Closed Sets of Right Order Space on Real Numbers

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Theorem

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


Then $H \subseteq S$ is closed in $T$ if and only if:

$H = \O$ 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 $T$ if and only if:

$U = \O$ 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.

$\blacksquare$