If A and B are totally ordered, then the result is a total order as well.

Because the a are totally ordered, this is a well-founded definition.

In a totally ordered set, like the real numbers mentioned above, the concepts are the same.

First, since the real numbers are totally ordered, they carry an order topology.

If a set is totally ordered, then the following are equivalent to each other:

Thus in a totally ordered set we can simply use the terms minimum and maximum.

Z is a totally ordered set without upper or lower bound.

A sequence is usually indexed by the natural numbers, which are a totally ordered set.

The right order topology on a totally ordered set is a related example.

Any such interval is well defined only because the real numbers are totally ordered.