Open Real Interval is not Compact/Proof 1
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Theorem
Let $\R$ be the real number line considered as an Euclidean space.
Let $I = \openint a b$ be an open real interval.
Then $I$ is not compact.
Proof
From Open Real Interval is not Closed Set, $I$ is not a closed set of $\R$.
The result follows by definition of compact.
$\blacksquare$