Total Order - Examples

Examples

• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

• Any subset of a totally ordered set, with the restriction of the order on the whole set.

• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).

• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on X by setting x₁ < x₂ if and only if f(x₁) < f(x₂).

• The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, is itself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as a subset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol to the alphabet (and defining a space to be less than any letter).

• The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hence also the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique (to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A is the smallest with a certain property if whenever B has the property, there is an order isomorphism from A to a subset of B):
  • The natural numbers comprise the smallest totally ordered set with no upper bound.
  • The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.
  • The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. The definition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there is a 'q' in the rational numbers such that 'a' < 'q' < 'b'.
  • The real numbers comprise the smallest unbounded totally ordered set that is connected in the order topology (defined below).