Definition

A subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset, or for subset $U\subset V$: $$\begin{align*} u+v&\in U ,\space &\forall \space u,v\in U \newline av&\in U ,\space &u\in U \text{ and } a\in \mathbb{F} \end{align*}$$

Example: The natural numbers $\mathbb{N}$, are closed under addition but under subtraction: $1-2$ is not a natural number.