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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. -
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 . -
On Α-Normal and Β-Normal Spaces
Comment.Math.Univ.Carolin. 42,3 (2001)507–519 507 On α-normal and β-normal spaces A.V. Arhangel’skii, L. Ludwig Abstract. We define two natural normality type properties, α-normality and β-normality, and compare these notions to normality. A natural weakening of Jones Lemma immedi- ately leads to generalizations of some important results on normal spaces. We observe that every β-normal, pseudocompact space is countably compact, and show that if X is a dense subspace of a product of metrizable spaces, then X is normal if and only if X is β-normal. All hereditarily separable spaces are α-normal. A space is normal if and only if it is κ-normal and β-normal. Central results of the paper are contained in Sections 3 and 4. Several examples are given, including an example (identified by R.Z. Buzyakova) of an α-normal, κ-normal, and not β-normal space, which is, in fact, a pseudocompact topological group. We observe that under CH there exists a locally compact Hausdorff hereditarily α-normal non-normal space (Theorem 3.3). This example is related to the main result of Section 4, which is a version of the famous Katˇetov’s theorem on metrizability of a compactum the third power of which is hereditarily normal (Corollary 4.3). We also present a Tychonoff space X such that no dense subspace of X is α-normal (Section 3). Keywords: normal, α-normal, β-normal, κ-normal, weakly normal, extremally discon- nected, Cp(X), Lindel¨of, compact, pseudocompact, countably compact, hereditarily sep- arable, hereditarily α-normal, property wD, weakly perfect, first countable Classification: 54D15, 54D65, 54G20 §0. -
Distributive Contact Lattices: Topological Representations
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com The Journal of Logic and Algebraic Programming 76 (2008) 18–34 www.elsevier.com/locate/jlap Distributive contact lattices: Topological representations Ivo Düntsch a,1, Wendy MacCaull b,1, Dimiter Vakarelov c,2, Michael Winter a,1,∗ a Department of Computer Science, Brock University, St. Catharines, ON, Canada L2S 3A1 b Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS, Canada c Department of Mathematical Logic, Sofia University, Sofia, Bulgaria Available online 11 December 2007 Abstract In order to provide a region based theory of space the notion of Boolean contact algebras has been used. However, not all of the Boolean connectives, in particular complement, are well motivated in that context. A suitable generalization of this theory is to drop the notion of complement, thereby weakening the algebraic structure from a Boolean algebra to a distributive lattice. In this paper we investigate the representation theory of that weaker notion in order to determine whether it is still possible to represent each abstract algebra as a substructure of the regular closed sets of a suitable topological space with the standard (Whiteheadean) contact relation. Furthermore, we consider additional axioms for contact and the representation of those structures in topological spaces with richer structure. © 2007 Elsevier Inc. All rights reserved. 1. Introduction This paper investigates topological representation theory for a nonclassical approach to the region based theory of space. In contrast to the classical approach in which (the notion of) point is primitive and geometric figures are sets of points, this approach adopts as its primitive the more realistic spatial notion of region (as an abstraction of a “solid” spatial body), together with some basic relations and operations on regions. -
A Generalization of the Stone Duality Theorem
A Generalization of the Stone Duality Theorem G. Dimov, E. Ivanova-Dimova and D. Vakarelov Department of Mathematics and Informatics, University of Sofia, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria Abstract We prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in [16], the recent duality theorems from [3, 23], and some new duality theorems for the category of contact algebras and for the category of complete contact algebras. 1 Introduction This paper is a continuation of the papers [13, 15, 16] and, to some extent, of the papers [7, 8, 9, 10, 11, 12, 14, 29]. In it we prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version arXiv:1603.00932v1 [math.GN] 3 Mar 2016 obtained in [16], the recent duality theorems from [3, 23], and some new duality theorems for the category of contact algebras and for the category of complete contact algebras. More precisely, we show that there exists a duality functor Ga between the category PCA of all precontact algebras and suitable morphisms between them and the category PCS of all 2-precontact spaces and suitable morphisms between them. Then, clearly, fixing some full subcategory C of the category PCA, we obtain a duality between the categories C and Ga(C). Finding categories which are isomorphic or equivalent to the category C and (or) to the category Ga(C), we obtain as corollaries the Stone Duality and the other dualities mentioned above. -
Locally Compact Spaces of Measures
LOCALLY COMPACT SPACES OF MEASURES NORMAN Y. LUTHER Abstract. Under varying conditions on the topological space X, the spaces of <r-smooth, T-smooth, and tight measures on X, respectively, are each shown to be locally compact in the weak topology if, and only if, X is compact. 1. Introduction. Using the methods and results of Varadarajan [5], we give necessary and sufficient conditions on the topological space X for the local compactness of the spaces of ff-smooth, r- smooth, and tight measures [resp., signed measures] on X, respec- tively, when endowed with the weak topology. For such spaces of signed measures, the finiteness of X is easily seen to be necessary and sufficient (Remark 4). For such spaces of (nonnegative) measures, we find that, under various conditions on X, the compactness of X is necessary and sufficient (Corollaries 3 and 4); in fact, it is the dividing line between when the space is locally compact and when it has no compact neighborhoods whatsoever (Theorems 1, 2, and 5). 2. Preliminaries and main results. Our definitions, notation, and terminology follow quite closely those in the paper of Varadarajan [5]. In particular, our topological definitions and terminology co- incide with those of Kelley [3]. X will denote a topological space and C(X) the space of all real-valued, bounded, continuous functions on X. We use £ to denote the class of all zero-sets on X; i.e., $ = {/-1(0); fEC(X)}. The class of all co-zero sets (i.e., complements of zero-sets) in X shall be denoted by U. -
18 | Compactification
18 | Compactification We have seen that compact Hausdorff spaces have several interesting properties that make this class of spaces especially important in topology. If we are working with a space X which is not compact we can ask if X can be embedded into some compact Hausdorff space Y . If such embedding exists we can identify X with a subspace of Y , and some arguments that work for compact Hausdorff spaces will still apply to X. This approach leads to the notion of a compactification of a space. Our goal in this chapter is to determine which spaces have compactifications. We will also show that compactifications of a given space X can be ordered, and we will look for the largest and smallest compactifications of X. 18.1 Proposition. Let X be a topological space. If there exists an embedding j : X → Y such that Y is a compact Hausdorff space then there exists an embedding j1 : X → Z such that Z is compact Hausdorff and j1(X) = Z. Proof. Assume that we have an embedding j : X → Y where Y is a compact Hausdorff space. Let j(X) be the closure of j(X) in Y . The space j(X) is compact (by Proposition 14.13) and Hausdorff, so we can take Z = j(X) and define j1 : X → Z by j1(x) = j(x) for all x ∈ X. 18.2 Definition. A space Z is a compactification of X if Z is compact Hausdorff and there exists an embedding j : X → Z such that j(X) = Z. 18.3 Corollary. -
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. -
Completely Regular Spaces
Comment.Math.Univ.Carolin. 32,1 (1991)129–153 129 Completely regular spaces H. L. Bentley, E. Lowen-Colebunders Dedicated to the memory of Zdenˇek Frol´ık Abstract. We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the di- vergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy space; (2) a sep- arated filter space which is a z-regular space but not a nearness space; (3) a separated, Cauchy, zero space which is z-regular but not regular; (4) a separated, Cauchy, zero space which is µ-regular but not regular and not z-regular; (5) a separated, Cauchy, zero space which is not weakly regular; (6) a topological space which is regular and µ-regular but not z-regular; (7) a filter, zero space which is regular and z-regular but not completely regu- lar; and, (8) a regular Hausdorff topological space which is z-regular but not completely regular. Keywords: merotopic space, nearness space, Cauchy space, filter merotopic space, pretopo- logical space, zero space, complete regularity, weak regularity, z-regularity, µ-regularity Classification: 54C30, 54C40, 54E17, 18B30 Introduction During the last three decades, topologists have studied several categories, e.g., • Conv, the category of convergence spaces [F59]. • Mer, the category of merotopic spaces [K63]. -
Which Spaces Have a Coarser Connected Hausdorff Topology?
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 125 (2002) 215–231 Which spaces have a coarser connected Hausdorff topology? William Fleissner, Jack Porter, Judith Roitman Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA Received 4 July 2000; received in revised form 2 October 2001 Abstract We present some answers to the title. For example, if K is compact, zero-dimensional and D | | is discrete, then K ⊕ D has a coarser connected topology iff w(K) 2 D . Similar theorems hold for ordinal spaces and spaces K ⊕ D where K is compact, not necessarily zero-dimensional. Every infinite cardinal has a coarser connected Hausdorff topology; so do Kunen lines, Ostaszewski spaces, and Ψ -spaces; but spaces X with X ⊂ βω and |βω \ X| < 2c do not. The statement “every locally countable, locally compact extension of ω with cardinality ω1 has a coarser connected topology” is consistent with and independent of ZFC. If X is a Hausdorff space and w(X) 2κ ,thenX can be embedded in a Hausdorff space of density κ. 2001 Elsevier Science B.V. All rights reserved. MSC: 54C10; 54D05; 54D35 Keywords: Connected; Condense; Ordinal; Coarser topology; Extension; Minimal Hausdorff; Franklin–Rajagopolan space 1. Goals All spaces in this paper are assumed to be Hausdorff. All ordinals (hence all cardinals) are assumed to have the order topology. For a space X,letτ(X) denote the collection of open sets of X. A Hausdorff topology σ ⊆ τ(X) is called a coarser topology. -
7560 THEOREMS with PROOFS Definition a Compactification of A
7560 THEOREMS WITH PROOFS De¯nition A compacti¯cation of a completely regular space X is a compact T2- space ®X and a homeomorphic embedding ® : X ! ®X such that ®(X) is dense in ®X. Remark. To explain the separation assumptions in the above de¯nition: we will only be concerned with compacti¯cations which are Hausdor®. Note that only completely regular spaces can have a T2-compacti¯cation. Remark. A few of the concepts and results below are from ¯rst year topology, and are included here for completeness. De¯nition. Let X be a locally compact non-compact Hausdor® space. The one-point compacti¯cation !X of X is the space X [ f1g, where points of X have their usual neighborhoods, and a neighborhood of 1 has the form f1g [ (X n K) where K is a compact subset of X. In the framework of the previous de¯nition, the map ! : X ! !X is the identity map on X. Of course, X is open in its one-point compacti¯cation. We'll see that this is the case for any compacti¯cation of a locally compact X. Lemma 1. If X is a dense subset of a T2 space Z, and X is locally compact, then X is open in Z. Proof. If x 2 X, let Nx be a compact neighborhood x in X and Ux = int(Nx). Then Ux = X \ Vx for some Vx open in Z. Thus Ux is dense in Vx and so Vx ½ V x = U x ½ N x = Nx where closures are taken in Z.