"Closure" is a property which a set either has or lacks with respect to a given operation. A set is closed with respect to that operation if the operation can always be completed with elements in the set.

For example, the set of even natural numbers, 2, 4, 6, 8,..., is closed with respect to addition because the sum of any two of them is another even natural number. It is not closed with respect to division because the quotients 6/2 and 4/8, for instance, cannot be computed without using odd numbers or fractions.

Knowing the operations for which a given set is closed helps one understand the nature of the set. Thus one knows that the set of natural numbers is less versatile than the set of integers because the latter is closed with respect to subtraction, but the former is not. Similarly one knows that the set of polynomials is...