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A metrizable space is a topological space which is homeomorphic to a metric space. Metrization theorems are theorems which give sufficient conditions for a topological space to be metrizable. |
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A metrizable space is a topological space which is homeomorphic to a metric space. Metrization theorems are theorems which give sufficient conditions for a topological space to be metrizable. |
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For explanations of many of the terms used in this article, the reader should see the article on topology. |
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For explanations of many of the terms used in this article, the reader should see the Topology Glossary. |
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The first really useful metrization theorem was Urysohn's Metrization Theorem. This states that every second-countable regular Hausdorff space is metrizable. So, for example, every second-countable manifold is metrizable. |
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The first really useful metrization theorem was Urysohn's Metrization Theorem. This states that every second-countable regular Hausdorff space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical note: The form of the theorem shown here was in fact proved by Tychonoff? in 1926. What Urysohn had shown, in a paper published posthumously in 1925, was the slightly weaker result that every second-countable normal Hausdorff space is metrizable.) |
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