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Directed Sets and Topological Spaces Definable in O-Minimal Structures
Directed sets and topological spaces definable in o-minimal structures. Pablo And´ujarGuerrero∗ Margaret E. M. Thomas ∗ Erik Walsbergy 2010 Mathematics Subject Classification. 03C64 (Primary), 54A20, 54A05, 54D30 (Secondary). Key words. o-minimality, directed sets, definable topological spaces. Abstract We study directed sets definable in o-minimal structures, show- ing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all de- finable topological spaces display properties akin to first countability, and give several characterizations of a notion of definable compactness due to Peterzil and Steinhorn [PS99] generalized to this setting. 1 Introduction The study of objects definable in o-minimal structures is motivated by the notion that o-minimality provides a rich but \tame" setting for the theories of said objects. In this paper we study directed sets definable in o-minimal structures, focusing on expansions of groups and fields. By \directed set" we mean a preordered set in which every finite subset has a lower (if downward ∗Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, U.S.A. E-mail addresses: [email protected] (And´ujarGuer- rero), [email protected] (Thomas) yDepartment of Mathematics, Statistics, and Computer Science, Department of Math- ematics, University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, CA 92697-3875, U.S.A. -
Scott Spaces and the Dcpo Category
SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology. -
A Guide to Topology
i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
On Dimension and Weight of a Local Contact Algebra
Filomat 32:15 (2018), 5481–5500 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1815481D University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Dimension and Weight of a Local Contact Algebra G. Dimova, E. Ivanova-Dimovaa, I. D ¨untschb aFaculty of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria bCollege of Maths. and Informatics, Fujian Normal University, Fuzhou, China. Permanent address: Dept. of Computer Science, Brock University, St. Catharines, Canada Abstract. As proved in [16], there exists a duality Λt between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then t t w(X) = wa(Λ (X)), and if, in addition, X is normal, then dim(X) = dima(Λ (X)). 1. Introduction According to Stone’s famous duality theorem [43], the Boolean algebra CO(X) of all clopen (= closed and open) subsets of a zero-dimensional compact Hausdorff space X carries the whole information about the space X, i.e. the space X can be reconstructed from CO(X), up to homeomorphism. It is natural to ask whether the Boolean algebra RC(X) of all regular closed subsets of a compact Hausdorff space X carries the full information about the space X (see Example 2.5 below for RC(X)). -
Lecture II - Preliminaries from General Topology
Lecture II - Preliminaries from general topology: We discuss in this lecture a few notions of general topology that are covered in earlier courses but are of frequent use in algebraic topology. We shall prove the existence of Lebesgue number for a covering, introduce the notion of proper maps and discuss in some detail the stereographic projection and Alexandroff's one point compactification. We shall also discuss an important example based on the fact that the sphere Sn is the one point compactifiaction of Rn. Let us begin by recalling the basic definition of compactness and the statement of the Heine Borel theorem. Definition 2.1: A space X is said to be compact if every open cover of X has a finite sub-cover. If X is a metric space, this is equivalent to the statement that every sequence has a convergent subsequence. If X is a topological space and A is a subset of X we say that A is compact if it is so as a topological space with the subspace topology. This is the same as saying that every covering of A by open sets in X admits a finite subcovering. It is clear that a closed subset of a compact subset is necessarily compact. However a compact set need not be closed as can be seen by looking at X endowed the indiscrete topology, where every subset of X is compact. However, if X is a Hausdorff space then compact subsets are necessarily closed. We shall always work with Hausdorff spaces in this course. For subsets of Rn we have the following powerful result. -
Proper Maps and Universally Closed Maps
Proper Maps and Universally Closed Maps Reinhard Schultz Continuous functions defined on compact spaces generally have special properties. For exam- ple, if X is a compact space and Y is a Hausdorff space, then f is a closed mapping. There is a useful class of continuous mappings called proper, perfect, or compact maps that satisfy many properties of continuous maps with compact domains. In this note we shall study a few basic properties of these maps. Further information about proper maps can be found in Bourbaki [B2], Section I.10), Dugundji [D] (Sections XI.5{6, pages 235{240), and Kasriel [K] (Sections 95 and 105, pages 214{217 and 243{247). There is a slight difference between the notions of perfect and compact mappings in [B2], [D], and [K] and the notion of proper map considered here. Specifically, the definitions in [D] and [K] include a requirement that the maps in question be surjective. The definition in [B2] is entirely different and corresponds to the notion of univer- sally closed map discussed later in these notes; the equivalence of this definition with ours for a reasonable class of spaces is proved both in [B2] and in these notes (see the section Universally closed maps). PROPER MAPS If f : X ! Y is a continuous map of topological spaces, then f is said to be proper if for each compact subset K ⊂ Y the inverse image f −1[K] is also compact. EXAMPLE. Suppose that A is a finite subset of Rn and f : Rn − A ! Rm is a continuous function such that for all a 2 A we have lim jf(x)j = lim jf(x)j = 1: x!a x!1 Then f is proper. -
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0. -
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 . -
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). -
Directed Sets and Cofinal Types by Stevo Todorcevic
transactions of the american mathematical society Volume 290, Number 2, August 1985 DIRECTED SETS AND COFINAL TYPES BY STEVO TODORCEVIC Abstract. We show that 1, w, ax, u x ux and ["iF" are the only cofinal types of directed sets of size S,, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set D is directed if every two elements of D have an upper bound in D. In this note we consider some basic problems concerning directed sets which have their origin in the theory of Moore-Smith convergence in topology [12, 3, 19, 9]. One such problem is to determine "all essential kind of directed sets" needed for defining the closure operator in a given class of spaces [3, p. 47]. Concerning this problem, the following important notion was introduced by J. Tukey [19]. Two directed sets D and E are cofinally similar if there is a partially ordered set C in which both can be embedded as cofinal subsets. He showed that this is an equivalence relation and that D and E are cofinally similar iff there is a convergent map from D into E and also a convergent map from E into D. The equivalence classes of this relation are called cofinal types. This concept has been extensively studied since then by various authors [4, 13, 7, 8]. Already, from the first introduc- tion of this concept, it has been known that 1, w, ccx, w X cox and [w1]<" represent different cofinal types of directed sets of size < Kls but no more than five such types were known. -
MATH 3210 Metric Spaces
MATH 3210 Metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1. Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences. Continuity of mappings. (6) 2. Real inner-product spaces, orthonormal sequences, perpendicular distance to a subspace, applications in approximation theory. (7) 3. Cauchy sequences, completeness of R with the standard metric; uniform convergence and completeness of C[a; b] with the uniform metric. (3) 4. The contraction mapping theorem, with applications in the solution of equations and differential equations. (5) 5. Connectedness and path-connectedness. Introduction to compactness and sequential compactness, including subsets of Rn. (6) LECTURE 1 Books: Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985. M. O.´ Searc´oid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006. D. Kreider, An introduction to linear analysis, Addison-Wesley, 1966. 1 Metrics, open and closed sets We want to generalise the idea of distance between two points in the real line, given by d(x; y) = jx − yj; and the distance between two points in the plane, given by p 2 2 d(x; y) = d((x1; x2); (y1; y2)) = (x1 − y1) + (x2 − y2) : to other settings. [DIAGRAM] This will include the ideas of distances between functions, for example. 1 1.1 Definition Let X be a non-empty set. A metric on X, or distance function, associates to each pair of elements x, y 2 X a real number d(x; y) such that (i) d(x; y) ≥ 0; and d(x; y) = 0 () x = y (positive definite); (ii) d(x; y) = d(y; x) (symmetric); (iii) d(x; z) ≤ d(x; y) + d(y; z) (triangle inequality).