\begin{definition}[Definition:Ordered Square]
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'''.
Category:Definitions/Examples of Topologies
\end{definition}