Metrizability is hereditary

Statement
If $$X$$ is a metrizable space, and $$Y$$ is a subset of $$X$$ given the subspace topology, then $$Y$$ is also a metrizable space.

Facts used

 * 1) uses::Topology from subspace metric equals subspace topology

Proof
Suppose $$d$$ is a metric on $$X$$ inducing the given topology on $$X$$. By fact (1), the subspace metric on $$Y$$ induces the subspace topology on $$Y$$. Thus, the subspace topology is induced by a metric, hence $$Y$$ is metrizable.