Definition:Generalized Ordered Space

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Definition

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

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


Definition 1

$\left({S, \preceq, \tau}\right)$ is a generalized ordered space if and only if:

$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space
$(2): \quad$ there exists a basis for $\left({S, \tau}\right)$ whose elements are convex in $S$.


Definition 2

$\left({S, \preceq, \tau}\right)$ is a generalized ordered space if and only if:

$(1): \quad$ there exists a linearly ordered space $\left({S', \preceq', \tau'}\right)$
$(2): \quad$ there exists a mapping $\phi: S \to S'$ such that $\phi$ is both an order embedding and a topological embedding.


Definition 3

$\left({S, \preceq, \tau}\right)$ is a generalized ordered space if and only if:

$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space
$(2): \quad$ there exists a sub-basis for $\left({S, \tau}\right)$ each of whose elements is an upper set or lower set 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.