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:

$\displaystyle \openint a b = \bigcup_{\epsilon \mathop > 0} \hointr {a + \epsilon} b$

Since $\hointr {a + \epsilon} b$ are open in $T$, $\openint a b$ is also open in $T$.

$\blacksquare$