Definition:Generalized Ordered Space
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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
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
- Results about generalized ordered spaces can be found here.
Linguistic Note
The British English spelling renders as generalised ordered space.