Second-countability is hereditary

Property-theoretic statement
The property of topological spaces of being a second-countable space satisfies the metaproperty of topological spaces of being hereditary.

Verbal statement
Any subspace of a second-countable space is second-countable under the subspace topology.

Second-countable space
A topological space is termed second-countable if it admits a countable basis.

Subspace topology
Given a topological space $$X$$ and a subspace $$A$$, with a basis $$\{ B_i \}_{i \in I}$$ for $$X$$, the subspace topology on $$A$$ is defined as a topology with basis $$B_i \cap A$$.

Proof
Given: A second-countable space $$X$$ with countable basis $$B_n, n \in \mathbb{N}$$. A subspace $$A$$ of $$X$$

To prove: $$A$$ has a countable basis.

Proof: By the definition of subspace topology, the sets $$B_n \cap A$$ form a basis for the subspace topology on $$A$$. This is a countable basis for $$A$$.