# Topology

## Volume II

- 1st Edition - January 1, 1969
- Author: K. Kuratowski
- Language: English
- 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. Groupโฆ Read more

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Request a sales quoteTopology, 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 ยง 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 ๐1-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 ยง 42. the Space 2๐ I. Compactness of the Space 2๐ II. Case of ๐ Compact Metric III. Families of Subsets of ๐. Operations on Sets IV. Irreducible Sets. Saturated Sets V. Operations ฮด(F) and ฯ (F1,F2) ยง 43. Semi-Continuity I. Semi-Continuity and the Assumption of Compactness of ๐ II. Case of ๐ 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 ยง 44, the Space Y๐ I. The Compact-Open Topology of Y๐ II. Joint Continuity and Related Problems III. The Restriction Operation. Inverse Systems IV. Relations Between the Spaces Y๐รT and (Y๐)T V. The Topology of Uniform Convergence of Y๐ VI. The Homeomorphisms VII. Case of ๐ Locally Compact VIII. The Pointwise Topology of Y๐ ยง 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 (โr)* for r โฅ 2 ยท dim ๐ + 1 VIII. Space (โr) for r > dim ๐ IX. Space (โr) for r โค dim ๐Chapter Five Connected Spaces ยง 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 ยง 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 2๐ VIII. Semi-Continua. Cuts of the Space IX. Hereditarily Discontinuous Spaces ยง 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 ยง 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 ยง 50. Locally Connected Metric Continua I. Arcwise Connectedness II. Characterization of Locally Connected Continua III. Regions and Subcontinua of a Locally Connected Continuum ๐ IV. Continua Hereditarily Locally Connected (h.l.c.) ยง 51. Theory of Curves. the Order of a Space at a Point I. Definitions and Examples II. General Properties III. Order ๐0 and C IV. Regular Spaces, Rational Spaces V. Points of Finite Order. Characterization of Arcs and Simple Closed Curves VI. Dendrites VII. Local Dendrites ยง 52. Cyclic Elements of a Locally Connected Metric Continuum I. Completely Arcwise Connected Sets II. Cyclic Elements III. Extensible Properties IV. ฮธ-CurvesChapter Seven Absolute Retracts. Spaces Connected in Dimension n Gontractible Spaces ยง 53. Extending of Continuous Functions. Retraction I. Relations ฯ and ฯv II. Operations III. Absolute Retracts IV. Connectedness in Dimension n. The Case Where โnฯY V. Operations VI. Characterization of Dimension VII. The Space LCn(Y) ยง 54. Homotopy. Contractibility I. Homotopic Functions II. Homotopy with Respect to l.c. n Spaces III. Relation F0irrnon โf IV. Deformation V. Contractibility VI. Spaces Contractible in Themselves VII. Local Contractibility VIII. The Components of Y๐ Where Y is ANR IX. The Space ๐ฎ(Y๐) of Components of Y๐Chapter Eight Groups ๐๐, L๐ and ๐ธ(๐) ยง 55. Groups ๐๐ and ๐ญ0(๐) I. General Properties of Commutative Groups II. Homomorphism. Isomorphism III. Factor Groups IV. Operation ร V. Linear Independence, Rank, Basis VI. Linear Independence ModG VII. Cartesian Products VIII. Group Y๐ IX. Group ๐๐ X. Addition Theorems XI. Relations to the Connectedness Between Sets ยง 56. The Groups L๐ and P๐ I. General Properties II. Group ฮ(A) III. Group ๐ญ1(๐) 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 ยง 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 ยง 58. The Group ๐ธ(๐) 0. Introduction. The Family (0,1)๐ I. ๐ธ(๐) as a Topological Space II. ๐ธ(๐) as a Topological Group III. Normed Measures IV. Extension of MeasuresChapter Nine Some Theorems on the Disconnection of the Sphere Ln ยง 59. Qualitative Problems I. Polygonal Arcs in ๐n II. Cuts of Ln III. Irreducible Cuts IV. Invariants V. Remarks Connected with the Borsuk-Ulam Theorem ยง 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 ๐ฎ(LXn) for dimX โค 2n โ 2 VI. The Group ๐ฎ(PXn) for X โ ๐n and n โฅ 2 VII. The Group ๐ฎ(PXn) Where X is a Compact Subset of ๐n VIII. Duality Theorems for Compact X โ ๐n (n โฅ 2) IX. Duality Theorems for Arbitrary X โ ๐n X. Duality Theorems for Locally X Compact โ ๐nChapter Ten Topology of the Plane ยง 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 ยง 62. Quantitative Problems. The Group PA I. General Properties and Notation II. Cuts of L2 III. Groups PF and ๐ญ1 (F) for F = F โ L2 IV. Addition Theorems V. Irreducible Cuts VI. Groups PA and ๐ญ1(A) for Locally Connected A VII. Groups PG and ๐ญ1(G) for Open G VIII. Multiplicity of a Set with Respect to a Continuous Function f: F โ P Where F is Closed IX. Multiplicity with Respect to a Continuous Function f: G โ P Where G Is Open X. Characterization of the Group ๐ญ1(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

- No. of pages: 622
- Language: English
- Edition: 1
- Published: January 1, 1969
- Imprint: Academic Press
- eBook ISBN: 9781483271798

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