Complement of an open set is closed

Statement

Let $X$ be a topological space and let $U \subseteq X$. Then $U$ is open in $X$ if and only if $X \setminus U$ is closed in $X$.

Remarks

Proof: $X \setminus (X \setminus U) = U$, so the claim is the definition of a closed set applied to $X \setminus U$. Thus the closed sets determine the open sets, and the open sets determine the closed sets.

Explanations

L1 human

Taking the complement turns open sets into closed sets and closed sets back into open sets. For example, $(0, 1)$ is open in $\mathbb{R}$, so $(-\infty, 0] \cup [1, \infty)$ is closed.