If $Z\subset Y\subset X$ and $Z$ is closed in $Y$, then $Z=Y\cap\overline{Z}$

Question

Suppose $X,Y,Z$ are topological spaces such that $Z\subset Y\subset X$ and $Z$ is closed in $Y$. Is it then true that $Z=Y\cap\overline{Z}$? (Here $\overline{Z}$ is the closure of $Z$ in $X$).

If $Z$ is closed in $Y$, then $Z=C\cap Y$ for some set $C$ closed in $X$. I'm not sure where to go from here though.

Answer 1

Since $\def\Bar{\overline} Z=C\cap Y$ then we have $\Bar Z\subseteq \Bar C\cap \Bar Y=C\cap\Bar Y$, so $$\Bar Z\cap Y\ \subseteq\ C\cap\Bar Y\,\cap\, Y\ =\ C\cap Y\ =\ Z\,,$$ and clearly $Z\subseteq \Bar Z\cap Y$.