A given set is closed under an operation if performing that operation on members of the set always produces a member of that set.

Formally, a set $S$ is closed under an operation $\ast$ if it satisfies the magma property:

Example

For example, consider the natural numbers $\mathbb{N}$ ($\{1,2,3,\dots \}$).
Adding two natural numbers will always produce a natural number, therefore the natural numbers are closed under addition.
Subtracting two natural numbers will sometimes produce a number which is not a natural number, for example subtracting the natural number $2$ from the natural number $1$ will produce the number $-1$ which is not a natural number, therefore the natural numbers are not closed under subtraction.

Example

The polynomials are closed under addition, subtraction, and multiplication, but not division.