Nagata–Smirnov metrization theorem

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The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space $X$ is metrizable if and only if it is regular and Hausdorff and has a countably locally finite basis.

Unlike the Urysohn's metrization theorem which provides a sufficient condition for metrization, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable.