Closed Sets of Right Order Space on Real Numbers

Theorem
Let $T = \struct {\R, \tau}$ be the right order space on $\R$.

Then $H \subseteq S$ is closed in $T$ :
 * $H = \O$ or $\R$

or
 * $H = \hointl {-\infty} a$ for some $a \in \R$.

Proof
By definition of the right order space on $\R$, $U \subseteq S$ is open in $T$ :
 * $U = \O$ or $\R$

or
 * $U = \openint a \infty$ for some $a \in \R$.

Note that:
 * $\R \setminus \O = \R$
 * $\R \setminus \R = \O$
 * $\R \setminus \openint a \infty = \hointl {-\infty} a$

The result follows from the definition of closed set.