Topology
1st Edition
Volume II
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Description
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
Details
- No. of pages:
- 622
- Language:
- English
- Copyright:
- © Academic Press 1968
- Published:
- 1st January 1969
- Imprint:
- Academic Press
- eBook ISBN:
- 9781483271798
About the Author
K. Kuratowski
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