Topology

Volume II

1st Edition - January 1, 1969
Imprint: Academic Press
Author: K. Kuratowski
Language: English
Paperback ISBN:
9 7 8 - 1 - 4 8 3 2 - 4 2 1 2 - 5
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 7 1 7 9 - 8

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… Read more

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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.

Preface to the Second VolumeChapter 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 𝖃Chapter 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 RemarksChapter 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-CurvesChapter 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 Y𝖃Chapter 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 MeasuresChapter 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 ⊂ 𝓔nChapter 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 TopologyList of Important SymbolsAuthor IndexSubject Index

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