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.

Let there be:
 * a linearly ordered space $\left({X', \preceq', \tau'}\right)$

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:
 * $p^\prec$ and $p^\succ$ are $\tau$-open

where:
 * $p^\prec$ is the strict lower closure of $p$
 * $p^\succ$ is the strict upper closure of $p$.

Proof
We will prove that $p^\succ$ is open.

That $p^\prec$ is open will follow by duality.

By Inverse Image under Order Embedding of Strict Upper Closure of Image of Point:


 * $\phi^{-1} \left({\phi \left({p}\right)^\succ}\right) = p^\succ$


 * $\phi \left({p}\right)^\succ$ is an open ray in $X'$

Therefore $\tau'$-open by the definition of the order topology.

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

Thus $p^\succ$ is $\tau$-open.