Elementary Theory of Metric Spaces

0. Some Ideas of Logic. - I. Sets and Mappings. - 1. Some Concepts of Set Theory. - 2. Some Further Operations on Sets. - 3. Mappings. - 4. Surjective and Injective Mappings. - 5. Bijective Mappings and Inverses. - II. Metric Spaces. - 1. Definition of Metric Space and Some Examples. - 2. Closed and Open Balls; Spheres. - 3. Open Sets. - 4. Closed Sets. - 5. Closure of a Set. - 6. Diameter of a Set; Bounded Sets. - 7. Subspaces of a Metric Space. - 8. Interior of a Set. - 9. Boundary of a Set. - 10. Dense Sets. - 11. Afterword. - III. Mappings of Metric Spaces. - 1. Continuous Mappings. - 2. Continuous Mappings and Subspaces. - 3. Uniform Continuity. - IV. Sequences in Metric Spaces. - 1. Sequences. - 2. Sequences in Metric Spaces. - 3. Cluster Points of a Sequence. - 4. Cauchy Sequences. - 5. Complete Metric Spaces. - V. Connectedness. - 1. Connected Spaces and Sets. - 2. Connected Sets in R. - 3. Mappings of Connected Spaces and Sets. - VI. Compactness. - 1. Compact Spaces and Sets. - 2. Mappings of Compact Spaces. - 3. Sequential Compactness. - 4. Compact Subsets of R. - Afterword. - Appendix M. Mathematical Induction. - Appendix S. Solutions. Language: English