Definition:Generalized Ordered Space/Definition 2

Definition
Let $\left({X, \preceq}\right)$ be a totally ordered set.

Let $\tau$ be a topology for $X$.

Then $\left({X, \preceq, \tau}\right)$ is a generalized ordered space there exists:
 * $(1): \quad$ a linearly ordered space $\left({X', \preceq', \tau'}\right)$


 * $(2): \quad$ a mapping $\phi: X \to X'$ such that $\phi$ is both an order embedding and a topological embedding.

Also see

 * Equivalence of Definitions of Generalized Ordered Space