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 ??1-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 2??


I. Compactness of the Space 2??


II. Case of ?? Compact Metric


III. Families of Subsets of ??. Operations on Sets


IV. Irreducible Sets. Saturated Sets


V. Operations d(F) and (F1,F2)


§ 43. Semi-Continuity


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


II. Case of ?? 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 Y??


I. The Compact-Open Topology of Y??


II. Joint Continuity and Related Problems


III. The Restriction Operation. Inverse Systems


IV. Relations Between the Spaces Y??×T and (Y??)T


V. The Topology of Uniform Convergence of Y??


VI. The Homeomorphisms


VII. Case of ?? Locally Compact


VIII. The Pointwise Topology of Y??


§ 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 ?? + 1


> dim ??


IX. Space (Ir) for r = dim ??


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 2??


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 ??


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 ??0 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. -Curves


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


§ 53. Extending of Continuous Functions. Retraction


I. Relations t and tv


II. Operations


III. Absolute Retracts


IV. Connectedness in Dimension n. The Case Where IntY


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 Y?? Where Y is ANR


IX. The Space ??(Y??) of Components of Y??


Chapter Eight Groups ????, L?? and ??(??)


§ 55. Groups ???? and ??0(??)


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 Produc…