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.
$\blacksquare$