Sorgenfrey Line is Expansion of Real Line
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Theorem
Let $\R = \struct {\R, d}$ be the metric space defined in Real Number Line is Metric Space.
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.
Then $T$ is an expansion of $\R$ as a topological space.
Proof
It is enough to prove that any open set in $\R$ is open in $T$.
Let $a, b \in \R$.
Then:
- $\ds \openint a b = \bigcup_{\epsilon \mathop > 0} \hointr {a + \epsilon} b$
Since $\hointr {a + \epsilon} b$ are open in $T$, $\openint a b$ is also open in $T$.
$\blacksquare$