Proposition 1

If $A$ and $B$ are two subsets of a topological space $X$, and $B \subset A$, then the closure of $B$ in the subspace $A$ is the intersection with $A$ of the closure $\overline{B}$ of $B$ in $X$.

If $x \in A$, every neighbourhood of $x$ in $A$ is of the form $V \cap A$, where $V$ is a neighbourhood of $x$ in $X$. Since $V \cap B = (V \cap A) \cap B$, it follows that $x$ lies in the closure of $B$ with respect to $A$ if and only if $x$ lies in the closure of $B$ with respect to $X$.