Nonempty

The axiom of choice asserts the existence of such elements; it is therefore equivalent to: :Given any family of nonempty sets, their Cartesian product is a nonempty set.

A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a linear subspace of V, or simply a subspace of V, when the ambient space is unambiguously a vector space.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.

A topological space X is disconnected if there exist disjoint, nonempty, closed subsets A and B of X whose union is X. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.

Basic properties of subgroups *A subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses.

Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection.

Comments * The descending chain condition on P is equivalent to P being well-founded : every nonempty subset of P has a minimal element (also called the minimal condition).

Connected components The maximal connected subsets (ordered by inclusion ) of a nonempty topological space are called the connected components of the space.

Elements The convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex.

Every bounded nonempty polytope is pointed.

Examples The nature of the individual nonempty sets in the collection may make it possible to avoid the axiom of choice even for certain infinite collections.

For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent.

For example, to show that the naturals are well-ordered —every nonempty subset of N has a least element —one can reason as follows.

For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1, although the measure cannot be exactly one (else its complement would be a nonempty open set with measure zero, which is impossible).

Hence it is commonly stipulated that all of the domains be nonempty.

If an ordered set has the property that every nonempty subset of having an upper bound also has a least upper bound, then is said to have the least-upper-bound property.

In all cases, the first condition can be replaced by the following well-known criterion that ensures a nonempty subset of a group is a subgroup: :1'.

It follows that every nonempty periodic semigroup has at least one idempotent.

Let a nonempty X ⊆ N be given and assume X has no least element.