Category Archives: Topology

Application of Urysohn’s metrization theorem

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Theorem. Let be a compact metric space, a Hausdorff space, and a continuous function from onto . Then is metrizable. Proof. Note that since and is continuous and is compact, is compact. Since is Hausdorff space, is . So is . It suffices to show is second countable. Since is compact, it is second countable.… Read More »