Topology

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 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