Definition:Ordered Square

Definition

Let:

$S = \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right]$

where $\left[{0 \,.\,.\, 1}\right]$ is the closed real interval from $0$ to $1$.

Let $\preceq$ be the lexicographic ordering on $S$ induced by the usual ordering of $\left[{0 \,.\,.\, 1}\right]$.

Let $\tau$ be the order topology on $S$ induced by $\preceq$.

Then $\left({S, \tau}\right)$ is the ordered square.