# Equivalence of Definitions of Generalized Ordered Space/Definition 1 implies Definition 3

This article needs proofreading.Please check it for mathematical errors.If you believe there are none, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Proofread}}` from the code. |

## Theorem

Let $\struct {S, \preceq, \tau}$ be a generalized ordered space by Definition 1:

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

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

Then $\struct {S, \preceq, \tau}$ is a generalized ordered space by 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 $S$.

## Proof

Let $\BB$ be a basis for $\tau$ consisting of convex sets.

Let:

- $\SS = \set {U^\succeq: U \in \BB} \cup \set {U^\preceq: U \in \BB}$

where $U^\succeq$ and $U^\preceq$ denote the upper closure and lower closure respectively of $U$.

By Upper Closure is Upper Section and Lower Closure is Lower Section, the elements of $\SS$ are upper and lower sections.

It is to be shown that $\SS$ is a sub-basis for $\tau$.

By Upper and Lower Closures of Open Set in GO-Space are Open:

- $\SS \subseteq \tau$

By Convex Set Characterization (Order Theory), each element of $\BB$ is the intersection of its upper closure with its lower closure.

Thus each element of $\BB$ is generated by $\SS$.

The term Definition:Generated as used here has been identified as being ambiguous.If you are familiar with this area of mathematics, you may be able to help improve $\mathsf{Pr} \infty \mathsf{fWiki}$ by determining the precise term which is to be used.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Disambiguate}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Thus $\SS$ is a sub-basis for $\tau$.

$\blacksquare$