Closed sets are complements of open sets

Closed sets are complements of open sets: In a metric space $(X,d)$, a set $F\subseteq X$ is closed if and only if $X\setminus F$ is open .

Consequently:

• arbitrary intersections of closed sets are closed, and
• finite unions of closed sets are closed.

This duality between open and closed sets is a basic tool in topology and analysis, especially for closure , limit points , and compactness arguments.

Proof sketch (optional): If $X\setminus F$ is open, then points outside $F$ have neighborhoods disjoint from $F$, so all limit points of $F$ must lie in $F$. Conversely, if $F$ contains all its limit points, then each point outside $F$ has positive distance to $F$, giving an open ball contained in the complement.