# Open Real Interval is not Closed Set/Corollary

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## Theorem

Let $\R$ be the real number line with the usual (Euclidean) metric.

Let $a, b \in \R$.

Let:

- $I_a = \openint \gets a$
- $I_b = \openint b \to$

be unbounded open real intervals.

Then neither $I_a$ nor $I_b$ are closed sets of $\R$.

## Proof

Consider the relative complement of $I_a$ in $\R$:

- $J = \relcomp \R I = \R \setminus I = \hointr a \to$

Let $\epsilon \in \R_{>0}$.

Consider the open $\epsilon$-ball $\map {B_\epsilon} a$.

Whatever the value of $\epsilon$ is, $a - \epsilon$ is not in $\map {B_\epsilon} a$.

So, by definition, $J$ is not an open set of $\R$.

By Relative Complement of Relative Complement, $\relcomp \R J = I$.

By definition, it follows that $I_a$ is not a closed set of $\R$.

Mutatis mutandis, the same proof shows that neither is $I_b$ a closed set of $\R$.

$\blacksquare$