DOCSLIB.ORG

Chapter 6 Products and Quotients

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. We define the product set E \œ Ö\αααα Àαα −EלÖB− \ ÀBÐÑ−\ ×. The \'s are called the factors of \. For each th α1α, the function ααÀÖ\À−E×Ä\ α defined by 1 α ÐBÑœB α is called the α -projection map . th For B−\, we write more informally Bαα œBÐα Ñœthe α -coordinate of B and write BœÐB ). Note: the index set EBœÐBÑ might not be ordered. So, even though we use the informal notation α , such phrases as “the first coordinate of BBB,” “the next coordinate in after α,” and “the coordinate in BB preceding αα” may not make sense. The notation ÐBÑis handy but can lead you into errors if you're not careful. 239 A point in is, by definition, a function that “chooses” a coordinate from each set BÖ\À−E× α α Bα in the collection Ö\α Àα − E×. To say that such a “choice function” must exist if all the \α 's are nonempty is precisely the Axiom of Choice. (See the discussion following Theorem I.6.8 .) Theorem 2.2 The Axiom of Choice (AC) is equivalent to the statement that every product of nonempty sets is nonempty. Note: In ZF set theory, certain special products can be shown to be nonempty without using AC. For example, if then ∞ . Without using AC, we can \œ88 ß8œ" \œÖB− ÀBÐ8Ñ−ל precisely describe a point (œ function) in the product for example, the identity function so ∞ . 3œÖÐ7ß8Ñ− ‚ À7œ8× œ8œ" \8 Ág We will often write as . If the indexing set is clearly understood, we may Ö\αα Àα − E×α−E \ E simply write . \α Example 2.3 1) If , then E E œ g Ö\ααα Àαα − E× œ ÖÐα−E \ Ñ À BÐ Ñ − \ × œ Ög×Þ 2) Suppose \œg for some αα −E. Then \œÖÐ \ÑÀBÐÑ−\×ÞE αααα! ! αα−E −E Since BÐα Ñ − \ is impossible, \ œ g. ! αα! αα−E 3) Strictly speaking, we have two different definitions for a finite product\‚\"# : i) \"# ‚\ œÖÐBßBÑÀB "#""## −\ßB −\× (a set of ordered pairs) Ö"ß#× ii) \‚\œÖB−Ð\∪\Ñ"# "# ÀBÐ3Ñ−\× 3 (a set of functions) But there is an obvious way to identify these two sets: the ordered pair ÐB"# ß B Ñ corresponds to the Ö"ß#× function B œ ÖÐ"ß B"# Ñß Ð#ß B Ñ× − Ð\ "# ∪ \ Ñ Þ 4) If , then ∞∞ Eœ Ö\88888 À8− ל8œ" \ œÖB−Ð 8œ" \ Ñ À B −\ × œ ÖÐB"# ß B ß ÞÞÞß B 8 ß ÞÞÞß Ñ À B 8 − \ 8 × œthe set of all sequences ÐB 8 Ñ where B 8 − \ 8 Þ 5) Suppose the 's are identical, say for all . Then \\œ]−EÖ\À−E×αααα α EEE. If , we will sometimes œÖB−Ðα−E \αα Ñ ÀBÐα Ñ−\ לÖB−] ÀB α −]ל] lElœ7 write this product simply as ]œ7 “the product of 7 copies of ]” because the number of factors 7 is often important while the specific index set E is not. Now that we have a definition of the set , we can think about a product topology. We Ö\α Àα − E× begin by recalling the definition and a few basic facts about the “weak topology.” (See Example III.8.6.) 240 Definition 2.4 Let \−EÐ\ßÑ be a set. For each αg, suppose αα is a topological space and that 0À\Ä\αα. The weak topology on \ generated by the collection Y œÖ0Àα α −E× is the smallest topology on \0 that makes all the α's continuous. Certainly, there is at least one topology on \0 that makes all the α's continuous: the discrete topology. Since he intersection of a collection of topologies on \ is a topology (why? ), the weak topology exists we can describe it “from the top down” as { : is a topology on making all the 's \0 gg α continuous×Þ However, this slick description of the weak topology doesn't give a useful description of what sets are open. Usually it is more useful to describe the weak topology on \ “from the bottom up.” to make all the 0α's continuous it is necessary and sufficient that " for each α −E and for each open set Yαα ©\, the set 0α ÒYÓ α must be open. " Therefore the weak topology g is the smallest topology that contains all such sets 0ÒYÓα α and that is " the topology for which ÆαœÖ0α ÒYÓÀααα −EßYopen in \ × is a subbase. (See Example III.8.6. ) Therefore a base for the weak topology consists of all finite intersections of sets from Æ. A typical basic open set has form "] " ... " where each and each is 0ÒY∩0ÒYÓ∩∩0ÒYÓαα"#αα"# α 8 α 8α3 −E Y α 3 open in \α3 . To cut down on symbols, we will use a special notation for these subbasic and basic open sets: " We will write Yαα for 0α ÒY ÓÞ A typical basic open set is then "] " ... " Yœ0ÒY∩0ÒYÓ∩∩0ÒYÓαα"#αα"# α 8 α 8 which we further abbreviate as Y œ Yαα"# ß Y ß ÞÞÞß Y α 8 So B − Y œ Yαα"# ß Y ß ÞÞÞY α 8 iff 0 α 3 ÐBÑ − Y α 3 for each 3 œ "ß ÞÞÞß 8Þ This notation is not standard but it should be because it's very handy. You should verify that to get a base for the weak topology g on \, it is sufficient to use only the sets Yαα"# ß Y ß ÞÞÞß Y α 8 where each Y\is a basic (or even subbasic) open set in . αα3 3 Example 2.5 Suppose Ð\ßg Ñ is any topological space and E©\Þ Let 3ÀEÄ\ be the inclusion map 3Ð+Ñ œ +Þ Then the subspace topology on E is the same as the weak topology generated by Y œÖ3×ÞTo see this, just note that a base for the weak topology is Ö3" ÒYÓÀY open in \× and that 3ÒYÓœY∩E" which is a typical open set in the subspace topology. The following theorem tells us that a map 0\ into a space with the weak topology is continuous iff each composition 0‰0α is continuous. Theorem 2.6 For αg−E, suppose 0ααα À\ÄÐ\ß Ñ and that \ has the weak topology generated by the functions 0^α. Let be a topological space and 0À^Ä\0. Then is continuous iff 0‰0À^Ä\αα is continuous for every α. Proof If 00‰0 is continuous, then each composition α is continuous. Conversely, suppose each " 0‰0α is continuous. To show that 0 is continuous, it is sufficient to show that 0 ÒZÓis open in ^ whenever Z\ZœYY\ is a subbasic open set in (why? ). So let ααwith open in α. Then " " " " 0 ÒZ Ó œ 0 Ò0α ÒYαα ÓÓ œ Ð0 ‰ 0Ñ ÒY α Ó which is open since 0 α ‰ 0 is continuous. ñ 241 Definition 2.7 For each αg−Eßlet Ð\ßαα Ñ be a topological space. The product topology g on the set is the weak topology generated by the collection of projection maps \œÖÀ−E×Þα Y1αα The product topology is sometimes called the “Tychonoff topology.” We always assume that a product space has the product topology unless something else is explicitly stated. \α Since the product topology is a weak topology, a subbase for the product topology consists of all sets " of form Yαα œ1αα ÒYÓ, where −E and Y α is open in \ αÞ A base consists of all finite intersections of such sets À "] ... " Yαααα"8"8 ∩ ÞÞÞ ∩ Y œ11αα"8 ÒY ∩ ∩ ÒY Ó œ ÖB − \ À B − Y for each 3 œ "ß ÞÞÞß 8× αα33 α where for (*) œα−E Yααα Y œ \ααα Á"# ß ß ÞÞÞß α 8 (It is sufficient to use only Y\αα' s which are basic (or even subbasic) open sets in Þ Why?) A basic open set in “depends on only finitely many coordinates” in the following sense: \α B − Yαα"# ß Y ß ÞÞÞß Y α 8 iff B satisfies the finitely many restrictions B α33 − Y α Ð3 œ "ß ÞÞÞß 8Ñ.
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]
  • Chapter 7 Separation Properties
    Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and .
    [Show full text]
  • On Compactness in Locally Convex Spaces
    Math. Z. 195, 365-381 (1987) Mathematische Zeitschrift Springer-Verlag 1987 On Compactness in Locally Convex Spaces B. Cascales and J. Orihuela Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Murcia, E-30.001-Murcia-Spain 1. Introduction and Terminology The purpose of this paper is to show that the behaviour of compact subsets in many of the locally convex spaces that usually appear in Functional Analysis is as good as the corresponding behaviour of compact subsets in Banach spaces. Our results can be intuitively formulated in the following terms: Deal- ing with metrizable spaces or their strong duals, and carrying out any of the usual operations of countable type with them, we ever obtain spaces with their precompact subsets metrizable, and they even give good performance for the weak topology, indeed they are weakly angelic, [-14], and their weakly compact subsets are metrizable if and only if they are separable. The first attempt to clarify the sequential behaviour of the weakly compact subsets in a Banach space was made by V.L. Smulian [26] and R.S. Phillips [23]. Their results are based on an argument of metrizability of weakly compact subsets (see Floret [14], pp. 29-30). Smulian showed in [26] that a relatively compact subset is relatively sequentially compact for the weak to- pology of a Banach space. He also proved that the concepts of relatively countably compact and relatively sequentially compact coincide if the weak-* dual is separable. The last result was extended by J. Dieudonn6 and L. Schwartz in [-9] to submetrizable locally convex spaces.
    [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]
  • September 2020 INTRODUCTION to TOPOLOGY (SOME ADDITIONAL
    September 2020 INTRODUCTION TO TOPOLOGY (SOME ADDITIONAL BASIC EXERCISES) MICHAEL MEGRELISHVILI Abstract. We provide some additional exercises in the course Topology-8822205 (Bar-Ilan University). We are going to update this file several times (during the current semester). Contents 1. Metric spaces 1 2. Topological spaces 5 3. Topological products 8 4. Compactness 9 5. Quotients 11 6. Hints and solutions 12 References 25 1. Metric spaces Exercise 1.1. Give an example of a metric space (X; d) containing two balls such that B(a1; r1) ( B(a2; r2) but r2 < r1: Hint: think about metric subspaces of R. Exercise 1.2. Let (X; d) be a metric space and 0 < r + d(a; b) < R: Prove that B(b; r) ⊆ B(a; R). Conclude that the ball B(a; R) is an open subset of X. Exercise 1.3. Let X be a set. Define ( 0 for x = y d (x; y) := ∆ 1 for x 6= y Show that (X; d∆) is an ultrametric space and describe the balls and spheres according to their radii. Date: September 14, 2020. 1 2 Exercise 1.4. Find non-isometric metric spaces X; Y such that there exist isometric embeddings f : X,! Y , g : Y,! X. Exercise 1.5. Let (V; jj · jj) be a normed space. Show that every translation Tz : V ! V; Tz(x) = z + x is an isometry. Conclude that all open balls B(a; r) with the same radius r (and a 2 V ) are isometric. Exercise 1.6. Give geometric descriptions of B[v; r];B(v; r);S(v; r) in R2 for v = (1; 2), r = 3 with respect to the following metrics: (a) Euclidean d; (b) d1; (c) dmax.
    [Show full text]
  • Section 11.3. Countability and Separability
    11.3. Countability and Separability 1 Section 11.3. Countability and Separability Note. In this brief section, we introduce sequences and their limits. This is applied to topological spaces with certain types of bases. Definition. A sequence {xn} in a topological space X converges to a point x ∈ X provided for each neighborhood U of x, there is an index N such that n ≥ N, then xn belongs to U. The point x is a limit of the sequence. Note. We can now elaborate on some of the results you see early in a senior level real analysis class. You prove, for example, that the limit of a sequence of real numbers (under the usual topology) is unique. In a topological space, this may not be the case. For example, under the trivial topology every sequence converges to every point (at the other extreme, under the discrete topology, the only convergent sequences are those which are eventually constant). For an ad- ditional example, see Bonus 1 in my notes for Analysis 1 (MATH 4217/5217) at: http://faculty.etsu.edu/gardnerr/4217/notes/3-1.pdf. In a Hausdorff space (as in a metric space), points can be separated and limits of sequences are unique. Definition. A topological space (X, T ) is first countable provided there is a count- able base at each point. The space (X, T ) is second countable provided there is a countable base for the topology. 11.3. Countability and Separability 2 Example. Every metric space (X, ρ) is first countable since for all x ∈ X, the ∞ countable collection of open balls {B(x, 1/n)}n=1 is a base at x for the topology induced by the metric.
    [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]
  • On Separability of the Functional Space with the Open-Point and Bi-Point-Open Topologies, Arxiv:1602.02374V2[Math.GN] 12 Feb 2016
    On separability of the functional space with the open-point and bi-point-open topologies, II Alexander V. Osipov Ural Federal University, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaja street, 620219, Ekaterinburg, Russia Abstract In this paper we continue to study the property of separability of functional space C(X) with the open-point and bi-point-open topologies. We show that for every perfect Polish space X a set C(X) with bi-point-open topology is a separable space. We also show in a set model (the iterated perfect set model) that for every regular space X with countable network a set C(X) with bi-point-open topology is a separable space if and only if a dispersion character ∆(X)= c. Keywords: open-point topology, bi-point-open topology, separability 2000 MSC: 54C40, 54C35, 54D60, 54H11, 46E10 1. Introduction The space C(X) with the topology of pointwise convergence is denoted by Cp(X). It has a subbase consisting of sets of the form [x, U]+ = {f ∈ C(X): f(x) ∈ U}, where x ∈ X and U is an open subset of real line R. arXiv:1604.04609v1 [math.GN] 15 Apr 2016 In paper [3] was introduced two new topologies on C(X) that we call the open-point topology and the bi-point-open topology. The open-point topology on C(X) has a subbase consisting of sets of the form [V,r]− = {f ∈ C(X): f −1(r) V =6 ∅}, where V is an open subset ofTX and r ∈ R.
    [Show full text]
  • Which Topologies Can Have Immediate Successors in the Lattice of T1-Topologies?
    Applied General Topology c Universidad Polit´ecnica de Valencia @ Volume 5, No. 2, 2004 pp. 231-242 Which topologies can have immediate successors in the lattice of T 1-topologies? Ofelia T. Alas and Richard G. Wilson∗ Abstract. We give a new characterization of those topologies which have an immediate successor or cover in the lattice of T1-topologies on a set and show that certain classes of compact and countably compact topologies do not have covers. Keywords: lattice of T1-topologies, maximal point, KC-space, cover of a topology, upper topology, lower topology 2000 AMS Classification: Primary 54A10; Secondary 06A06 1. Introduction and Preliminary Results The lattice L1(X) of T1-topologies on a set X has been studied, with differing emphasis, by many authors. Most articles on the subject have considered com- plementation in the lattice and properties of mutually complementary spaces (see for example, [12, 13, 15]. Here we consider a facet of the order structure of the lattice L1(X), specifically the problem of when a jump can occur in the order; that is to say, when there exist topologies τ and τ + on a set X such that whenever µ is a topology on X such that τ ⊆ µ ⊆ τ + then µ = τ or µ = τ +. The existence of jumps in L1(X) has been studied in [10] and [16] where the immediate successor τ + was said to be a cover of (or simply to cover) τ. We prefer order-theoretic terminology and then call τ a lower topology for X; the topology τ + will then be termed an upper topology.
    [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]
  • TOPOLOGICAL SPACES DEFINITIONS and BASIC RESULTS 1.Topological Space
    PART I: TOPOLOGICAL SPACES DEFINITIONS AND BASIC RESULTS 1.Topological space/ open sets, closed sets, interior and closure/ basis of a topology, subbasis/ 1st ad 2nd countable spaces/ Limits of sequences, Hausdorff property. Prop: (i) condition on a family of subsets so it defines the basis of a topology on X. (ii) Condition on a family of open subsets to define a basis of a given topology. 2. Continuous map between spaces (equivalent definitions)/ finer topolo- gies and continuity/homeomorphisms/metric and metrizable spaces/ equiv- alent metrics vs. quasi-isometry. 3. Topological subspace/ relatively open (or closed) subsets. Product topology: finitely many factors, arbitrarily many factors. Reference: Munkres, Ch. 2 sect. 12 to 21 (skip 14) and Ch. 4, sect. 30. PROBLEMS PART (A) 1. Define a non-Hausdorff topology on R (other than the trivial topol- ogy) 1.5 Is the ‘finite complement topology' on R Hausdorff? Is this topology finer or coarser than the usual one? What does lim xn = a mean in this topology? (Hint: if b 6= a, there is no constant subsequence equal to b.) 2. Let Y ⊂ X have the induced topology. C ⊂ Y is closed in Y iff C = A \ Y , for some A ⊂ X closed in X. 2.5 Let E ⊂ Y ⊂ X, where X is a topological space and Y has the Y induced topology. Thenn E (the closure of E in the induced topology on Y ) equals E \ Y , the intersection of the closure of E in X with the subset Y . 3. Let E ⊂ X, E0 be the set of cluster points of E.
    [Show full text]