# Definition:Generalized Ordered Space

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## Contents

## 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

### 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**.