Let $X$ be a metric space and $Y \subseteq X$. $Y$ is closed if $X \setdif Y$ is open.
Let $X$ be a metric space. If $\mathcal{S}$ is a set of closed sets from $X$, then $\bigcap \mathcal{S}$ is closed.
Let $X$ be a metric space and $Y \subseteq X$. $Y$ is closed if $X \setdif Y$ is open.
Let $X$ be a metric space. If $\mathcal{S}$ is a set of closed sets from $X$, then $\bigcap \mathcal{S}$ is closed.