Definition:Generalized Ordered Space/Definition 2

Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $\tau$ be a topology on $S$.

$\struct {S, \preceq, \tau}$ is a generalized ordered space if and only if:

$(1): \quad$ there exists a linearly ordered space $\struct {S', \preceq', \tau'}$

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

Also see

• Equivalence of Definitions of Generalized Ordered Space