Open Ray is Open in GO-Space/Definition 2

Theorem
Let $\left({X, \preceq, \tau}\right)$ be a generalized ordered space by Definition 2.

That is:


 * Let $\left({X, \preceq}\right)$ be a totally ordered set.


 * Let $\left({X, \tau}\right)$ be a topological space.


 * Suppose that there is a Definition:Linearly Ordered Space $(X', \preceq', \tau')$ and a mapping $\phi: X \to X'$ which is both a $\preceq$-$\preceq$ order embedding and a $\tau$-$\tau'$ topological embedding.

Let $p \in X$.

Then ${\dot\downarrow} p$ and ${\dot\uparrow} p$ are $\tau$-open.

Here, ${\dot\downarrow}p$ and ${\dot\uparrow}p$ are the strict down-set and strict up-set of $p$, respectively.

Proof
We will prove that ${\dot\uparrow} p$ is open.

That ${\dot\downarrow}p$ is open will follow by duality.

By Inverse Image under Order Embedding of Strict Up-Set of Image of Point:


 * $\phi^{-1} \left({ {\dot\uparrow'} \phi(p) }\right) = {\dot\uparrow}p$,

${\dot\uparrow'} \phi(p)$ is an open ray in $X'$, and therefore $\tau'$-open by the definition of the order topology.

Since $\phi$ is a topological embedding, it is continuous.

Thus ${\dot\uparrow} p$ is $\tau$-open.