$\Mathfrak G $-Bases in Free (Locally Convex) Topological Vector Spaces
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Or Dense in Itself)Ifx Has No Isolated Points
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 11, Pages 3607{3616 S 0002-9939(03)06660-7 Article electronically published on February 24, 2003 TYCHONOFF EXPANSIONS BY INDEPENDENT FAMILIES WANJUN HU (Communicated by Alan Dow) Abstract. A method for Tychonoff expansions using independent families is introduced. Using this method we prove that every countable Tychonoff space which admits a partition into infinitely many open-hereditarily irre- solvable dense subspaces has a Tychonoff expansion that is !-resolvable but not strongly extraresolvable. We also show that, under Luzin's Hypothesis ! ! (2 1 =2 ), there exists an !-resolvable Tychonoff space of size !1 which is not maximally resolvable. Introduction A topological space X is called crowded (or dense in itself)ifX has no isolated points. E. Hewitt defined a resolvable space as a space which has two disjoint dense subsets (see [17]). A space is called κ-resolvable [3] if it admits κ-many pairwise disjoint dense sets. The cardinal ∆(X):=minfjUj : U is a non-empty open subset of Xg is called the dispersion character of the space X.ThespaceX is called maximally resolvable [2] if it is ∆(X)-resolvable; if every crowded subspace of X is not resolvable, then X is called hereditarily irresolvable (HI). Analogously, X is open-hereditarily irresolvable (or OHI) if every non-empty open subset of X is irresolvable. V.I. Malykhin introduced a similar concept of extraresolvable (see [20] + and [6]): a space is extraresolvable if there exists a family fDα : α<∆(X) g of dense susbets of X such that Dα \ Dβ is nowhere dense in X for distinct α and β. -
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. -
12 | Urysohn Metrization Theorem
12 | Urysohn Metrization Theorem In this chapter we return to the problem of determining which topological spaces are metrizable i.e. can be equipped with a metric which is compatible with their topology. We have seen already that any metrizable space must be normal, but that not every normal space is metrizable. We will show, however, that if a normal space space satisfies one extra condition then it is metrizable. Recall that a space X is second countable if it has a countable basis. We have: 12.1 Urysohn Metrization Theorem. Every second countable normal space is metrizable. The main idea of the proof is to show that any space as in the theorem can be identified with a subspace of some metric space. To make this more precise we need the following: 12.2 Definition. A continuous function i: X → Y is an embedding if its restriction i: X → i(X) is a homeomorphism (where i(X) has the topology of a subspace of Y ). 12.3 Example. The function i: (0; 1) → R given by i(x) = x is an embedding. The function j : (0; 1) → R given by j(x) = 2x is another embedding of the interval (0; 1) into R. 12.4 Note. 1) If j : X → Y is an embedding then j must be 1-1. 2) Not every continuous 1-1 function is an embedding. For example, take N = {0; 1; 2;::: } with the discrete topology, and let f : N → R be given ( n f n 0 if = 0 ( ) = 1 n if n > 0 75 12. -
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.......................... -
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 . -
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. -
Intrinsic Volumes and Polar Sets in Spherical Space
1 INTRINSIC VOLUMES AND POLAR SETS IN SPHERICAL SPACE FUCHANG GAO, DANIEL HUG, ROLF SCHNEIDER Abstract. For a convex body of given volume in spherical space, the total invariant measure of hitting great subspheres becomes minimal, equivalently the volume of the polar body becomes maximal, if and only if the body is a spherical cap. This result can be considered as a spherical counterpart of two Euclidean inequalities, the Urysohn inequality connecting mean width and volume, and the Blaschke-Santal´oinequality for the volumes of polar convex bodies. Two proofs are given; the first one can be adapted to hyperbolic space. MSC 2000: 52A40 (primary); 52A55, 52A22 (secondary) Keywords: spherical convexity, Urysohn inequality, Blaschke- Santal´oinequality, intrinsic volume, quermassintegral, polar convex bodies, two-point symmetrization In the classical geometry of convex bodies in Euclidean space, the intrinsic volumes (or quermassintegrals) play a central role. These functionals have counterparts in spherical and hyperbolic geometry, and it is a challenge for a geometer to find analogues in these spaces for the known results about Euclidean intrinsic volumes. In the present paper, we prove one such result, an inequality of isoperimetric type in spherical (or hyperbolic) space, which can be considered as a counterpart to the Urysohn inequality and is, in the spherical case, also related to the Blaschke-Santal´oinequality. Before stating the result (at the beginning of Section 2), we want to recall and collect the basic facts about intrinsic volumes in Euclidean space, describe their noneuclidean counterparts, and state those open problems about the latter which we consider to be of particular interest. -
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). -
An Exploration of the Metrizability of Topological Spaces
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov Metrization The- orem Contents 1. Introduction 1 2. Product Topology 2 3. More Product Topology and R! 3 4. Separation Axioms 4 5. Urysohn's Lemma 5 6. Urysohn's Metrization Theorem 6 7. Local Finiteness and Gδ sets 8 8. Nagata-Smirnov Metrization Theorem 11 References 13 1. Introduction In this paper we will be exploring a basic topological notion known as metriz- ability, or whether or not a given topology can be understood through a distance function. We first give the reader some basic definitions. As an outline, we will be using these notions to first prove Urysohn's lemma, which we then use to prove Urysohn's metrization theorem, and we culminate by proving the Nagata Smirnov Metrization Theorem. Definition 1.1. Let X be a topological space. The collection of subsets B ⊂ X forms a basis for X if for any open U ⊂ X can be written as the union of elements of B Definition 1.2. Let X be a set. Let B ⊂ X be a collection of subsets of X. The topology generated by B is the intersection of all topologies on X containing B. Definition 1.3. Let X and Y be topological spaces. A map f : X ! Y . If f is a homeomorphism if it is a continuous bijection with a continuous inverse. If there is a homeomorphism from X to Y , we say that X and Y are homeo- morphic. -
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 . -
On Modified L 1-Minimization Problems in Compressed Sensing Man Bahadur Basnet Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2013 On Modified l_1-Minimization Problems in Compressed Sensing Man Bahadur Basnet Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Basnet, Man Bahadur, "On Modified l_1-Minimization Problems in Compressed Sensing" (2013). Graduate Theses and Dissertations. 13473. https://lib.dr.iastate.edu/etd/13473 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On modified `-one minimization problems in compressed sensing by Man Bahadur Basnet A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics (Applied Mathematics) Program of Study Committee: Fritz Keinert, Co-major Professor Namrata Vaswani, Co-major Professor Eric Weber Jennifer Davidson Alexander Roitershtein Iowa State University Ames, Iowa 2013 Copyright © Man Bahadur Basnet, 2013. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my parents Jaya and Chandra and to my family (wife Sita, son Manas and daughter Manasi) without whose support, encouragement and love, I would not have been able to complete this work. iii TABLE OF CONTENTS LIST OF TABLES . vi LIST OF FIGURES . vii ACKNOWLEDGEMENTS . viii ABSTRACT . x CHAPTER 1. INTRODUCTION . -
Extension of Compact Operators from DF-Spaces to C(K) Spaces
Applied General Topology c Universidad Polit´ecnica de Valencia @ Volume 7, No. 2, 2006 pp. 165-170 Extension of Compact Operators from DF-spaces to C(K) spaces Fernando Garibay Bonales and Rigoberto Vera Mendoza Abstract. It is proved that every compact operator from a DF- space, closed subspace of another DF-space, into the space C(K) of continuous functions on a compact Hausdorff space K can be extended to a compact operator of the total DF-space. 2000 AMS Classification: Primary 46A04, 46A20; Secondary 46B25. Keywords: Topological vector spaces, DF-spaces and C(K) the spaces. 1. Introduction Let E and X be topological vector spaces with E a closed subspace of X. We are interested in finding out when a continuous operator T : E → C(K) has an extension T˜ : X → C(K), where C(K) is the space of continuous real functions on a compact Hausdorff space K and C(K) has the norm of the supremum. When this is the case we will say that (E,X) has the extension property. Several advances have been made in this direction, a basic resume and bibliography for this problem can be found in [5]. In this work we will focus in the case when the operator T is a compact operator. In [4], p.23 , it is proved that (E,X) has the extension property when E and X are Banach spaces and T : E → C(K) is a compact operator. In this paper we extend this result to the case when E and X are DF-spaces (to be defined below), for this, we use basic tools from topological vector spaces.