Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $Y\subset X$, and equip $Y$ with the subspace metric. For a subset $F\subset Y$, the set $F$ is closed in the [metric subspace](/page/Metric%20Subspace) $Y$ if and only if there exists a [closed set](/page/Closed%20Set) $C\subset X$ in the metric space $X$ such that $F=Y\cap C$.