At the start of Baby Rudin (Walter Rudin's Principles of Mathematical Analysis), Rudin defines an order $<$ on a set $S$ as being any relation which obeys the following two axioms:
(1) One and only one of $x<y, x=y, y<x$ is true.
(2) If $x<y$ and $y<z$ then $x<z$.
My question is, to what notion of order (in order theory) does this correspond? Is it a total order (which is defined as a relation which is reflexive, transitive, antisymmetric, and strongly connected)?
Best Answer
It is an alternative description of a total order.
Usually an order on a set is defined as relation $\le$ which has suitable properties. One can then define $x < y$ iff $x \le y $ nd $x \ne y$.
If $\le$ is a total order, then $<$ clearly satisfies Rudin's two axioms. Conversely, given Rudin's relation $<$, we can define $x \le y$ iff $x < y$ or $x = y$. It is an easy exercise to verify that $\le$ is a total order.