توپولوژی

Contents I. Elementary set theory Sets Boolean Algebra Cartesian Product Families of Sets Power set Functions, or Maps Binary relations; equivalence relations Axiomatics General Cartesian Products Problems II. Ordinals and Cardinals Orderings Zorn ́s Lemma; Zermelo ́s Theorem Ordinals Comparability of Ordinals Transfinite induction and Construction Ordinal numbers Cardinals Cardinal Arithmetic The ordinal number omega Problems III. Topological spaces Topological spaces Bassis for a given topology Topologizing of sets Elementary concepts Topologizing with preassigned elementary operations Gfi, Fsigma and Borel sets Relativization Continuous maps Piecewise definition of maps Continuous maps into E1 Open maps and colse maps Homeomorphism Problems IV. Cartesian products Cartesian product topology Continuity of maps Slices in Cartesian Products Peano curves Problems V. Connectedness Conectedness Applications Components Local Connectedness Path-Conectedness Problems VI. Identification Topology; weak topology Identification topology Subspaces General theorems Spaces with equivalence relations Cones and suspensions Attaching of spaces The relation K(f) for continuous maps Weak topologies Problems VII. Separation axioms Hausdorff spaces Regular spaces Normal spaces Urysohn ́s characterization of normality Tietze ́s characterization of normality Covering characterization fo normality Completely regular spaces Problems VIII. Covering axioms Coverings of spaces Paracomplact spaces Types of refinements Partitions of unity Complexes; Nerves of Coverings Second-countable spaces; Lindelöf spaces Separability Problems IX. Metric spaces Metrics on sets Topoloty induced by a metric Equivalent metrics Continuity of the distance Properties of metirc topologies Maps of metric spaces into affine spaces Cartesian products of metric spaces The space l2(A); Hilbert cube Metrization of topological spaces Gauge spaces Uniform spaces Problems X. Convergence Sequences and nets Filterbases in spaces Convergence properties of filterbases Closure in terms of filterbases Continuity; convergence in cartesian products Adequacy of sequences Maximal filterbases Problems XI. Compactness Compact spaces Special properties of compact spaces Countable compactness Compactness in metric spaces Perfect maps Local compactness sigma-compact spaces Compactification k-spaces Baire spaces; category Problems XII. Function spaces The compact-open topology Continuity of composition; the evaluation map Cartesian products Application to identification topologies Basis for Zy Compact subsets of Zy Sequential convergence in the c-Topology Metric topologies; relation to the c-topology Pointwise convergence Comparison of topologies in Zy Problems XIII. The spaces C(Y) Continuity of the algebraic operations Algebras in C(Y;c) Stone-Weierstrass theorem The metric space C(y) Embedding of Y in C(Y) The ring C(Y) Problems XIV. Complete spaces Cauchy sequences Complete metrics and complete spaces Cauchy filterbases; total boundedness Baire ́s Theorem for complete metric spaces Extension of uniformly continuous maps Completion of a metric space Fixed-point theorem for complete spaces Complete subspaces of complete spaces Complete gauge structures Problems XV. Homotopy Homotopy Homotopy classes Homotopy and function spaces Relative homotopy Retracts and extendability Deformation retraction and homotopy Homotopy and extendability Applications Problems XVI. Maps into spheres Degree of a map Sn a Sn Brouwer ́s theorem Further applications of the degree of a map Maps of spheres into Sn Maps of spaces into Sn Borsuk ́s antipodal theorem Degree and homotopy Problems XVII. Topology of En Components of compact sets in En+1 Borsuk ́s separation theorem Domain invarience Deformations of subsets of En+1 The jordan curve theorem Problems XVIII. Homotopy type Homotopy type Homotopy type invariants Homotopy of pairs Mapping cylinder Properties of X in C(f) Change of bases in C(f) Problems XIX. Path spaces; H-Spaces Path spaces H-structures H-Homomorphisms H-Spaces Units Inversion Associativity Path spaces on H-Spaces Problems XX. Fiber spaces Fiber spaces Fiber spaces for the class of all spaces The uniformization theorem of Hurewicz Locally trivial fiber structures Problems Appendix one: Vector spaces; polytopes Appendix two: Direct and inverse limits Index