Notes on Point-Set Topology
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Notes on Topology
Notes on Topology Notes on Topology These are links to PostScript files containing notes for various topics in topology. I learned topology at M.I.T. from Topology: A First Course by James Munkres. At the time, the first edition was just coming out; I still have the photocopies we were given before the printed version was ready! Hence, I'm a bit biased: I still think Munkres' book is the best book to learn from. The writing is clear and lively, the choice of topics is still pretty good, and the exercises are wonderful. Munkres also has a gift for naming things in useful ways (The Pasting Lemma, the Sequence Lemma, the Tube Lemma). (I like Paul Halmos's suggestion that things be named in descriptive ways. On the other hand, some names in topology are terrible --- "first countable" and "second countable" come to mind. They are almost as bad as "regions of type I" and "regions of type II" which you still sometimes encounter in books on multivariable calculus.) I've used Munkres both of the times I've taught topology, the most recent occasion being this summer (1999). The lecture notes below follow the order of the topics in the book, with a few minor variations. Here are some areas in which I decided to do things differently from Munkres: ● I prefer to motivate continuity by recalling the epsilon-delta definition which students see in analysis (or calculus); therefore, I took the pointwise definition as a starting point, and derived the inverse image version later. ● I stated the results on quotient maps so as to emphasize the universal property. -
Topological Description of Riemannian Foliations with Dense Leaves
TOPOLOGICAL DESCRIPTION OF RIEMANNIAN FOLIATIONS WITH DENSE LEAVES J. A. AL¶ VAREZ LOPEZ*¶ AND A. CANDELy Contents Introduction 1 1. Local groups and local actions 2 2. Equicontinuous pseudogroups 5 3. Riemannian pseudogroups 9 4. Equicontinuous pseudogroups and Hilbert's 5th problem 10 5. A description of transitive, compactly generated, strongly equicontinuous and quasi-e®ective pseudogroups 14 6. Quasi-analyticity of pseudogroups 15 References 16 Introduction Riemannian foliations occupy an important place in geometry. An excellent survey is A. Haefliger’s Bourbaki seminar [6], and the book of P. Molino [13] is the standard reference for riemannian foliations. In one of the appendices to this book, E. Ghys proposes the problem of developing a theory of equicontinuous foliated spaces parallel- ing that of riemannian foliations; he uses the suggestive term \qualitative riemannian foliations" for such foliated spaces. In our previous paper [1], we discussed the structure of equicontinuous foliated spaces and, more generally, of equicontinuous pseudogroups of local homeomorphisms of topological spaces. This concept was di±cult to develop because of the nature of pseudogroups and the failure of having an in¯nitesimal characterization of local isome- tries, as one does have in the riemannian case. These di±culties give rise to two versions of equicontinuity: a weaker version seems to be more natural, but a stronger version is more useful to generalize topological properties of riemannian foliations. Another relevant property for this purpose is quasi-e®ectiveness, which is a generalization to pseudogroups of e®ectiveness for group actions. In the case of locally connected fo- liated spaces, quasi-e®ectiveness is equivalent to the quasi-analyticity introduced by Date: July 19, 2002. -
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.......................... -
Dense Embeddings of Sigma-Compact, Nowhere Locallycompact Metric Spaces
proceedings of the american mathematical society Volume 95. Number 1, September 1985 DENSE EMBEDDINGS OF SIGMA-COMPACT, NOWHERE LOCALLYCOMPACT METRIC SPACES PHILIP L. BOWERS Abstract. It is proved that a connected complete separable ANR Z that satisfies the discrete n-cells property admits dense embeddings of every «-dimensional o-compact, nowhere locally compact metric space X(n e N U {0, oo}). More gener- ally, the collection of dense embeddings forms a dense Gs-subset of the collection of dense maps of X into Z. In particular, the collection of dense embeddings of an arbitrary o-compact, nowhere locally compact metric space into Hubert space forms such a dense Cs-subset. This generalizes and extends a result of Curtis [Cu,]. 0. Introduction. In [Cu,], D. W. Curtis constructs dense embeddings of a-compact, nowhere locally compact metric spaces into the separable Hilbert space l2. In this paper, we extend and generalize this result in two distinct ways: first, we show that any dense map of a a-compact, nowhere locally compact space into Hilbert space is strongly approximable by dense embeddings, and second, we extend this result to finite-dimensional analogs of Hilbert space. Specifically, we prove Theorem. Let Z be a complete separable ANR that satisfies the discrete n-cells property for some w e ÍV U {0, oo} and let X be a o-compact, nowhere locally compact metric space of dimension at most n. Then for every map f: X -» Z such that f(X) is dense in Z and for every open cover <%of Z, there exists an embedding g: X —>Z such that g(X) is dense in Z and g is °U-close to f. -
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 . -
Solutions to Exercises in Munkres
April 21, 2006 Munkres §29 Ex. 29.1. Closed intervals [a, b] ∩ Q in Q are not compact for they are not even sequentially compact [Thm 28.2]. It follows that all compact subsets of Q have empty interior (are nowhere dense) so Q can not be locally compact. To see that compact subsets of Q are nowhere dense we may argue as follows: If C ⊂ Q is compact and C has an interior point then there is a whole open interval (a, b) ∩ Q ⊂ C and also [a, b] ∩ Q ⊂ C for C is closed (as a compact subset of a Hausdorff space [Thm 26.3]). The closed subspace [a, b] ∩ Q of C is compact [Thm 26.2]. This contradicts that no closed intervals of Q are compact. Ex. 29.2. Q (a). Assume that the product Xα is locally compact. Projections are continuous and open [Ex 16.4], so Xα is locally compact for all α [Ex 29.3]. Furthermore, there are subspaces U ⊂ C such that U is nonempty and open and C is compact. Since πα(U) = Xα for all but finitely many α, also πα(C) = Xα for all but finitely many α. But C is compact so also πα(C) is compact. Q (b). We have Xα = X1 × X2 where X1 is a finite product of locally compact spaces and X2 is a product of compact spaces. It is clear that finite products of locally compact spaces are locally compact for finite products of open sets are open and all products of compact spaces are compact by Tychonoff. -
Completeness Properties of the Generalized Compact-Open Topology on Partial Functions with Closed Domains
Topology and its Applications 110 (2001) 303–321 Completeness properties of the generalized compact-open topology on partial functions with closed domains L’ubica Holá a, László Zsilinszky b;∗ a Academy of Sciences, Institute of Mathematics, Štefánikova 49, 81473 Bratislava, Slovakia b Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA Received 4 June 1999; received in revised form 20 August 1999 Abstract The primary goal of the paper is to investigate the Baire property and weak α-favorability for the generalized compact-open topology τC on the space P of continuous partial functions f : A ! Y with a closed domain A ⊂ X. Various sufficient and necessary conditions are given. It is shown, e.g., that .P,τC/ is weakly α-favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α-favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τC, the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results. 2001 Elsevier Science B.V. All rights reserved. Keywords: Generalized compact-open topology; Fell topology; Compact-open topology; Topological games; Banach–Mazur game; Weakly α-favorable spaces; Baire spaces; Graph topology; π-base; Restriction mapping AMS classification: Primary 54C35, Secondary 54B20; 90D44; 54E52 0. -
Introduction to Algebraic Topology MAST31023 Instructor: Marja Kankaanrinta Lectures: Monday 14:15 - 16:00, Wednesday 14:15 - 16:00 Exercises: Tuesday 14:15 - 16:00
Introduction to Algebraic Topology MAST31023 Instructor: Marja Kankaanrinta Lectures: Monday 14:15 - 16:00, Wednesday 14:15 - 16:00 Exercises: Tuesday 14:15 - 16:00 August 12, 2019 1 2 Contents 0. Introduction 3 1. Categories and Functors 3 2. Homotopy 7 3. Convexity, contractibility and cones 9 4. Paths and path components 14 5. Simplexes and affine spaces 16 6. On retracts, deformation retracts and strong deformation retracts 23 7. The fundamental groupoid 25 8. The functor π1 29 9. The fundamental group of a circle 33 10. Seifert - van Kampen theorem 38 11. Topological groups and H-spaces 41 12. Eilenberg - Steenrod axioms 43 13. Singular homology theory 44 14. Dimension axiom and examples 49 15. Chain complexes 52 16. Chain homotopy 59 17. Relative homology groups 61 18. Homotopy invariance of homology 67 19. Reduced homology 74 20. Excision and Mayer-Vietoris sequences 79 21. Applications of excision and Mayer - Vietoris sequences 83 22. The proof of excision 86 23. Homology of a wedge sum 97 24. Jordan separation theorem and invariance of domain 98 25. Appendix: Free abelian groups 105 26. English-Finnish dictionary 108 References 110 3 0. Introduction These notes cover a one-semester basic course in algebraic topology. The course begins by introducing some fundamental notions as categories, functors, homotopy, contractibility, paths, path components and simplexes. After that we will study the fundamental group; the Fundamental Theorem of Algebra will be proved as an application. This will take roughly the first half of the semester. During the second half of the semester we will study singular homology. -
General Topology
General Topology Andrew Kobin Fall 2013 | Spring 2014 Contents Contents Contents 0 Introduction 1 1 Topological Spaces 3 1.1 Topology . .3 1.2 Basis . .5 1.3 The Continuum Hypothesis . .8 1.4 Closed Sets . 10 1.5 The Separation Axioms . 11 1.6 Interior and Closure . 12 1.7 Limit Points . 15 1.8 Sequences . 16 1.9 Boundaries . 18 1.10 Applications to GIS . 20 2 Creating New Topological Spaces 22 2.1 The Subspace Topology . 22 2.2 The Product Topology . 25 2.3 The Quotient Topology . 27 2.4 Configuration Spaces . 31 3 Topological Equivalence 34 3.1 Continuity . 34 3.2 Homeomorphisms . 38 4 Metric Spaces 43 4.1 Metric Spaces . 43 4.2 Error-Checking Codes . 45 4.3 Properties of Metric Spaces . 47 4.4 Metrizability . 50 5 Connectedness 51 5.1 Connected Sets . 51 5.2 Applications of Connectedness . 56 5.3 Path Connectedness . 59 6 Compactness 63 6.1 Compact Sets . 63 6.2 Results in Analysis . 66 7 Manifolds 69 7.1 Topological Manifolds . 69 7.2 Classification of Surfaces . 70 7.3 Euler Characteristic and Proof of the Classification Theorem . 74 i Contents Contents 8 Homotopy Theory 80 8.1 Homotopy . 80 8.2 The Fundamental Group . 81 8.3 The Fundamental Group of the Circle . 88 8.4 The Seifert-van Kampen Theorem . 92 8.5 The Fundamental Group and Knots . 95 8.6 Covering Spaces . 97 9 Surfaces Revisited 101 9.1 Surfaces With Boundary . 101 9.2 Euler Characteristic Revisited . 104 9.3 Constructing Surfaces and Manifolds of Higher Dimension . -
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. -
Essays on Topological Vector Spaces Quasi-Complete
Last revised 11:47 a.m. January 7, 2017 Essays on topological vector spaces Bill Casselman University of British Columbia [email protected] Quasi-complete TVS Suppose G to be a locally compact group. In the theory of representations of G, an indispensable role is played by an action of the convolution algebra Cc(G) on the space V of a continuous representation of G. In order to define this, one must know how to make sense of integrals f(x) dx ZG where f is a continuous function of compact support on G with values in V . This is not quite a trivial matter. If V is assumed to be complete, a definition by Riemann sums works quite well, but this is an unnecessarily strong requirement. The natural condition on V seems to be that it be quasi-complete, a somewhat technical but nonetheless valuable condition. Continuous representations of a locally compact group are almost always assumed to be on a quasi•complete topological vector space. I can illustrate here quite briefly what the problem is. Suppose M to be a manifold assigned a smooth measure dm, and suppose f a continuous, compactly supported function on M with values in the TVS (i.e. locally convex, Hausdorff, topological vector space) V . How is one to define the integral I = f(m) dm ? ZM If V is complete, as I say, this is not difficult. In general, what one can do, according to the Hahn•Banach Theorem, is characterize the integral uniquely by the condition that F, I = F, f(m) dm = F,f(m) dm D ZM E ZM for any F in the continuous linear dual of V . -
Locally Compact Simple Rings Having Minimal Left Ideals
LOCALLY COMPACT SIMPLE RINGS HAVING MINIMAL LEFT IDEALS BY SETH WARNERC) Let A be an indiscrete locally compact primitive ring having minimal left ideals. Algebraically, A may be regarded as a dense ring of linear operators containing nonzero linear operators of finite rank on a vector space E over a division ring K, and furthermore if A is simple, then A contains only linear operators of finite rank. The topology of A induces topologies on F and A"in a natural way, so that K becomes a locally compact division ring and E a locally compact vector space over K. Theorems about locally compact division rings and locally compact vector spaces imply that if K is indiscrete, then E is necessarily finite dimensional over K, and thus A is the ring of all linear operators on a finite-dimensional vector space over a locally compact division ring. Kaplansky proved that if A has characteristic zero, then K is indeed indiscrete, but he showed by an example that E could be infinite dimensional. Kaplansky remarked [9, p. 459], however, that if A is simple, it seemed unlikely that F could be infinite dimensional, since otherwise "complete- ness appears to require the presence of linear transformations with infinite- dimensional range." Our principal result in §1 is the verification of this conjecture; however, our argument is based not on the fact that a locally compact ring is complete, but rather on the fact that a locally compact space is a Baire space and on the symmetry between minimal left and right ideals in a simple ring.