Page 231 - DMTH503_TOPOLOGY

Unit 27: Complete Metric Spaces

3. Show that the metric space (, d) is complete, where d is usual metric on . Notes
4. Show that the set  of complex numbers with usual metric is complete metric space.
5. Prove that every closed subset of a complete metric space is complete.
6. Prove that Frechet space is complete.

7. Show that a metric space is complete iff every infinite totally bounded subset has a limit
point.

27.7 Further Readings

Books Dmitre Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American
Mathematical Society, 2004.
Victor Bryant, Metric Spaces; Iteration and Application, Cambridge University Press,
1985.

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