Notes on Introductory Point-Set Topology
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De Rham Cohomology
De Rham Cohomology 1. Definition of De Rham Cohomology Let X be an open subset of the plane. If we denote by C0(X) the set of smooth (i. e. infinitely differentiable functions) on X and C1(X), the smooth 1-forms on X (i. e. expressions of the form fdx + gdy where f; g 2 C0(X)), we have natural differntiation map d : C0(X) !C1(X) given by @f @f f 7! dx + dy; @x @y usually denoted by df. The kernel for this map (i. e. set of f with df = 0) is called the zeroth De Rham Cohomology of X and denoted by H0(X). It is clear that these are precisely the set of locally constant functions on X and it is a vector space over R, whose dimension is precisley the number of connected components of X. The image of d is called the set of exact forms on X. The set of pdx + qdy 2 C1(X) @p @q such that @y = @x are called closed forms. It is clear that exact forms and closed forms are vector spaces and any exact form is a closed form. The quotient vector space of closed forms modulo exact forms is called the first De Rham Cohomology and denoted by H1(X). A path for this discussion would mean piecewise smooth. That is, if γ : I ! X is a path (a continuous map), there exists a subdivision, 0 = t0 < t1 < ··· < tn = 1 and γ(t) is continuously differentiable in the open intervals (ti; ti+1) for all i. -
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. -
The Topology of Subsets of Rn
Lecture 1 The topology of subsets of Rn The basic material of this lecture should be familiar to you from Advanced Calculus courses, but we shall revise it in detail to ensure that you are comfortable with its main notions (the notions of open set and continuous map) and know how to work with them. 1.1. Continuous maps “Topology is the mathematics of continuity” Let R be the set of real numbers. A function f : R −→ R is called continuous at the point x0 ∈ R if for any ε> 0 there exists a δ> 0 such that the inequality |f(x0) − f(x)| < ε holds for all x ∈ R whenever |x0 − x| <δ. The function f is called contin- uous if it is continuous at all points x ∈ R. This is basic one-variable calculus. n Let R be n-dimensional space. By Or(p) denote the open ball of radius r> 0 and center p ∈ Rn, i.e., the set n Or(p) := {q ∈ R : d(p, q) < r}, where d is the distance in Rn. A function f : Rn −→ R is called contin- n uous at the point p0 ∈ R if for any ε> 0 there exists a δ> 0 such that f(p) ∈ Oε(f(p0)) for all p ∈ Oδ(p0). The function f is called continuous if it is continuous at all points p ∈ Rn. This is (more advanced) calculus in several variables. A set G ⊂ Rn is called open in Rn if for any point g ∈ G there exists a n δ> 0 such that Oδ(g) ⊂ G. -
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)). -
TOPOLOGY and ITS APPLICATIONS the Number of Complements in The
TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 55 (1994) 101-125 The number of complements in the lattice of topologies on a fixed set Stephen Watson Department of Mathematics, York Uniuersity, 4700 Keele Street, North York, Ont., Canada M3J IP3 (Received 3 May 1989) (Revised 14 November 1989 and 2 June 1992) Abstract In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies 7 and (T are complementary if and only if 7 A c = 0 and 7 V (T = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality K has at least K distinct complements and at most 2” many distinct complements. -
Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X). -
Open Sets in Topological Spaces
International Mathematical Forum, Vol. 14, 2019, no. 1, 41 - 48 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.913 ii – Open Sets in Topological Spaces Amir A. Mohammed and Beyda S. Abdullah Department of Mathematics College of Education University of Mosul, Mosul, Iraq Copyright © 2019 Amir A. Mohammed and Beyda S. Abdullah. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce a new class of open sets in a topological space called 푖푖 − open sets. We study some properties and several characterizations of this class, also we explain the relation of 푖푖 − open sets with many other classes of open sets. Furthermore, we define 푖푤 − closed sets and 푖푖푤 − closed sets and we give some fundamental properties and relations between these classes and other classes such as 푤 − closed and 훼푤 − closed sets. Keywords: 훼 − open set, 푤 − closed set, 푖 − open set 1. Introduction Throughout this paper we introduce and study the concept of 푖푖 − open sets in topological space (푋, 휏). The 푖푖 − open set is defined as follows: A subset 퐴 of a topological space (푋, 휏) is said to be 푖푖 − open if there exist an open set 퐺 in the topology 휏 of X, such that i. 퐺 ≠ ∅,푋 ii. A is contained in the closure of (A∩ 퐺) iii. interior points of A equal G. One of the classes of open sets that produce a topological space is 훼 − open. -
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). -
The Seifert-Van Kampen Theorem Via Covering Spaces
Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona The Seifert-Van Kampen theorem via covering spaces Autor: Roberto Lara Martín Director: Dr. Javier José Gutiérrez Marín Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, 29 de juny de 2017 Contents Introduction ii 1 Category theory 1 1.1 Basic terminology . .1 1.2 Coproducts . .6 1.3 Pushouts . .7 1.4 Pullbacks . .9 1.5 Strict comma category . 10 1.6 Initial objects . 12 2 Groups actions 13 2.1 Groups acting on sets . 13 2.2 The category of G-sets . 13 3 Homotopy theory 15 3.1 Homotopy of spaces . 15 3.2 The fundamental group . 15 4 Covering spaces 17 4.1 Definition and basic properties . 17 4.2 The category of covering spaces . 20 4.3 Universal covering spaces . 20 4.4 Galois covering spaces . 25 4.5 A relation between covering spaces and the fundamental group . 26 5 The Seifert–van Kampen theorem 29 Bibliography 33 i Introduction The Seifert-Van Kampen theorem describes a way of computing the fundamen- tal group of a space X from the fundamental groups of two open subspaces that cover X, and the fundamental group of their intersection. The classical proof of this result is done by analyzing the loops in the space X and deforming them into loops in the subspaces. For all the details of such proof see [1, Chapter I]. The aim of this work is to provide an alternative proof of this theorem using covering spaces, sets with actions of groups and category theory. -
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 .