DOCSLIB.ORG

Chapter 5 Connectedness

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Chapter V Connected Spaces 1. Introduction In this chapter we introduce the idea of connectedness. Connectedness is a topological property quite different from any property we considered in Chapters 1-4. A connected space \ need not have any of the other topological properties we have discussed so far. Conversely, the only topological properties that imply “\l\lŸ\ is connected” are very extreme such as “ 1” or “ has the trivial topology.” 2. Connectedness Intuitively, a space is connected if it is all in one piece; equivalently a space is disconnected if it can be written as the union of two nonempty “separated” pieces. To make this precise, we need to decide what “separated” should mean. For example, we think of ‘‘ as connected even though can be written as the union of two disjoint pieces: for example, ‘ œE∪Fß where E œ Ð ∞ß !Ó and F œ Ð!ß ∞Ñ. Evidently, “separated” should mean something more than “disjoint.” On the other hand, if we remove the point ! to “cut” ‘, then we probably think of the remaining space \ œ‘ Ö!× as “disconnected.” Here, we can write \ œ E ∪ F, where E œ( ∞ß !Ñ and FœÐ!ß∞Ñ. E and F are disjoint, nonempty sets and (unlike EF and in the preceding paragraph) they satisfy the following (equivalent) conditions: i) EFand are open in \ ii) EFand are closed in \ iii) ÐF ∩cl\\ EÑ ∪ ÐE ∩ cl FÑ œ g that is, each of E and F is disjoint from the closure of the other. (This is true, in fact, even if we use cl‘ instead of cl\ .) Condition iii) is important enough to deserve a name. Definition 2.1 Suppose EF and are subspaces of Ð\ßÑEFg . and are called separated if each is disjoint from the closure of the otherÐF∩EÑ∪ÐE∩FÑœg that is, if cl\\ cl . It follows immediately from the definition that i) separated sets must be disjoint, and ii) subsets of separated sets are separated: if Eß F are separated, G © E and H©F, then Gand H are also separated. Example 2.2 1) In‘ , the sets E œ Ð ∞ß !Ó and F œ Ð!ß ∞Ñ are disjoint but not separated. Likewise in ‘#####, the sets EœÖÐBßCÑÀB C Ÿ"×and FœÖÐBßCÑÀÐB 2 Ñ C "× are disjoint but not separatedÞ 213 2) The intervals E œ Ð ∞ß !Ñ and F œ Ð!ß ∞Ñare separated in ‘ but cl‘‘ E ∩ cl F Á gÞ The same is true for the open balls EœÖÐBßCÑÀB## C "×and F œ ÖÐBß CÑ À ÐB 2 Ñ## C "× in ‘ #. The condition that two sets are separated is stronger than saying they are disjoint, but weaker than saying that the sets have disjoint closures. Theorem 2.3 In any space Ð\ßg Ñ, the following statements are equivalent: 1) g\ and are the only clopen sets in \ 2) if E©\ and Fr Eœg , then Eœg or Eœ\ 3) \ is not the union of two disjoint nonempty open sets 4) \ is not the union of two disjoint nonempty closed sets 5) \ is not the union of two nonempty separated sets. Note: Condition 2) is not frequently used. However it is fairly expressive: to say that FrEœg says that no point B in \ can be “approximated arbitrarily closely” from both inside and outside Eso, in that sense, Eand Fœ\E are pieces of \ that are “separated” from each other. Proof 1)ÍEEœg 2) This follows because is clopen iff Fr (see Theorem II.4.5.3 ). 1)Ê\œE∪FEF 3) Suppose 3) is false and that where , are disjoint, nonempty and open. Since \EœF is open and nonempty, we have that E is a nonempty proper clopen set in \, which shows that 1) is false. 3)Í 4) This is clear. 4)Ê\œE∪FEßF 5) If 5) is false, then , where are nonempty and separated. Since clF∩Eœgß we conclude that cl F©F , so F is closed. Similarly, E must be closed. Therefore 4) is false. 5)ÊE 1) Suppose 1) is false and that is a nonempty proper clopen subset of \. Then Fœ\E is nonempty and clopen, so E and F are separated. Since \œE∪F, 5) is false. ñ Definition 2.4 A space Ð\ßg Ñ is connected if any (therefore all) of the conditions 1) - 5) in Theorem 2.3 hold. If G©\, we say that G is connected if G is connected in the subspace topology. According to the definition, a subspace G©\ is disconnected if we can write GœE∪F, where the following (equivalent) statements are true: 1) EFand are disjoint, nonempty and open in G 2) EFand are disjoint, nonempty and closed in G 3) EF and are nonempty and separated in G. If GEßFG is disconnected, such a pair of sets will be called a disconnection or separation of . The following technical theorem and its corollary are very useful in working with connectedness in subspaces. 214 Theorem 2.5 Suppose EßF©G©\ÞEFThen and are separated in G iff EF and are separated in \. Proof clG\FœG∩ cl F (see Theorem III.7.6 ), so E∩cl G Fœg iff E∩Ðcl \ F∩GÑœg iffÐE∩GÑ∩ cl\\ Fœg iff E∩cl FœgÞ Similarly, F∩cl G\ Eœg iff F∩cl EœgÞ ñ Caution: According to Theorem 2.5, GGœE∪FEF is disconnected iff where and are nonempty separated set in GGœE∪FEF iff where and are nonempty separated set in \. Theorem 2.5 is very useful because it means that we don't have to distinguish here between “separated in G\” and “separated in ” because these are equivalent. In contrast, when we say that GG is disconnected if is the union of two disjoint, nonempty open (or closed ) sets EßF in G, then phrase “in GEF\”cannot be omitted: the sets , might not be open (or closed ) in . "" " For example, suppose \ œ Ò!ß "Ó and G œ Ò!ß## Ñ ∪ Ð ß "Ó . The sets E œ Ò!ß # Ñ and " FœÐ# ß"Ó are open, closed and separated in G. By Theorem 2.5, they are also separated in ‘ but they are neither open nor closed in ‘. Example 2.6 1) Clearly, connectedness is a topological property. More generally, suppose 0À\Ä] is continuous and onto. If F]0ÒFÓ is proper nonempty clopen set in , then " is a proper nonempty clopen set in \. Therefore a continuous image of a connected space is connected. 2) A discrete space \l\lŸ" is connected iff . In particular, ™ and are not connected. 3) is not connected since we can write as the union of two nonempty separated sets: œÖ;− À;## #×∪Ö;− À; #×. Similarly, we can show is not connected. More generally suppose G©‘ and that G is not an interval. Then there are points +D, where +ß,−G but DÂGÞThen ÖB−GÀBDלÖB−GÀBŸD× is a nonempty proper clopen set in GG. Therefore is not connected. In fact, a subset GG of ‘ is connected iff is an interval. It is not very hard, using the least upper bound property of ‘‘, to prove that every interval in is connected. (Try it as an exercise! ) We will give a short proof soon (Corollary 2.12) using a different argument. 4) (The Intermediate Value Theorem ) If \0À\Ä is connected and ‘ is continuous, then ranÐ0Ñ is connected (by part 1 ) so ran Ð0Ñ is an interval ( by part 3 ). Therefore if +ß , − \ and 0Ð+ÑD0Ð,Ñ, there must be a point -−\ for which 0Ð-ÑœDÞ 5) The Cantor set G is not connected (since it is not an interval). But much more is true. Suppose Bß C − E © G and that B CÞSince G is nowhere dense (see IV.10 ), the interval ÐBß CÑ ©Î G, so we can choose DÂG with BDC. Then FœÐ∞ßDÑ∩E œÐ∞ßDÓ∩E is clopen in E, and F contains B but not Cso E is not connected. It follows that every connected subset of G contains at most one point. A space Ð\ßg Ñ is called totally disconnected every connected subset E satisfies lEl Ÿ "Þ The spaces ™ ß ß and are other examples of totally disconnected spaces. 6) \ is connected iff every continuous 0 À \ Ä Ö!ß "× is constant: certainly, if 0 is continuous and not constant, then 0" ÒÖ!×Ó is a proper nonempty clopen set in \ so \ is not 215 connected. Conversely, if \Eis not connected and is a proper nonempty clopen set, then the characteristic function ;E À \ Ä Ö!ß "× is continuous but not constant. Theorem 2.7 Suppose 0 À \ Ä ]. Let > œ ÖÐBß CÑ − \ ‚ ] À C œ 0ÐBÑ× œ“the graph of 0Þ” If 00 is continuous, then graph of is homeomorphic to the domain of 0; in particular, the graph of a continuous function is connected iff its domain is connected. Proof We want to show that \2À\Ä is homeomorphic to >>. Let be defined by 2ÐBÑ œ ÐBß 0ÐBÑÑÞ Clearly 2 is a one-to-one map from \ onto >. Let + − \ and suppose ÐY ‚ Z Ñ ∩> is a basic open set containing 2Ð+Ñ œ Ð+ß 0Ð+ÑÑÞ Since 00Ð+Ñ−ZßS\+ is continuous and there exists an open set in containing and such that 0ÒSÓ © Z Þ Then + − Y ∩ S, and 2ÒY ∩ SÓ © ÐY ‚ Z Ñ ∩>, so 2 is continuous at +. If Y\2ÒYÓœÐY‚]Ñ∩2 is open in , then >> is open in , so is open. Therefore 2 is a homeomorphism. ñ Note: It is not true that a function 0 with a connected graph must be continuous. See Example 2.22. The following lemma makes a simple but very useful observation.
Recommended publications
  • A Guide to Topology

    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.
  • Open 3-Manifolds Which Are Simply Connected at Infinity

    Open 3-Manifolds Which Are Simply Connected at Infinity

    OPEN 3-MANIFOLDS WHICH ARE SIMPLY CONNECTED AT INFINITY C H. EDWARDS, JR.1 A triangulated open manifold M will be called l-connected at infin- ity il each compact subset C of M is contained in a compact poly- hedron P in M such that M—P is connected and simply connected. Stallings has shown that, if M is a contractible open combinatorial manifold which is l-connected at infinity and is of dimension ra^5, then M is piecewise-linearly homeomorphic to Euclidean re-space E" [5]. Theorem 1. Let M be a contractible open 3-manifold, each of whose compact subsets can be imbedded in £3. If M is l-connected at infinity, then M is homeomorphic to £3. Notice that, in order to prove the 3-dimensional Poincaré conjec- ture, it would suffice to prove Theorem 1 without the hypothesis that each compact subset of M can be imbedded in £3. For, if M is a simply connected closed 3-manifold and p is a point of M, then M —p is a contractible open 3-manifold which is clearly l-connected at infinity. Conversely, if the 3-dimensional Poincaré conjecture were known, then the hypothesis that each compact subset of M can be imbedded in £3 would be unnecessary. All spaces and mappings in this paper are considered in the poly- hedral or piecewise-linear sense, unless otherwise stated. As usual, by an open re-manifold is meant a noncompact connected space triangu- lated by a countable simplicial complex without boundary, such that the link of each vertex is piecewise-linearly homeomorphic to the usual (re—1)-sphere.
  • MS Word Template for CAD Conference Papers

    MS Word Template for CAD Conference Papers

    382 Title: Regularized Set Operations in Solid Modeling Revisited Authors: K. M. Yu, [email protected], The Hong Kong Polytechnic University K. M. Lee, [email protected], The Hong Kong Polytechnic University K. M. Au, [email protected], HKU SPACE Community College Keywords: Regularised Set Operations, CSG, Solid Modeling, Manufacturing, General Topology DOI: 10.14733/cadconfP.2019.382-386 Introduction: The seminal works of Tilove and Requicha [8],[13],[14] first introduced the use of general topology concepts in solid modeling. However, their works are brute-force approach from mathematicians’ perspective and are not easy to be comprehended by engineering trained personnels. This paper reviewed the point set topological approach using operational formulation. Essential topological concepts are described and visualized in easy to understand directed graphs. These facilitate subtle differentiation of key concepts to be recognized. In particular, some results are restated in a more mathematically correct version. Examples to use the prefix unary operators are demonstrated in solid modeling properties derivations and proofs as well as engineering applications. Main Idea: This paper is motivated by confusion in the use of mathematical terms like boundedness, boundary, interior, open set, exterior, complement, closed set, closure, regular set, etc. Hasse diagrams are drawn by mathematicians to study different topologies. Instead, directed graph using elementary operators of closure and complement are drawn to show their inter-relationship in the Euclidean three- dimensional space (E^3 is a special metric space formed from Cartesian product R^3 with Euclid’s postulates [11] and inheriting all general topology properties.) In mathematics, regular, regular open and regular closed are similar but different concepts.
  • MTH 304: General Topology Semester 2, 2017-2018

    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 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

    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).
  • Daniel Irvine June 20, 2014 Lecture 6: Topological Manifolds 1. Local

    Daniel Irvine June 20, 2014 Lecture 6: Topological Manifolds 1. Local

    Daniel Irvine June 20, 2014 Lecture 6: Topological Manifolds 1. Local Topological Properties Exercise 7 of the problem set asked you to understand the notion of path compo- nents. We assume that knowledge here. We have done a little work in understanding the notions of connectedness, path connectedness, and compactness. It turns out that even if a space doesn't have these properties globally, they might still hold locally. Definition. A space X is locally path connected at x if for every neighborhood U of x, there is a path connected neighborhood V of x contained in U. If X is locally path connected at all of its points, then it is said to be locally path connected. Lemma 1.1. A space X is locally path connected if and only if for every open set V of X, each path component of V is open in X. Proof. Suppose that X is locally path connected. Let V be an open set in X; let C be a path component of V . If x is a point of V , we can (by definition) choose a path connected neighborhood W of x such that W ⊂ V . Since W is path connected, it must lie entirely within the path component C. Therefore C is open. Conversely, suppose that the path components of open sets of X are themselves open. Given a point x of X and a neighborhood V of x, let C be the path component of V containing x. Now C is path connected, and since it is open in X by hypothesis, we now have that X is locally path connected at x.
  • Discrete Mathematics Panconnected Index of Graphs

    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.
  • On the Simple Connectedness of Hyperplane Complements in Dual Polar Spaces

    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.
  • METRIC SPACES and SOME BASIC TOPOLOGY

    METRIC SPACES and SOME BASIC TOPOLOGY

    Chapter 3 METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under- standing and ability to make use of properties of U U1. The next goal is to generalize our work to Un and, eventually, to study functions on Un. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. For completeness, we include the following De¿nition 3.1.1 Let S and T be sets. The Cartesian product of S and T , denoted by S T,is p q : p + S F q + T . The Cartesian product of any ¿nite number of sets S1 S2 SN , denoted by S1 S2 SN ,is j b ck p1 p2 pN : 1 j j + M F 1 n j n N " p j + S j . The object p1 p2pN is called an N-tuple. Our primary interest is going to be the case where each set is the set of real numbers. 73 74 CHAPTER 3. METRIC SPACES AND SOME BASIC TOPOLOGY De¿nition 3.1.2 Real n-space,denotedUn, is the set all ordered n-tuples of real numbers i.e., n U x1 x2 xn : x1 x2 xn + U . Un U U U U Thus, _ ^] `, the Cartesian product of with itself n times. nofthem Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a b a a b . def This de¿nition leads to the more familiar statement that a b c d if and only if a bandc d.
  • A Concise Form of Continuity in Fine Topological Space

    A Concise Form of Continuity in Fine Topological Space

    Advances in Computational Sciences and Technology (ACST). ISSN 0973-6107 Volume 10, Number 6 (2017), pp. 1785–1805 © Research India Publications http://www.ripublication.com/acst.htm A Concise form of Continuity in Fine Topological Space P.L. Powar and Pratibha Dubey Department of Mathematics and Computer Science, R.D.V.V., Jabalpur, India. Abstract By using the topology on a space X, a wide class of sets called fine open sets have been studied earlier. In the present paper, it has been noticed and verified that the class of fine open sets contains the entire class of A-sets, AC-sets and αAB-sets, which are already defined. Further, this observation leads to define a more general continuous function which in tern reduces to the four continuous functions namely A-continuity, AB-continuity, AC-continuity and αAB-continuity as special cases. AMS subject classification: primary54XX; secondry 54CXX. Keywords: fine αAB-sets, fine A-sets, fine topology. 1. Introduction The concept of fine space has been introduced by Powar and Rajak in [1] is the special case of generalized topological space (see[2]). The major difference between the two spaces is that the open sets in the fine space are not the random collection of subsets of X satisfying certain conditions but the open sets (known as fine sets) have been generated with the help of topology already defined over X. It has been studied in [1] that this collection of fine open sets is really a magical class of subsets of X containing semi-open sets, pre-open sets, α-open sets, β-open sets etc.
  • Maximally Edge-Connected and Vertex-Connected Graphs and Digraphs: a Survey Angelika Hellwiga, Lutz Volkmannb,∗

    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.