Compact and Totally Bounded Metric Spaces
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K-THEORY of FUNCTION RINGS Theorem 7.3. If X Is A
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 69 (1990) 33-50 33 North-Holland K-THEORY OF FUNCTION RINGS Thomas FISCHER Johannes Gutenberg-Universitit Mainz, Saarstrage 21, D-6500 Mainz, FRG Communicated by C.A. Weibel Received 15 December 1987 Revised 9 November 1989 The ring R of continuous functions on a compact topological space X with values in IR or 0Z is considered. It is shown that the algebraic K-theory of such rings with coefficients in iZ/kH, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from R6 (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in Z/kh of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added. 0. Introduction For a topological space X, let C(X, C) be the ring of continuous complex valued functions on X, C(X, IR) the ring of continuous real valued functions on X. KU*(X) is the topological complex K-theory of X, KO*(X) the topological real K-theory of X. The main goal of this paper is the following result: Theorem 7.3. If X is a compact space, k any positive integer, then there are natural isomorphisms: K,(C(X, C), Z/kZ) = KU-‘(X, Z’/kZ), K;(C(X, lR), Z/kZ) = KO -‘(X, UkZ) for all i 2 0. -
Boundedness in Linear Topological Spaces
BOUNDEDNESS IN LINEAR TOPOLOGICAL SPACES BY S. SIMONS Introduction. Throughout this paper we use the symbol X for a (real or complex) linear space, and the symbol F to represent the basic field in question. We write R+ for the set of positive (i.e., ^ 0) real numbers. We use the term linear topological space with its usual meaning (not necessarily Tx), but we exclude the case where the space has the indiscrete topology (see [1, 3.3, pp. 123-127]). A linear topological space is said to be a locally bounded space if there is a bounded neighbourhood of 0—which comes to the same thing as saying that there is a neighbourhood U of 0 such that the sets {(1/n) U} (n = 1,2,—) form a base at 0. In §1 we give a necessary and sufficient condition, in terms of invariant pseudo- metrics, for a linear topological space to be locally bounded. In §2 we discuss the relationship of our results with other results known on the subject. In §3 we introduce two ways of classifying the locally bounded spaces into types in such a way that each type contains exactly one of the F spaces (0 < p ^ 1), and show that these two methods of classification turn out to be identical. Also in §3 we prove a metrization theorem for locally bounded spaces, which is related to the normal metrization theorem for uniform spaces, but which uses a different induction procedure. In §4 we introduce a large class of linear topological spaces which includes the locally convex spaces and the locally bounded spaces, and for which one of the more important results on boundedness in locally convex spaces is valid. -
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.......................... -
Compact Non-Hausdorff Manifolds
A.G.M. Hommelberg Compact non-Hausdorff Manifolds Bachelor Thesis Supervisor: dr. D. Holmes Date Bachelor Exam: June 5th, 2014 Mathematisch Instituut, Universiteit Leiden 1 Contents 1 Introduction 3 2 The First Question: \Can every compact manifold be obtained by glueing together a finite number of compact Hausdorff manifolds?" 4 2.1 Construction of (Z; TZ ).................................5 2.2 The space (Z; TZ ) is a Compact Manifold . .6 2.3 A Negative Answer to our Question . .8 3 The Second Question: \If (R; TR) is a compact manifold, is there always a surjective ´etale map from a compact Hausdorff manifold to R?" 10 3.1 Construction of (R; TR)................................. 10 3.2 Etale´ maps and Covering Spaces . 12 3.3 A Negative Answer to our Question . 14 4 The Fundamental Groups of (R; TR) and (Z; TZ ) 16 4.1 Seifert - Van Kampen's Theorem for Fundamental Groupoids . 16 4.2 The Fundamental Group of (R; TR)........................... 19 4.3 The Fundamental Group of (Z; TZ )........................... 23 2 Chapter 1 Introduction One of the simplest examples of a compact manifold (definitions 2.0.1 and 2.0.2) that is not Hausdorff (definition 2.0.3), is the space X obtained by glueing two circles together in all but one point. Since the circle is a compact Hausdorff manifold, it is easy to prove that X is indeed a compact manifold. It is also easy to show X is not Hausdorff. Many other examples of compact manifolds that are not Hausdorff can also be created by glueing together compact Hausdorff manifolds. -
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). -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
1 Definition
Math 125B, Winter 2015. Multidimensional integration: definitions and main theorems 1 Definition A(closed) rectangle R ⊆ Rn is a set of the form R = [a1; b1] × · · · × [an; bn] = f(x1; : : : ; xn): a1 ≤ x1 ≤ b1; : : : ; an ≤ x1 ≤ bng: Here ak ≤ bk, for k = 1; : : : ; n. The (n-dimensional) volume of R is Vol(R) = Voln(R) = (b1 − a1) ··· (bn − an): We say that the rectangle is nondegenerate if Vol(R) > 0. A partition P of R is induced by partition of each of the n intervals in each dimension. We identify the partition with the resulting set of closed subrectangles of R. Assume R ⊆ Rn and f : R ! R is a bounded function. Then we define upper sum of f with respect to a partition P, and upper integral of f on R, X Z U(f; P) = sup f · Vol(B); (U) f = inf U(f; P); P B2P B R and analogously lower sum of f with respect to a partition P, and lower integral of f on R, X Z L(f; P) = inf f · Vol(B); (L) f = sup L(f; P): B P B2P R R R RWe callR f integrable on R if (U) R f = (L) R f and in this case denote the common value by R f = R f(x) dx. As in one-dimensional case, f is integrable on R if and only if Cauchy condition is satisfied. The Cauchy condition states that, for every ϵ > 0, there exists a partition P so that U(f; P) − L(f; P) < ϵ. -
Classification of Compact 2-Manifolds
Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2016 Classification of Compact 2-manifolds George H. Winslow Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Geometry and Topology Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/4291 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. Abstract Classification of Compact 2-manifolds George Winslow It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then ob- serve that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we delve into algebraic topology to determine that the aforementioned sur- faces are not homeomorphic to one another, and thus we can place each surface into exactly one of these equivalence classes. Thesis Director: Dr. Marco Aldi Classification of Compact 2-manifolds by George Winslow Bachelor of Science University of Mary Washington Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mathematics Virginia Commonwealth University 2016 Dedication For Lily ii Abstract It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. -
A Representation of Partially Ordered Preferences Teddy Seidenfeld
A Representation of Partially Ordered Preferences Teddy Seidenfeld; Mark J. Schervish; Joseph B. Kadane The Annals of Statistics, Vol. 23, No. 6. (Dec., 1995), pp. 2168-2217. Stable URL: http://links.jstor.org/sici?sici=0090-5364%28199512%2923%3A6%3C2168%3AAROPOP%3E2.0.CO%3B2-C The Annals of Statistics is currently published by Institute of Mathematical Statistics. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ims.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Tue Mar 4 10:44:11 2008 The Annals of Statistics 1995, Vol. -
16. Compactness
16. Compactness 1 Motivation While metrizability is the analyst's favourite topological property, compactness is surely the topologist's favourite topological property. Metric spaces have many nice properties, like being first countable, very separative, and so on, but compact spaces facilitate easy proofs. They allow you to do all the proofs you wished you could do, but never could. The definition of compactness, which we will see shortly, is quite innocuous looking. What compactness does for us is allow us to turn infinite collections of open sets into finite collections of open sets that do essentially the same thing. Compact spaces can be very large, as we will see in the next section, but in a strong sense every compact space acts like a finite space. This behaviour allows us to do a lot of hands-on, constructive proofs in compact spaces. For example, we can often take maxima and minima where in a non-compact space we would have to take suprema and infima. We will be able to intersect \all the open sets" in certain situations and end up with an open set, because finitely many open sets capture all the information in the whole collection. We will specifically prove an important result from analysis called the Heine-Borel theorem n that characterizes the compact subsets of R . This result is so fundamental to early analysis courses that it is often given as the definition of compactness in that context. 2 Basic definitions and examples Compactness is defined in terms of open covers, which we have talked about before in the context of bases but which we formally define here. -
Chapter 7. Complete Metric Spaces and Function Spaces
43. Complete Metric Spaces 1 Chapter 7. Complete Metric Spaces and Function Spaces Note. Recall from your Analysis 1 (MATH 4217/5217) class that the real numbers R are a “complete ordered field” (in fact, up to isomorphism there is only one such structure; see my online notes at http://faculty.etsu.edu/gardnerr/4217/ notes/1-3.pdf). The Axiom of Completeness in this setting requires that ev- ery set of real numbers with an upper bound have a least upper bound. But this idea (which dates from the mid 19th century and the work of Richard Dedekind) depends on the ordering of R (as evidenced by the use of the terms “upper” and “least”). In a metric space, there is no such ordering and so the completeness idea (which is fundamental to all of analysis) must be dealt with in an alternate way. Munkres makes a nice comment on page 263 declaring that“completeness is a metric property rather than a topological one.” Section 43. Complete Metric Spaces Note. In this section, we define Cauchy sequences and use them to define complete- ness. The motivation for these ideas comes from the fact that a sequence of real numbers is Cauchy if and only if it is convergent (see my online notes for Analysis 1 [MATH 4217/5217] http://faculty.etsu.edu/gardnerr/4217/notes/2-3.pdf; notice Exercise 2.3.13). 43. Complete Metric Spaces 2 Definition. Let (X, d) be a metric space. A sequence (xn) of points of X is a Cauchy sequence on (X, d) if for all ε > 0 there is N N such that if m, n N ∈ ≥ then d(xn, xm) < ε. -
Cohomology of Generalized Configuration Spaces Immediate How the Functor Λ Should Be Modified
Cohomology of generalized configuration spaces Dan Petersen Abstract Let X be a topological space. We consider certain generalized configuration spaces of points on X, obtained from the cartesian product Xn by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call “tcdga models” for the cochains on X. We prove the following theorem: suppose that X is a “nice” topological space, R is any commutative ring, • • • Hc (X, R) → H (X, R) is the zero map, and that Hc (X, R) is a projective R-module. Then the compact support cohomology of any generalized configuration space of points • on X depends only on the graded R-module Hc (X, R). This generalizes a theorem of Arabia. 1. Introduction § 1.1. If X is a Hausdorff space, let F (X,n) denote the configuration space of n distinct ordered points on X. A basic question in the study of configuration spaces is the following: How do the (co)homology groups of F (X,n) depend on the (co)homology groups of X? This question has a slightly nicer answer if one considers cohomology with compact support instead of the usual cohomology. This can be seen already for n = 2, in which case there is a Gysin long exact sequence k k 2 k k+1 . → Hc (F (X, 2)) → Hc (X ) → Hc (X) → Hc (F (X, 2)) → . associated to the inclusion of F (X, 2) as an open subspace of X2. If we work with field coefficients, k 2 ∼ p q k 2 k then Hc (X ) = p+q=k Hc (X) ⊗ Hc (X) and the map Hc (X ) → Hc (X) is given by multiplica- • arXiv:1807.07293v2 [math.AT] 5 Dec 2019 tion in the cohomology ring H (X); consequently, the compactly supported cohomology groups L c of F (X, 2) are completely determined by the compactly supported cohomology of X, with its ring structure.