Indeed, for each index let be the set of all nonempty subsets of .

Theorem: The set of all finite subsets of the natural numbers is countable.

Let be the set of all subsets of the NR bits i.e.

We denote by the set of all infinite subsets of .

On the set of all non-empty subsets of M, d yields an extended pseudometric.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

The set of all linearly independent subsets of a vector space V, ordered by inclusion.

By we denote the set of all subsets of is called the power set of A.

Let denote the set of all compact subsets of .

The set of all subsets of Z is uncountable.