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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 . -
[Math.GN] 25 Dec 2003
Problems from Topology Proceedings Edited by Elliott Pearl arXiv:math/0312456v1 [math.GN] 25 Dec 2003 Topology Atlas, Toronto, 2003 Topology Atlas Toronto, Ontario, Canada http://at.yorku.ca/topology/ [email protected] Cataloguing in Publication Data Problems from topology proceedings / edited by Elliott Pearl. vi, 216 p. Includes bibliographical references. ISBN 0-9730867-1-8 1. Topology—Problems, exercises, etc. I. Pearl, Elliott. II. Title. Dewey 514 20 LC QA611 MSC (2000) 54-06 Copyright c 2003 Topology Atlas. All rights reserved. Users of this publication are permitted to make fair use of the material in teaching, research and reviewing. No part of this publication may be distributed for commercial purposes without the prior permission of the publisher. ISBN 0-9730867-1-8 Produced November 2003. Preliminary versions of this publication were distributed on the Topology Atlas website. This publication is available in several electronic formats on the Topology Atlas website. Produced in Canada Contents Preface ............................................ ............................v Contributed Problems in Topology Proceedings .................................1 Edited by Peter J. Nyikos and Elliott Pearl. Classic Problems ....................................... ......................69 By Peter J. Nyikos. New Classic Problems .................................... ....................91 Contributions by Z.T. Balogh, S.W. Davis, A. Dow, G. Gruenhage, P.J. Nyikos, M.E. Rudin, F.D. Tall, S. Watson. Problems from M.E. Rudin’s Lecture notes in set-theoretic topology ..........103 By Elliott Pearl. Problems from A.V. Arhangel′ski˘ı’s Structure and classification of topological spaces and cardinal invariants ................................................... ...123 By A.V. Arhangel′ski˘ıand Elliott Pearl. A note on P. Nyikos’s A survey of two problems in topology ..................135 By Elliott Pearl. -
Chapter 7 Separation Properties
Chapter VII Separation Properties 1. Introduction “Separation” refers here to whether objects such as points or disjoint closed sets can be enclosed in disjoint open sets. In spite of the similarity of terminology, “separation properties” have no direct connection to the idea of “separated sets” that appeared in Chapter 5 in the context of connected spaces. 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 hypotheses for “more separation” are added, spaces 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 original literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , ,and X % , each one stronger than its predecessor in the list. Once these had become common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: X" Þ 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 YB−YCÂY with , or there exists an open set Zwith C−ZBÂZ, 2) X"-space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set Zwith BÂYßC−Z 3) X#-space (or, Hausdorff space) if, whenever BÁC−\, there exist disjoint open sets Y and Z in \ such that B−Y and C−Z . -
THREE DIMES of TOPOLOGY A. Candel Class Notes for Math 262, Winter 95-96, the University of Chicago. Preface Chapter I. Topologi
THREE DIMES OF TOPOLOGY A. Candel Class Notes for Math 262, Winter 95-96, The University of Chicago. Preface Chapter I. Topological spaces 1. The concept of topological space 2. Bases and subbases 3. Subspaces, unions and hyperspaces 4. Pseudometric and metric spaces 5. The order topology 6. More examples Chapter II. Continuous maps 1. Continuous maps 2. Continuity and convergence Chapter III. Product and quotient spaces 1. The product topology 2. Quotient spaces 3. Cutting and pasting Chapter IV. Connected and path-connected spaces 1. Connected spaces 2. Path-connected spaces 3. Local connectedness and path-connectedness Chapter V. Convergence 1. Hausdor® spaces 2. Countability axioms 3. Filters 4. Ultra¯lters Chapter VI. Compact spaces 1. Compact spaces 2. The Tichonov theorem 3. Local compactness 4. Compacti¯cation 5. Components of a compact hausdor® space Chapter VII. The Urysohn lemma 1. Regular and normal spaces Typeset by AMS-TEX 1 2 TOPOLOGY 2. Urysohn and Tietze lemmas 3. Embeddings 4. Stone-Cec· h compacti¯cation 5. Metrization Chapter VIII. Paracompact spaces 1. Paracompact spaces 2. Partitions of unity Chapter IX. Function spaces 1. Topologies on function spaces 2. The evaluation map 3. Uniform spaces 4. Ascoli's theorem Chapter X. Topological groups 1. Topological groups 2. Topological vector spaces 3. Haar measure 4. Stone-Weierstrass theorem 5. Topological spaces and Banach algebras Homeworks and exams Bibliography NOTE: These notes are being o®ered without warranty. Bibliography and references have not been collected. Chapter X does not exist. 1. THE CONCEPT OF TOPOLOGICAL SPACE 1 CHAPTER I TOPOLOGICAL SPACES 1. -
Topology T : the Union, and the finite Intersection, of Open Sets Are Open
Joseph Muscat 2015 1 Topological Spaces [email protected] 9 August 2014 A review on filters of subsets and nets is found as an appendix. 1 Convergence Spaces A filter of subsets captures the idea of a refinement process, or gathering of information; The filter sets may represent imperfect knowledge or a fuzzy `mea- surement'. Which set of points remains in the `limit' of taking smaller sets of the filter? It need not only consist of the points common to all the filter sets (which may be empty), but may include other `cluster' points. A cluster space is a set X with a cluster function mapping proper filters F of X to subsets of X, such that cl(F\G) = cl(F) [ cl(G); \ F ⊆ cl(F) Some notation: F 99K A means A = cl(F); a cluster point of F is an element X x 2 cl(F), here denoted by F 99K x. The cluster set of the improper filter 2 can be taken to be ?. The set of cluster points of a subset is cl(A) := clF(A). The morphisms, called continuous maps, preserve clustering, F 99K x ) f(F) 99K f(x) i.e., fcl(F) ⊆ clf(F) and clf −1F ⊆ f −1cl(F), e.g. constant maps. An isomor- phism is called a homeomorphism, and the set of morphisms X ! Y is denoted C(X; Y ). Examples: A trivial space has cl(F) := X; a discrete space has cl(F) := T F, so that F 99K x , F ⊆ F(x). The standard clustering of N is taken to be discrete. -
Chapter 3: Topological Spaces
Chapter III Topological Spaces 1. Introduction In Chapter I we looked at properties of sets, and in Chapter II we added some additional structure to a set a distance function . to create a pseudometric space. We then saw some of the most basic fundamental definitions and properties of pseudometric spaces. There is much more, and some of the most useful and interesting properties of pseudometric spaces will be discussed in Chapter IV. But in Chapter III we look at an important generalization. We observed, early in Chapter II, that the idea of continuity (in calculus) depends on talking about “nearness,” so we used a distance function . to make the idea of “nearness” precise. In that way, we were able to extend the definition of continuity from ‘8 to pseudometric spaces. The distance function . also led us to the idea of open sets in a pseudometric space. From there we developed properties of closed sets, closures, interiors, frontiers, dense sets, continuity, and sequential convergence. An important observation in Chapter II was that open (or closed) sets are really all that we require to talk about many of these ideas. In other words, we can often do what's necessary using the open sets without knowing which specific . generated the open sets: the topology g. is what really matters. For example, intE is defined in Ð\ß.Ñin terms of the open sets ßE so int doesn't change if . is replaced with a different but equivalent metric .w one that generates the same open sets. Similarly, changing . to an equivalent metric . -
Topological Spaces
Chapter III Topological Spaces 1. Introduction In Chapter I we looked at properties of sets, and in Chapter II we added some additional structure to a set. a distance function to create a pseudometric space. We then looked at some of the most basic definitions and properties of pseudometric spaces. There is much more, and some of the most useful and interesting properties of pseudometric spaces will be discussed in Chapter IV. But in Chapter III we look at an important generalization. Early in Chapter II we observed that the idea of continuity (in calculus) depends on talking about “nearness,” so we used a distance function . to make the idea of “nearness” precise. The result was that we could carry over the definition of continuity from calculus to pseudometric spaces. The distance function . also led us to the idea of an open set in a pseudometric space. From there we developed properties of closed sets, closures, interiors, frontiers, dense sets, continuity, and sequential convergence. One important observation was that open (or closed) sets are all we need to work with many of these concepts; that is, we can often do what we need using the open sets without knowing what specific . that generated these open sets: the topology g. is what really matters. For example, clEÐ\ß.ÑßE. is defined in in terms of the closed sets so cl doesn't change if is replaced with a different but equivalent metric .w one that generates the same open sets. Changing . to an equivalent metric .w also doesn't affect interiors, continuity, or convergent sequences. -
MA-231 Topology 14
Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore August-December 2004 MA-231 Topology 14. Separation Axioms1) —————————— November 29, 2004 Frigyes Riesz† Andrey Nikolaevich Kolmogorov†† (1880-1956) (1903-1987) First recall the following definitions and results : 2 N14.1 . (Regular spaces ; T3-spaces )) Let X be a topological space. We say that X is regular space ifwhenever A is a closed subset in X and x ∈ A, then there are disjoint open subsets U and V with x ∈ U and A ⊆ V . A regular T1 space is called a T3-space. It is clear that every T3-space is T2. But not every T2-space is T3, for example, a). For a topological space the following statements are equivalent : (i) X is regular. (ii) If U is open in X and if x ∈ U, then there is an open subset V containing x such that V ⊆ U . (iii) Each x ∈ X has a nhood base consisting of closed sets. b). (1) Every subspace of a regular (resp. T3 ) space is completely regular (resp. T3 ). (2) A non-empty product space is regular (resp. T3 ) space if and only if each factor space is regular (resp. T3 ). (3) Quotients of T3 spaces need not be regular. 3 N14.2 . (Completely regular spaces ; T 1 – Tychonoff spaces )) Let X be a topological space. 3 2 We say that X is completely regular space ifwhenever A is a closed subset in X and x ∈ A, then there is a continuous function f : X → [0, 1] such that f(x)= 0 and f(a)= 1 for all a ∈ A .