Definition:Ordered Square

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

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

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

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

Significance
The ordered square is one of many spaces used primarily to construct counterexamples to conjectures in topology.