Can someone present an example of linear ordering less trivial than $A = [1,2,3,4,5,6...]$

Question

A linear ordering (loset) is a poset that also satisfies the trichotomy law.

For any $x,y \in A$, we have $x \leq y$ or $y \leq x$

A common example is presented as $A = [1,2,3,4,5,6...]$

Can someone present an example that is less trivial than the set presented above so I can see the full utility of linear ordering?

Answer 1

Some examples include:

• The set $\mathbb Q$ of rational numbers with the usual ordering, which is interesting in that it has the same cardinality as your $A$ but a very different kind of order (a dense order).
• The long line, which is interesting in that it has the same cardinality as the set $\mathbb R$ of real numbers but a substantially different kind of order than the standard order on $\mathbb R$, leading to a different topology than the standard topology on $\mathbb R$.
• The first uncountable ordinal $\omega_1$ (which is used in defining the long line above).