Closure of a subset of a subspace of a topological space Let $X$ be a topological space.
Let $Y$ be a subspace of $X$.
Let $T$ be a subset of $Y$.
Let $\overline T$ be the closure of $T$ in $X$.
Then $Y \cap \overline T$ is the closure of $T$ in $Y$?.

Hint: Use the characterization of the closure of $T$ in $X$ as
$$\overline{T}=\bigcap_{\substack{\text{closed}\, C\,\subseteq X\\ C\,\supseteq T}} C$$
together with the definition of the subspace topology on $Y$ to prove that the statement is true, i.e. that the closure of $T$ in $Y$ is $\overline{T}\cap Y$.
