Closed Real Interval is not Open Set

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Theorem

Let $\R$ be the real number line considered as an Euclidean space.

Let $\left[{a \,.\,.\, b}\right] \subset \R$ be a closed interval of $\R$.


Then $\left[{a \,.\,.\, b}\right]$ is not an open set of $\R$.


Proof

From Closed Real Interval is Neighborhood Except at Endpoints, $a$ and $b$ have no open $\epsilon$-ball lying entirely in $\left[{a \,.\,.\, b}\right]$.

The result follows by definition of open set.

$\blacksquare$


Sources