Elementary Theory of Metric Spaces: A Course in Constructing Mathematical Proofs (Universitext)

Reisel, Robert B.

ISBN 10: 0387907068 / ISBN 13: 9780387907062

Editore: Springer, 1998

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Dati bibliografici

• Titolo: Elementary Theory of Metric Spaces: A Course in Constructing Mathematical Proofs (Universitext)
• Editore: Springer
• Data di pubblicazione: 1998
• Lingua: Inglese
• ISBN 10: 0387907068
• ISBN 13: 9780387907062
• Rilegatura: Brossura
• Condizione: New

Descrizione articolo

Riassunto

Science students have to spend much of their time learning how to do laboratory work, even if they intend to become theoretical, rather than experimental, scientists. Mathematics students must try to prove results and then have their work criticized by experienced mathematicians.

Contenuti

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.