Nagata–Smirnov metrization theorem

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 (i.e., σ-locally finite) basis.

Unlike 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. The theorem is named after Junichi Nagata and Yuriĭ Mikhaĭlovich Smirnov.

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