Compactness 25 Lecture 18: Local Compactness 27 Lecture 19: More on Local Compactness 28 Lecture 20: Countability Axioms 29 Lecture 21: Regular Spaces 30 Lecture 22: Normal spaces 31 Lecture 23: Urysohn’s Lemma 31 Lecture 24: More on Urysohn 32 Lecture 25: Tietze Extension Theorem 32 Lecture 26: Tychonoff’s Theorem 33 Lecture 27: Alexander Subbase Theorem 36 Lecture 1: Topological Spaces Why topology? Topology provides the most general setting in which we can talk about continuity, which is good because continuous functions are amazing things to have available. Topology does this by providing a general setting in which we can talk about the notion of “near” or “close”, and it is this perspective which I hope to make more precise as we go on.
    [Show full text]
  • MAT327H1: Introduction to Topology
    MAT327H1: Introduction to Topology Topological Spaces and Continuous Functions TOPOLOGICAL SPACES Definition: Topology A topology on a set X is a collection T of subsets of X , with the following properties: 1. ∅ , X ∈T . ∈ ∈ ∪u ∈T 2. If u T , A , then . ∈A n ∈ = 3. If ui T ,i 1, , n , then ∩ui ∈T . i=1 The elements of T are called open sets. Examples 1. T={∅ , X } is an indiscrete topology. 2. T=2X =set of all subsets of X is a discrete topology. Definition: Finer, Coarser T 1 is finer than T 2 if T 2 ⊂T 1 . T 2 is coarser than T 1 . BASIS FOR A TOPOLOGY Definition: Basis A collection of subsets B of X is called a bais for a topology if: 1. The union of the elements of B is X . 2. If x∈B1 ∩B2 , B1, B2∈B , then there exists a B3 of B such that x∈B3⊂B1∩B2 . Examples 1 1 1. B is the set of open intervals a ,b in ℝ with ab . For each x∈ℝ , x∈ x− , x . 2 2 2. B is the set of all open intervals a ,b in ℝ where ab and a and b are rational numbers. 3. Let T be the collection of subsets of ℝ which are either empty or are the complements of finite sets. Note that c A∪B = Ac∩Bc and A∩Bc= Ac∪Bc . This topology does not have a countable basis. Claim A basis B generates a topology T whose elements are all possible unions of elements of B .
    [Show full text]
  • Chapter 6 Products and Quotients
    Chapter VI Products and Quotients 1. Introduction In Chapter III we defined the product of two topological spaces \] and and developed some simple properties of the product. (See Examples III.5.10-5.12 and Exercise IIIE20. ) Using simple induction, that work generalized easy to finite products. It turns out that looking at infinite products leads to some very nice theorems. For example, infinite products will eventually help us decide which topological spaces are metrizable. 2. Infinite Products and the Product Topology The set \‚] was defined as ÖÐBßCÑÀB−\ßC −]×. How can we define an “infinite product” set ? Informally, we want to say something like \œ Ö\α Àα −E× \œ Ö\αααα Àα −EלÖÐBÑÀB −\ × so that a point BB\ in the product consists of “coordinates αα” chosen from the 's. But what exactly does a symbol like BœÐBÑα mean if there are “uncountably many coordinates?” We can get an idea by first thinking about a countable product. For sets \"# ß \ ß ÞÞÞß \ 8,... we can informally define the product set as a certain set of sequences: ∞ . \œ8œ" \8888 œÖÐBÑÀB −\ × But if we want to be careful about set theory, then a legal definition of \ should have the form \œÖÐBÑ−Y888 ÀB −\ ×. From what “pre-existing set” Y will the sequences in \ be chosen? The answer is easy: we write ∞∞ \œ8œ" \8888 œÖB−Ð 8œ" \Ñ ÀBÐ8ÑœB −\×Þ Thus the elements of ∞ are certain functions (sequences) defined on the index set . This idea 8œ"\8 generalizes naturally to any productÞ Definition 2.1 Let Ö\α Àα − E× be a collection of sets.
    [Show full text]
  • Topology: Notes and Problems
    TOPOLOGY: NOTES AND PROBLEMS Abstract. These are the notes prepared for the course MTH 304 to be offered to undergraduate students at IIT Kanpur. Contents 1. Topology of Metric Spaces 1 2. Topological Spaces 3 3. Basis for a Topology 4 4. Topology Generated by a Basis 4 4.1. Infinitude of Prime Numbers 6 5. Product Topology 6 6. Subspace Topology 7 7. Closed Sets, Hausdorff Spaces, and Closure of a Set 9 8. Continuous Functions 12 8.1. A Theorem of Volterra Vito 15 9. Homeomorphisms 16 10. Product, Box, and Uniform Topologies 18 11. Compact Spaces 21 12. Quotient Topology 23 13. Connected and Path-connected Spaces 27 14. Compactness Revisited 30 15. Countability Axioms 31 16. Separation Axioms 33 17. Tychonoff's Theorem 36 References 37 1. Topology of Metric Spaces A function d : X × X ! R+ is a metric if for any x; y; z 2 X; (1) d(x; y) = 0 iff x = y. (2) d(x; y) = d(y; x). (3) d(x; y) ≤ d(x; z) + d(z; y): We refer to (X; d) as a metric space. 2 Exercise 1.1 : Give five of your favourite metrics on R : Exercise 1.2 : Show that C[0; 1] is a metric space with metric d1(f; g) := kf − gk1: 1 2 TOPOLOGY: NOTES AND PROBLEMS An open ball in a metric space (X; d) is given by Bd(x; R) := fy 2 X : d(y; x) < Rg: Exercise 1.3 : Let (X; d) be your favourite metric (X; d). How does open ball in (X; d) look like ? Exercise 1.4 : Visualize the open ball B(f; R) in (C[0; 1]; d1); where f is the identity function.
    [Show full text]
  • Topology Semester II, 2014–15
    Topology B.Math (Hons.) II Year Semester II, 2014{15 http://www.isibang.ac.in/~statmath/oldqp/QP/Topology%20MS%202014-15.pdf Midterm Solution ! 1 ! Question 1. Let R be the countably infinite product of R with itself, and let R be the subset of R consisting 1 of sequences which are eventually zero, that is, (ai)i=1 such that only finitely many ai's are nonzero. Determine 1 ! ! the closure of R in R with respect to the box topology, the uniform topology, and the product topology on R . Answer. ! (i) The product topology: A basic open set of R is of the form U = U1 × U2 × · · · × Un × R × R × · · · ! where Ui are open sets in R. Now take any x = (x1; x2; : : : ; xn; xn+1; xn+2;:::) 2 R , and any basic open set U as above containing x. Let y = (x1; x2; : : : ; xn; 0; 0; 0;:::). Now, it is easy to see that y 2 U. Also, 1 1 ! y 2 R . Hence R is dense in R in the product topology. (ii) The box topology : Every basic open set is of the form W = W1 × W2 × · · · × Wi × Wi+1 × · · · ! 1 Now take any x = (x1; x2; : : : ; xi; xi+1; xi+2;:::) 2 R nR , then xn 6= 0 for infinitely many n. In particular, if ( R n f0g if xn 6= 0 Wn = (−1; 1) if xn = 0 Q then if W = Wn as above, then x 2 W and 1 W \ R = ; 1 Hence R is closed in the box topology. ! ! (iii) The uniform topology: Consider R with the uniform topology and let d be the uniform metric.
    [Show full text]
  • Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Math Department: Class Notes and Learning Materials Mathematics, Department of 2010 Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe Laura Lynch University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathclass Part of the Science and Mathematics Education Commons Lynch, Laura, "Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe" (2010). Math Department: Class Notes and Learning Materials. 10. https://digitalcommons.unl.edu/mathclass/10 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Math Department: Class Notes and Learning Materials by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path connected, locally connected), compactness, completeness, countability, filters, and the fundamental group. Prepared by Laura Lynch, University of Nebraska-Lincoln August 2010 1 Topological Spaces and Continuous Functions Topology is the axiomatic study of continuity. We want to study the continuity of functions to and from the spaces C; Rn;C[0; 1] = ff : [0; 1] ! Rjf is continuousg; f0; 1gN; the collection of all in¯nite sequences of 0s and 1s, and H = 1 P 2 f(xi)1 : xi 2 R; xi < 1g; a Hilbert space. De¯nition 1.1. A subset A of R2 is open if for all x 2 A there exists ² > 0 such that jy ¡ xj < ² implies y 2 A: 2 In particular, the open balls B²(x) := fy 2 R : jy ¡ xj < ²g are open.
    [Show full text]
  • Review Tables
    Tables of properties: examples and constructions Topology FS 2019 Francesco Fournier Facio July 1, 2019 These two tables concern the behaviour of topological properties you have encountered this semester under certain constructions. The first table contains many examples, that should serve as a basis to complete the second table, containing the constructions. You are encouraged to write down the empty table and try to complete it yourself, coming to this document when you need help: it is an excellent way to prepare for the exam. Here is the list of properties, examples and constructions that appear in the table. Properties. • Compact. • Locally compact. • Connected. • Path-connected. • Locally path-connected. • First-countable. • Second-countable. (Remark. Whenever we consider the properties of being first- or second- countable, we will always assume that we are working in uncountable spaces. This is because, although there are countable spaces that are not first- or second-countable (notice that for a countable space, these properties are equivalent), the examples are quite technical and I didn't want to get into that.) • Hausdorff (T2). • Regular (T3). • Normal (T4). • Metrizable. • Complete. • Simply connected (Remark. In this document simply connected means path-connected and trivial fundamental group: some people don't require path-connectedness). 1 • Contractible. Examples • An infinite set X with the trivial topology Ttr. • An infinite set X with the discrete topology Tdisc. • An infinite set X with the cofinite topology Tcof (defined in Ex. 5.1). • An infinite set X with the particular point topology Tp, where p 2 X. This is the topology where a non-empty set is open if and only if it contains p.
    [Show full text]