Definition:Generalized Ordered Space

Definition

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

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

Definition 1

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

$(1): \quad \struct {S, \tau}$ is a Hausdorff space

$(2): \quad$ there exists a basis for $\struct {S, \tau}$ whose elements are convex in $S$.

Definition 2

$\struct {S, \preceq, \tau}$ 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.

Definition 3

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

$(1): \quad \struct {S, \tau}$ is a Hausdorff space

$(2): \quad$ there exists a sub-basis for $\struct {S, \tau}$ each of whose elements is an upper section or lower section in $S$.

Also known as

A generalized ordered space is often abbreviated as a GO-space.

Also see

• Equivalence of Definitions of Generalized Ordered Space

• Definition:Linearly Ordered Space

• Results about generalized ordered spaces can be found here.

Linguistic Note

The British English spelling renders as generalised ordered space.