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Unit 20: The Urysohn Metrization Theorem
5. A space X is locally metrizable if each point x of X has a neighbourhood that is metrizable Notes
in the subspace topology. Show that a compact Hausdorff space X is metrizable if it is
locally metrizable.
6. Let X be a locally compact Hausdorff space. Is it true that if X has a countable basis, then X
is metrizable? Is it true that if X is metrizable, then X has a countable basis?
7. Prove that the topological product of a finite family of metrizable spaces is metrizable.
8. Prove that every metrizable space is first countable.
20.5 Further Readings
Books Robert Canover, A first Course in Topology, The Williams and Wilkins Company
1975.
Michael Gemignani, Elementary Topology, Dover Publications 1990.
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