Open Real Interval is not Compact

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.

Also see

 * Closed Real Interval is Compact in Metric Space