What is the smallest set of numbers closed under subtraction?

Any one of the sets of the form:

{kz : where k is any fixed integer and z belongs to the set of all integers}

Thus, k = 1 gives the set of all integers,

k = 2 is the set of all even integers,

k = 3 is the set of all multiples of 3,

and so on.

You might think that as k gets larger the sets become smaller because the gaps between numbers in the set increases. However, it is easy to prove that the cardinality of each of these infinite sets is the same.