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Unit 28: Compactness in Metric Spaces

28.5 Review Questions Notes

1. A finite subset of a topological space is necessarily sequentially compact. Prove it.
2. Prove that if X is sequentially compact, then it is countably compact.

3. Let A be a compact subset of a metric space (X, d). Show that for every B  X,  p  A s.t.
d(p, B) = d(A, B).
4. Let A be a compact subset of a metric space (X, d) and let B  X, be closed. Show that
d(A, B) > 0 if A B = .

28.6 Further Readings

Books John Kelley (1955), General Topology, Graduate Texts in Mathematics, Springer-Verlag.
Dmitre Burago, Yu D Burgao, Sergei Ivanov, A course in Metric Geometry, American
Mathematical Society, 2004.

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