Definition:Unit Square under Lexicographic Ordering

Definition
Let $S$ be the unit square in the (real) Cartesian plane:

Let $\preccurlyeq$ be the lexicographical ordering applied to $S$:


 * $\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S: \tuple {x_1, y_1} \preccurlyeq \tuple {x_2, y_2} \iff \begin {cases} x_1 < x_2 \\ x_1 = x_2 \land y_1 \le y_2 \end {cases}$

Let $\tau$ be the order topology be applied to the ordered structure $\struct {S, \preccurlyeq}$.

Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the unit square under lexicographical ordering.