DOCSLIB.ORG

C-Space,Connectivity Space, C-Continuous Function, Quotient Space, Topologizable and Graphical C-Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
C-Space,Connectivity Space, C-Continuous Function, Quotient Space, Topologizable and Graphical C-Spaces Bol. Soc. Paran. Mat. (3s.) v. 35 1 (2017): 97–109. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v35i1.24592 On the quotients of c-spaces Santhosh P.K. abstract: In this paper, properties of quotient maps in c-spaces are studied in detail. A method of finding quotient space of topologizable and graphical c-spaces are described. Key Words: c-space,connectivity space, c-continuous function, quotient space, topologizable and graphical c-spaces Contents 1 Introduction 97 2 Preliminaries 98 3 Weak c-structure Generated by a Family of Functions 100 3.1 Examples of Quotient Spaces ......................101 4 Properties of Quotient Spaces 101 5 Quotient Space Verses Equivalence Classes 104 6 Quotient Space of Topologizable c-spaces 104 7 Quotient space of Graphical c-spaces 107 1. Introduction The concept of connected sets can be found in topology, graph theory, order structure and in their fuzzy analogues. A subset A of a topological space X is said to be connected if it cannot be separated into two disjoint non empty open subsets. In this definition, separation is emphasized. When working with graphs, connectivity is defined in terms of paths.A graph G is said to be connected if for every vertices x and y there is a path in G with end vertices x and y. Although these classical definitions have been extensively applied in image pro- cessing and analysis, these two approaches( i.e., separation by open sets and joining points by path) are incompatible [9]. For the sake of completeness of this paper, let us illustrate this with an example [9] here. Consider the digital space Z2. Define two adjacency relations, the 4- adjacency and 8-adjacency, as follows. Let (i, j) and (i′, j′) ∈ Z2. 2000 Mathematics Subject Classification: 54A05, 05C40 BSP Typeset by Mstyle. 97 c Soc. Paran. de Mat. 98 Santhosh P.K. (i, j) is 4-adjacent to (i′, j′) if |i − i′| + |j − j′| =1 and (i, j) is 8-adjacent to (i′, j′) if max{|i − i′|, |j − j′|} =1. Thus we get two graphs with Z2 as set of vertices and edges linking pairs of adjacent points (4 or 8-adjacency respectively). This leads to the classical notion of 4-connectivity and 8-connectivity of subsets of Z2. But it is proved that there is a topology on Z2 for which connected sets are precisely 4-connected sets and there is no topology on Z2 for which connectedness is equivalent to 8-connectedness. Further, it is proved that [2], any odd cycle Cn, n ≥ 5 has no compatible topology. Thus there are graphs where connectivity does not arise from a topology. Conversely there are topological spaces which are not graphical. For example, in [9] it is noted that, in Rn with n ≥ 1, every pair of points is disconnected and that it is not graphical. That is, topological connectivity of Rn cannot arise from a graph. In general, topological connectivity is useful for images defined over a contin- uous space where as graph theoretic connectivity is useful for images defined over a discrete space. Compatibility is essential as discrete images are often obtained from the discretization of the continuous scenes. Thus topological or graph theo- retical approach to connectivity limits the application of connectivity or the use of connected sets in practical purposes. In 1983,Bo¨rger R. [1] proposed an axiomatic approach to connectivity, known as the theory of connectivity classes or c-structures, which overcomes the shortfalls of the classical definitions of connectivity. Here the axioms involve the minimum properties that connected sets can have. That is, one point sets are connected and that union of connected sets having a common point is connected. A systematic study of this space was further carried out by J. Serra [12], H. J. A. M. Hejimans [4] and C. Ronse [9] etc. This space found profound applications in the areas of Image segmentation, Image Filtering, Image coding, Digital Topology, Pattern Recognition, Mathematical Morphology etc [4,5,10,12,13]. In this paper, some ex- isting concepts are formalized and explored some properties of quotient spaces, which paves the foundation for our further work. 2. Preliminaries Let X be a non empty set and CX be a collection of subsets of X such that the following properties hold. (i) φ ∈ CX and {x} ∈ CX for every x ∈ X. (ii) If {Ci : i ∈ I} be a non empty collection of subsets in C with ∩ Ci 6= φ, then i∈I ∪ Ci ∈ CX . i∈I Then CX is called a c-structure on X and X with a c-structure on it is called a c-space. A c-space X is said to be a connective space [7] if CX satisfies two more conditions as given below. On the quotients of c-spaces 99 (iii) Given any non empty sets A, B ∈ CX with A ∪ B ∈ CX , then there exists x ∈ A ∪ B such that {x} ∪ A ∈ CX and {x} ∪ B ∈ CX . (iv) If A, B, Ci ∈ CX are disjoint and A ∪ B ∪ Ci ∈ CX , then there exists J ⊆ I iS∈I such that A ∪ Cj ∈ CX and B ∪ Ci ∈ CX . jS∈J i∈SI−J A c-structure satisfies above two conditions are called a connective structure or connectology on X. Other terminologies used for c-spaces are connectivity space [1,12] and integral connectivity space [3]. In his paper [1], Reinhard Bo¨rger have the same definition for the connectivity space except that empty set is connected. Another terminology used for a c-structure is connectivity class [4,5,12] of X. We like to follow the terminology used in [7]. Elements of a c-structure are called connected sets. DX = {φ}∪{{x} : x ∈ X} is a c-structure on X, called the discrete c-structure and the space (X, DX ) is called the discrete c-space. Also, IX = P(X), the power set of X is a c-structure on X, called the indiscrete c-structure and the corre- sponding space is called an indiscrete c-space. A c-space (Y, CY ) is said to be a sub c-space of the c-space (X, CX ) if Y ⊆ X and CY = {A ∈ CX : A ⊆ Y }. Let B ⊆ P(X), the power set of X. Then the smallest c-structure on X containing B is called the c-structure generated by B and is denoted by < B >. The c-space (X, CX ) is denoted by X if there is no ambiguity. Let X and Y be two c-spaces and f : X → Y be a function. f is called c- continuous [7] or catenuous [7] or a connectivity morphism [4,12] or a connectivity map [3], if it maps connected sets of X to connected sets of Y . In [7], it is proved that the restriction of a c-continuous function and composition of two c-continuous functions are c-continuous. Also, a bijection f is said to be a c-isomorphism or catenomorphism if both f and f −1 are c-continuous. In [11], it is proved that if f : X → Y is a bijection, then f is a c-isomorphism if and only if f(CA)= CB. Let {Xi : i ∈ I} be a family of c-spaces and {fi : X → Xi : i ∈ I} be a family of functions defined on a set X. Let C be a collection of subsets of X such that A ∈ C C C if and only if fi(A) ∈ Xi for every i. Then is a c-structure on X and is called the strong c-structure generated by the given family of functions [11] and is denoted by < {fi : i ∈ I} >S . The c-structure on the product space Xi is the strong iQ∈I c-structure generated by the family of projection functions {πi : i ∈ I}. Hence the connected subsets of Rn are precisely those subsets of Rn whose projections are intervals. For example, the following figures represents some connected subsets and a disconnected subset of R2. 100 Santhosh P.K. A chain of connected sets [7] is a map C from a finite index set I to CX such that each Ci intersects its successor Ci+1, where Ci = C(i) for i ∈ I. Proposition 2.1. [1,7] Let B be a collection of subsets of a set X and let < B > be a c-structure on X generated by this collection(That is, the smallest c-structure on X containing B). Then the non trivial connected sets in < B > are characterized by the condition that any two points of such a connected set C can be joined by a finite chain of basic connected sets( i.e. elements of B) in C. That is, for all x, y ∈ C, we can find elements Bi, i = 0 to n in B such that Bi ⊆ C, Bi ∩ Bi+1 6= φ for i = 0 to n − 1 and x ∈ B0, y ∈ Bn. 3. Weak c-structure Generated by a Family of Functions Quotient spaces plays a vital role in all branches of mathematics. Though not stated in a formal language, the smallest c-structure on a set which makes a family of functions c-continuous can be found in [1] and for a single function in [3]. Here we are stating it formally as weak c-structure generated by a family of functions and quotient spaces are introduced as a particular case of this. Formalizing the concepts found in [1,3], we state the following definitions.
Load more

Recommended publications
  • 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..........................
    [Show full text]
  • Discrete Mathematics Panconnected Index of Graphs
    Discrete Mathematics 340 (2017) 1092–1097 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Panconnected index of graphs Hao Li a,*, Hong-Jian Lai b, Yang Wu b, Shuzhen Zhuc a Department of Mathematics, Renmin University, Beijing, PR China b Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA c Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, PR China article info a b s t r a c t Article history: For a connected graph G not isomorphic to a path, a cycle or a K1;3, let pc(G) denote the Received 22 January 2016 smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is Received in revised form 25 October 2016 a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a Accepted 25 October 2016 cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and Available online 30 November 2016 t, respectively, then P is called a divalent (s; t)-path. Let `(G) D maxfm V G has a divalent g Keywords: path of length m that is not both of length 2 and in a K3 . We prove the following. Iterated line graphs (i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is Panconnectedness essentially 3-edge-connected. Panconnected index of graphs (ii) pc(G) ≤ `(G) C 2.
    [Show full text]
  • On the Simple Connectedness of Hyperplane Complements in Dual Polar Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Ghent University Academic Bibliography On the simple connectedness of hyperplane complements in dual polar spaces I. Cardinali, B. De Bruyn∗ and A. Pasini Abstract Let ∆ be a dual polar space of rank n ≥ 4, H be a hyperplane of ∆ and Γ := ∆\H be the complement of H in ∆. We shall prove that, if all lines of ∆ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings. Keywords: diagram geometry, dual polar spaces, hyperplanes, simple connected- ness, universal embedding MSC: 51E24, 51A50 1 Introduction We presume the reader is familiar with notions as simple connectedness, hy- perplanes, hyperplane complements, full projective embeddings, hyperplanes arising from an embedding, and other concepts involved in the theorems to be stated in this introduction. If not, the reader may see Section 2 of this paper, where those concepts are recalled. The following is the main theorem of this paper. We shall prove it in Section 3. Theorem 1.1 Let ∆ be a dual polar space of rank ≥ 4 with at least 4 points on each line. If H is a hyperplane of ∆ then the complement ∆ \ H of H in ∆ is simply connected. ∗Postdoctoral Fellow on the Research Foundation - Flanders 1 Not so much is known on ∆ \ H when rank(∆) = 3. The next theorem is one of the few results obtained so far for that case.
    [Show full text]
  • Maximally Edge-Connected and Vertex-Connected Graphs and Digraphs: a Survey Angelika Hellwiga, Lutz Volkmannb,∗
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 3265–3296 www.elsevier.com/locate/disc Maximally edge-connected and vertex-connected graphs and digraphs: A survey Angelika Hellwiga, Lutz Volkmannb,∗ aInstitut für Medizinische Statistik, Universitätsklinikum der RWTH Aachen, PauwelsstraYe 30, 52074 Aachen, Germany bLehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany Received 8 December 2005; received in revised form 15 June 2007; accepted 24 June 2007 Available online 24 August 2007 Abstract Let D be a graph or a digraph. If (D) is the minimum degree, (D) the edge-connectivity and (D) the vertex-connectivity, then (D) (D) (D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge- connected if (D) = (D) and maximally vertex-connected if (D) = (D). In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively. © 2007 Elsevier B.V. All rights reserved. Keywords: Connectivity; Edge-connectivity; Vertex-connectivity; Restricted connectivity; Conditional connectivity; Super-edge-connectivity; Local-edge-connectivity; Minimum degree; Degree sequence; Diameter; Girth; Clique number; Line graph 1. Introduction Numerous networks as, for example, transport networks, road networks, electrical networks, telecommunication systems or networks of servers can be modeled by a graph or a digraph. Many attempts have been made to determine how well such a network is ‘connected’ or stated differently how much effort is required to break down communication in the system between some vertices.
    [Show full text]
  • Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space
    S S symmetry Article Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space Susmit Bagchi Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660701, Korea; [email protected] Abstract: The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. Citation: Bagchi, S. Connected However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex Fundamental Groups and Homotopy subspaces retain the path-connection property irrespective of the projective points on real subspace.
    [Show full text]
  • Solutions to Problem Set 4: Connectedness
    Solutions to Problem Set 4: Connectedness Problem 1 (8). Let X be a set, and T0 and T1 topologies on X. If T0 ⊂ T1, we say that T1 is finer than T0 (and that T0 is coarser than T1). a. Let Y be a set with topologies T0 and T1, and suppose idY :(Y; T1) ! (Y; T0) is continuous. What is the relationship between T0 and T1? Justify your claim. b. Let Y be a set with topologies T0 and T1 and suppose that T0 ⊂ T1. What does connectedness in one topology imply about connectedness in the other? c. Let Y be a set with topologies T0 and T1 and suppose that T0 ⊂ T1. What does one topology being Hausdorff imply about the other? d. Let Y be a set with topologies T0 and T1 and suppose that T0 ⊂ T1. What does convergence of a sequence in one topology imply about convergence in the other? Solution 1. a. By definition, idY is continuous if and only if preimages of open subsets of (Y; T0) are open subsets of (Y; T1). But, the preimage of U ⊂ Y is U itself. Hence, the map is continuous if and only if open subsets of (Y; T0) are open in (Y; T1), i.e. T0 ⊂ T1, i.e. T1 is finer than T0. ` b. If Y is disconnected in T0, it can be written as Y = U V , where U; V 2 T0 are non-empty. Then U and V are also open in T1; hence, Y is disconnected in T1 too. Thus, connectedness in T1 imply connectedness in T0.
    [Show full text]
  • CONNECTEDNESS-Notes Def. a Topological Space X Is
    CONNECTEDNESS-Notes Def. A topological space X is disconnected if it admits a non-trivial splitting: X = A [ B; A \ B = ;; A; B open in X; and non-empty. (We'll abbreviate `disjoint union' of two subsets A and B {meaning A\B = ;{ by A t B.) X is connected if no such splitting exists. A subset C ⊂ X is connected if it is a connected topological space, when endowed with the induced topology (the open subsets of C are the intersections of C with open subsets of X.) Note that X is connected if and only if the only subsets of X that are simultaneously open and closed are ; and X. Example. The real line R is connected. The connected subsets of R are exactly the intervals. (See [Fleming] for a proof.) Def. A topological space X is path-connected if any two points p; q 2 X can be joined by a continuous path in X, the image of a continuous map γ : [0; 1] ! X, γ(0) = p; γ(1) = q. Path-connected implies connected: If X = AtB is a non-trivial splitting, taking p 2 A; q 2 B and a path γ in X from p to q would lead to a non- trivial splitting [0; 1] = γ−1(A) t γ−1(B) (by continuity of γ), contradicting the connectedness of [0; 1]. Example: Any normed vector space is path-connected (connect two vec- tors v; w by the line segment t 7! tw + (1 − t)v; t 2 [0; 1]), hence connected. The continuous image of a connected space is connected: Let f : X ! Y be continuous, with X connected.
    [Show full text]
  • A Connectedness Principle in the Geometry of Positive Curvature Fuquan Fang,1 Sergio´ Mendonc¸A, Xiaochun Rong2
    communications in analysis and geometry Volume 13, Number 4, 671-695, 2005 A Connectedness principle in the geometry of positive curvature Fuquan Fang,1 Sergio´ Mendonc¸a, Xiaochun Rong2 The main purpose of this paper is to develop a connectedness prin- ciple in the geometry of positive curvature. In the form, this is a surprising analog of the classical connectedness principle in com- plex algebraic geometry. The connectedness principle, when ap- plied to totally geodesic immersions, provides not only a uniform formulation for the classical Synge theorem, the Frankel theorem and a recent theorem of Wilking for totally geodesic submanifolds, but also new connectedness theorems for totally geodesic immer- sions in the geometry of positive curvature. However, the connect- edness principle may apply in certain cases which do not require the existence of totally geodesic immersions. 0. Introduction. In algebraic geometry, the connectedness principle is a theme that may be stated as follows (cf. [14]): Given a suitably “positive” embedding of manifold of codimension d, D→ P , and a proper morphism, f : N n → P , f −1(D) −−−−→ N n f f D −−−−→ P, n −1 ∼ the homotopy groups should satisfy that πi(N ,f (D)) = πi(P, D)for i ≤ n − d− “defect”, where the defect can be measured by the lack of positivity of N in P , the singularities of N and the dimension of the fibers of f. The most basic case is when P = Pm × Pm, the product of the projective space, D = ∆, the diagonal submanifold P ,andN is the product 1Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China 2Supported partially by NSF Grant DMS 0203164 and a research found from Capital normal university 671 672 F.
    [Show full text]
  • Path Connectedness and Invertible Matrices
    PATH CONNECTEDNESS AND INVERTIBLE MATRICES JOSEPH BREEN 1. Path Connectedness Given a space,1 it is often of interest to know whether or not it is path-connected. Informally, a space X is path-connected if, given any two points in X, we can draw a path between the points which stays inside X. For example, a disc is path-connected, because any two points inside a disc can be connected with a straight line. The space which is the disjoint union of two discs is not path-connected, because it is impossible to draw a path from a point in one disc to a point in the other disc. Any attempt to do so would result in a path that is not entirely contained in the space: Though path-connectedness is a very geometric and visual property, math lets us formalize it and use it to gain geometric insight into spaces that we cannot visualize. In these notes, we will consider spaces of matrices, which (in general) we cannot draw as regions in R2 or R3. To begin studying these spaces, we first explicitly define the concept of a path. Definition 1.1. A path in X is a continuous function ' : [0; 1] ! X. In other words, to get a path in a space X, we take an interval and stick it inside X in a continuous way: X '(1) 0 1 '(0) 1Formally, a topological space. 1 2 JOSEPH BREEN Note that we don't actually have to use the interval [0; 1]; we could continuously map [1; 2], [0; 2], or any closed interval, and the result would be a path in X.
    [Show full text]
  • Connected Spaces and How to Use Them
    CONNECTED SPACES AND HOW TO USE THEM 1. How to prove X is connected Checking that a space X is NOT connected is typically easy: you just have to find two disjoint, non-empty subsets A and B in X, such that A [ B = X, and A and B are both open in X. (Notice n that when X is a subset in a bigger space, say R , \open in X" means relatively open with respected to X. Be sure to check carefully all required properties of A and B.) To prove that X is connected, you must show no such A and B can ever be found - and just showing that a particular decomposition doesn't work is not enough. Sometimes (but very rarely) open sets can be analyzed directly, for example when X is finite and the topology is given by an explicit list of open sets. (Anoth example is indiscrete topology on X: since the only open sets are X and ;, no decomposition of X into the union of two disjoint open sets can exist.) Typically, however, there are too many open sets to argue directly, so we develop several tools to establish connectedness. Theorem 1. Any closed interval [a; b] is connected. Proof. Several proofs are possible; one was given in class, another is Theorem 2.28 in the book. ALL proofs are a mix of analysis and topology and use a fundamental property of real numbers (Completeness Axiom). Corollary 1. All open and half-open intervals are connected; all rays are connected; a line is connected.
    [Show full text]
  • On Locally 1-Connectedness of Quotient Spaces and Its Applications to Fundamental Groups
    Filomat xx (yyyy), zzz–zzz Published by Faculty of Sciences and Mathematics, DOI (will be added later) University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat On locally 1-connectedness of quotient spaces and its applications to fundamental groups Ali Pakdamana, Hamid Torabib, Behrooz Mashayekhyb aDepartment of Mathematics, Faculty of Sciences, Golestan University, P.O.Box 155, Gorgan, Iran. bDepartment of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran. Abstract. Let X be a locally 1-connected metric space and A1; A2; :::; An be connected, locally path con- nected and compact pairwise disjoint subspaces of X. In this paper, we show that the quotient space X=(A1; A2; :::; An) obtained from X by collapsing each of the sets Ai’s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space X=(A1; A2; :::; An). 1. Introduction and Motivation Let be an equivalent relation on a topological space X. Then one can consider the quotient topological space X∼= and the quotient map p : X X= . In general, quotient spaces are not well behaved and it seems interesting∼ to determine which−! topological∼ properties of the space X may be transferred to the quotient space X= . For example, a quotient space of a simply connected or contractible space need not share those properties.∼ Also, a quotient space of a locally compact space need not be locally compact.
    [Show full text]
  • 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.
    [Show full text]