Sorgenfrey Line is Expansion of Real Line

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Theorem

Let $\R = \left({\R, d}\right)$ be the metric space defined in Real Number Line is Metric Space.

Let $T = \left({\R, \tau}\right)$ 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:

$\displaystyle \left({a \,.\,.\, b}\right) = \bigcup_{\varepsilon \mathop > 0} \left[{a + \varepsilon \,.\,.\, b}\right)$

Since $\left[{a + \varepsilon \,.\,.\, b}\right)$ are open in $T$, $\left({a \,.\,.\, b}\right)$ is also open in $T$.

$\blacksquare$