Metrizability is hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., metrizable space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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Suppose is a metric on inducing the given topology on . By fact (1), the subspace metric on induces the subspace topology on . Thus, the subspace topology is induced by a metric, hence is metrizable.