Definition:Unit Square under Lexicographic Ordering

Definition

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

\(\ds S\)

\(=\)

\(\ds \set {\tuple {x, y}: x, y \in \R, 0 \le x \le 1, 0 \le y \le 1}\)

\(\ds \)

\(=\)

\(\ds \closedint 0 1 \times \closedint 0 1\)

where $\closedint 0 1$ is the closed unit interval

Let $\preccurlyeq_l$ denote the lexicographic ordering applied to $S$:

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

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

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $48$. Lexicographic Ordering on the Unit Square