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When is a space metrizablePosted by DickT on June 27, 2003 at 06:46:07: Thanks to Kx21, we have this: Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is second-countable, regular and Hausdorff. The Nagata-Smirnov Metrization Theorem extends this to the non-separable case. It states that a topological space is metrizable if and only if it is regular and Hausdorff and has a ó-locally finite base. A ó-locally finite base is a base which is a union of countably many locally finite collections of open sets. You see that the ability to use a metric depends on several proerties, some local and some global. Hausdorff means that every two points in the space cbe put into disjoint open sets. Regular means that for every open set around a point, you can find the closure of another open set around the point inside it. This allows you to take limits. And finally locally finite in the definition means that each point is only in a finite number of open sets.
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