Sorgenfrey Line is Expansion of Real Line

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 basic open set in $\R$ is open in $T$.

Take $a, b \in \R$, then trivially $(a,b)=\bigcup_{\varepsilon>0}[a+\varepsilon,b)$.

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

$\blacksquare$