Metrizability
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A topological space \( (X,\mathcal{T}_X) \) is metrizable if there is a metric \(d \) on \( X\) such that \(\mathcal{T}_X = \mathcal{T}_d \), where \( \mathcal{T}_d \) is the metric topology on \( X\) induced by \( d\).
Metrizability is a homeomorphism invariant. Metrizability is not preserved by quotients, continuous images, or continuous preimages. Metrizable spaces are \( \mathcal{T}_4 \).
