Why is the Cartesian product $X\times X$ an ordering of $X$? 
It should be remarked that the definition of ordering is not very restrictive. For example, $X\times X$ is an ordering of $X,$ but a rather uninteresting one. Relative to this ordering, each member of $X$ is an upper bound, and in fact a supremum, of every subset. The more interesting  satisfy the further conditions: if $x$ is less than or equal to $y$ and $y$ is less than or equal to $x,$ than $y=x\,.$ In this case there is at most one supremum for a set, and at most infimum. 

This is excerpted from Orderings from John Kelley's General Topology.
Hmm; I stumbled upon the very first part of the excerpt:
What did Kelley mean by that the definition of ordering is not very restrictive?
How is the Cartesian Product $X\times X$ an ordering of $X\,?$

From page 13 of the book:

An ordering (partial ordering, quasi-ordering) is a transitive
  relation. A relation $<$ orders (partially orders) a set $X$ iff it is
  transitive on $X$.

Thus, a subset $R$ of $X\times X$ is an ordering of $X$ if and only if for $(x,y)\in R$ and $(y,z)\in R$ we always have $(x,z)\in R$. This is obviously satisfied by $R=X\times X$ and hence $X\times X$ is an ordering of $X$.
