Open Ray is Open in GO-Space/Definition 2

From ProofWiki
Jump to navigation Jump to search

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.


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.



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

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

$\blacksquare$