nLab order topology

Definition

Given a linearly ordered set $(S, \lt)$, its order topology is the topology on $S$ generated from the sub-basis $\beta \subset P(S)$

$$\beta \coloneqq \left\{ (a,\infty), (-\infty,a) \right\}_{a \in S} \,.$$

whose elements are the “open half rays”

$$(a, \infty) \coloneqq \left\{ s \in S \,\vert\, a \lt s \right\} \phantom{AAAA} (-\infty, a) \coloneqq \left\{ s \in S \,\vert\, s \lt a \right\} \,.$$

Examples

• The usual Euclidean space metric topology on the real numbers is equal to the order topology with respect to the canonical ordering of the real numbers. The analogous statement then holds for the rational numbers $\mathbb{Q} \subset \mathbb{R}$ equipped with their subspace topology.

Related concepts

• specialization topology,

• Scott topology

References

• Wikipedia, Order topology