Open Ray is Open in GO-Space/Definition 2

Theorem

Let $\struct {S, \preceq, \tau}$ be a generalized ordered space by Definition 2.

That is:

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

Let $\struct {S, \tau}$ be a topological space.

Let there be:

a linearly ordered space $\struct {S', \preceq', \tau'}$

and:

a mapping $\phi: S \to S'$ which is both:

a $\preceq$-$\preceq'$ order embedding

and:

a $\tau$-$\tau'$ topological embedding.

Let $p \in S$.

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.

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

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

$\map {\phi^{-1} } {\map \phi p^\succ} = p^\succ$

$\map \phi p^\succ$ is an open ray in $S'$

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

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

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

$\blacksquare$