Volume II

First published on January 1, 1969

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  • Author: K. Kuratowski
  • eBook ISBN: 9781483271798

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Topology, Volume II deals with topology and covers topics ranging from compact spaces and connected spaces to locally connected spaces, retracts, and neighborhood retracts. Group theory and some cutting problems are also discussed, along with the topology of the plane. Comprised of seven chapters, this volume begins with a discussion on the compactness of a topological space, paying particular attention to Borel, Lebesgue, Riesz, Cantor, and Bolzano-Weierstrass conditions. Semi-continuity and topics in dimension theory are also considered. The reader is then introduced to the connectedness of a space, with emphasis on the general properties and monotone mappings of connected spaces; local connectedness of a topological space; absolute retracts and contractible spaces; and general properties of commutative groups. Qualitative problems related to polygonal arcs are also examined, together with cohomotopic multiplication and duality theorems. The final chapter is devoted to the topology of a plane and evaluates the concept of the Janiszewski space. This monograph will be helpful to students and practitioners of algebra and mathematics.

Table of Contents

  • Preface to the Second Volume

    Chapter Four Compact Spaces

    § 41. Compactness

    I. Definitions. Conditions of Borel, Lebesgue, Eiesz, Cantor and Bolzano-Weierstrass.

    II. Normality and Related Properties of Compact Spaces

    III. Continuous Mappings

    IV. Cartesian Products

    V. Compactification of Completely Regular 𝔗1-Spaces

    VI. Relationships to Metric Spaces

    VII. Invariants Under Mappings with Small Point Inverses. Quasi-Homeomorphism

    VIII. Relationships to Boolean Rings

    IX. Dyadic Spaces

    X. Locally Compact Spaces

    § 42. the Space 2𝖃

    I. Compactness of the Space 2𝖃

    II. Case of 𝖃 Compact Metric

    III. Families of Subsets of 𝖃. Operations on Sets

    IV. Irreducible Sets. Saturated Sets

    V. Operations δ(F) and ρ (F1,F2)

    § 43. Semi-Continuity

    I. Semi-Continuity and the Assumption of Compactness of 𝖃

    II. Case of 𝖃 Compact Metric

    III. Decompositions of Compact Spaces

    IV. Decompositions of Compact Metric Spaces

    V. Continuous Decompositions of Compact Spaces

    VI. Examples. Identification of Points

    VII. Relationships of Semi-Continuous Mappings to the Mappings of Class 1

    VIII. Examples of Mappings of Class 2 Which Are Not of Class 1

    IX. Remarks Concerning Selectors

    § 44, the Space Y𝖃

    I. The Compact-Open Topology of Y𝖃

    II. Joint Continuity and Related Problems

    III. The Restriction Operation. Inverse Systems

    IV. Relations Between the Spaces Y𝖃×T and (Y𝖃)T

    V. The Topology of Uniform Convergence of Y𝖃

    VI. The Homeomorphisms

    VII. Case of 𝖃 Locally Compact

    VIII. The Pointwise Topology of Y𝖃

    § 45. Topics in Dimension Theory (Continued)

    I. Mappings of Order k

    II. Parametric Representation of n-Dimensional Perfect, Compact Spaces on the Cantor Set C

    III. Theorems of Decomposition

    IV. n-Dimensional Degree

    V. Dimensional Kernel of a Compact Space

    VI. Transformations with k-Dimensional Point Inverses

    VII. Space (ℐr)* for r ≥ 2 · dim 𝖃 + 1

    VIII. Space (ℐr) for r > dim 𝖃

    IX. Space (ℐr) for r ≤ dim 𝖃

    Chapter Five Connected Spaces

    § 46. Connectedness

    I. Definition. General Properties. Monotone Mappings

    II. Operations

    III. Components

    IV. Connectedness Between Sets

    V. Quasi-Components

    Va. the Space of Quasi-Components

    VI. Hereditarily Disconnected Spaces. Totally Disconnected Spaces

    VII. Separators

    VIII. Separation of Connected Spaces

    IX. Separating Points

    X. Unicoherence. Discoherence

    XI. n-Dimensional Connectedness

    XII. n-Dimensional Connectedness Between Two Sets

    § 47. Continua

    I. Definition. Immediate Consequences

    II. Connected Subsets of Compact Spaces

    III. Closed Subsets of a Continuum

    IV. Separation of Compact Metric Spaces

    V. Arcs. Simple Closed Curves

    VI. Decompositions of Compact Spaces Into Continua

    VII. The Space 2𝖃

    VIII. Semi-Continua. Cuts of the Space

    IX. Hereditarily Discontinuous Spaces

    § 48. Irreducible Spaces. Indecomposable Spaces

    I. Definition. Examples. General Properties

    II. Connected Subsets of Irreducible Spaces

    III. Closed Connected Subdomains

    IV. Layers of an Irreducible Space

    V. Indecomposable Spaces

    VI. Composants

    VII. Indecomposable Subsets of Irreducible Spaces

    VIII. Spaces Irreducibly Connected Between A and B

    IX. Irreducibly Connected Compact Spaces

    X. Additional Remarks

    Chapter Six Locally Connected Spaces

    § 49. Local Connectedness

    I. Points of Local Connectedness

    II. Locally Connected Spaces

    III. Properties of the Boundary

    IV. Separation of Locally Connected Spaces

    V. Irreducible Separators

    VI. The Set of Points at Which a Continuum Is Not l.c. Convergence Continua

    VII. Relative Distance. Oscillation

    § 50. Locally Connected Metric Continua

    I. Arcwise Connectedness

    II. Characterization of Locally Connected Continua

    III. Regions and Subcontinua of a Locally Connected Continuum 𝖃

    IV. Continua Hereditarily Locally Connected (h.l.c.)

    § 51. Theory of Curves. the Order of a Space at a Point

    I. Definitions and Examples

    II. General Properties

    III. Order 𝓝0 and C

    IV. Regular Spaces, Rational Spaces

    V. Points of Finite Order. Characterization of Arcs and Simple Closed Curves

    VI. Dendrites

    VII. Local Dendrites

    § 52. Cyclic Elements of a Locally Connected Metric Continuum

    I. Completely Arcwise Connected Sets

    II. Cyclic Elements

    III. Extensible Properties

    IV. θ-Curves

    Chapter Seven Absolute Retracts. Spaces Connected in Dimension n Gontractible Spaces

    § 53. Extending of Continuous Functions. Retraction

    I. Relations τ and τv

    II. Operations

    III. Absolute Retracts

    IV. Connectedness in Dimension n. The Case Where ℐnτY

    V. Operations

    VI. Characterization of Dimension

    VII. The Space LCn(Y)

    § 54. Homotopy. Contractibility

    I. Homotopic Functions

    II. Homotopy with Respect to l.c. n Spaces

    III. Relation F0irrnon ≃f

    IV. Deformation

    V. Contractibility

    VI. Spaces Contractible in Themselves

    VII. Local Contractibility

    VIII. The Components of Y𝖃 Where Y is ANR

    IX. The Space 𝕮(Y𝖃) of Components of Y𝖃

    Chapter Eight Groups 𝓖𝖃, L𝖃 and 𝕸(𝖃)

    § 55. Groups 𝓖𝖃 and 𝕭0(𝖃)

    I. General Properties of Commutative Groups

    II. Homomorphism. Isomorphism

    III. Factor Groups

    IV. Operation Â

    V. Linear Independence, Rank, Basis

    VI. Linear Independence ModG

    VII. Cartesian Products

    VIII. Group Y𝖃

    IX. Group 𝓖𝖃

    X. Addition Theorems

    XI. Relations to the Connectedness Between Sets

    § 56. The Groups L𝖃 and P𝖃

    I. General Properties

    II. Group Γ(A)

    III. Group 𝕭1(𝖃)

    IV. Addition Theorems

    V. Relations Between Factor Groups

    VI. Relations to Connectedness

    VII. Relation firrnon~1

    VIII. Compact Sets

    IX. Cartesian Products. Relations to Homotopy

    X. Locally Connected Sets

    XI. Mappings

    § 57. Spaces Contractible with Respect to L. Unicoherent Spaces

    I. Contractibility with Respect to L

    II. Properties of c.r. L Spaces

    III. Local Connectedness and Unicoherence

    IV. Remarks on Extending Homeomorphisms in c.r. L Continua

    § 58. The Group 𝕸(𝖃)

    0. Introduction. The Family (0,1)𝖃

    I. 𝕸(𝖃) as a Topological Space

    II. 𝕸(𝖃) as a Topological Group

    III. Normed Measures

    IV. Extension of Measures

    Chapter Nine Some Theorems on the Disconnection of the Sphere Ln

    § 59. Qualitative Problems

    I. Polygonal Arcs in 𝓔n

    II. Cuts of Ln

    III. Irreducible Cuts

    IV. Invariants

    V. Remarks Connected with the Borsuk-Ulam Theorem

    § 60. Quantitative Problems. Cohomotopic Multiplication. Duality Theorems

    I. Introduction

    II. Formulation of the Problem

    III. Auxiliary Homotopy Properties

    IV. Auxiliary Properties of the Sphere

    V. The Group 𝕮(LXn) for dimX ≤ 2n — 2

    VI. The Group 𝕮(PXn) for X ⊂ 𝓔n and n ≥ 2

    VII. The Group 𝕮(PXn) Where X is a Compact Subset of 𝓔n

    VIII. Duality Theorems for Compact X ⊂ 𝓔n (n ≥ 2)

    IX. Duality Theorems for Arbitrary X ⊂ 𝓔n

    X. Duality Theorems for Locally X Compact ⊂ 𝓔n

    Chapter Ten Topology of the Plane

    § 61. Qualitative Problems

    I. Janiszewski Spaces

    II. Locally Connected Subcontinua of L2

    III. Elementary Sets

    IV. Topological Characterization of L2. Consequences

    V. Extensions of Homeomorphisms. Topological Equivalence

    § 62. Quantitative Problems. The Group PA

    I. General Properties and Notation

    II. Cuts of L2

    III. Groups PF and 𝕭1 (F) for F = F ⊂ L2

    IV. Addition Theorems

    V. Irreducible Cuts

    VI. Groups PA and 𝕭1(A) for Locally Connected A

    VII. Groups PG and 𝕭1(G) for Open G

    VIII. Multiplicity of a Set with Respect to a Continuous Function f: F → P Where F is Closed

    IX. Multiplicity with Respect to a Continuous Function f: G → P Where G Is Open

    X. Characterization of the Group 𝕭1(G)

    XI. Increment of the Logarithm. Index

    XII. Relation to the Multiplicity. Kronecker Characteristic

    XIII. Positive and Negative Homeomorphisms. Oriented Topology

    List of Important Symbols

    Author Index

    Subject Index

Product details

  • No. of pages: 622
  • Language: English
  • Copyright: © Academic Press 1969
  • Published: January 1, 1969
  • Imprint: Academic Press
  • eBook ISBN: 9781483271798

About the Author

K. Kuratowski

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