Total order

An extension of a given partial order to a total order is called a linear extension of that partial order.

One may define a totally ordered set as a particular kind of lattice, namely one in which we have

The order topology induced by a total order may be shown to be hereditarily normal.

There are a number of results relating properties of the order topology to the completeness of X:

Intuitively, this means that the elements of the second set are added on top of the elements of the first set.

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are:

All three can similarly be defined for the Cartesian product of more than two sets.

A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.