Relatively Compact Subspace/Examples/Open Unit Interval in Reals

Example of Relatively Compact Subspace
The open unit interval $\openint 0 1$ is a relatively compact subspace of the real number line $\R$.

Proof
From Closure of Open Real Interval is Closed Real Interval, the closure of $\openint 0 1$ is $\closedint 0 1$.

From Closed Bounded Subset of Real Numbers is Compact, $\closedint 0 1$ is compact in $\R$.

The result follows by definition of relatively compact subspace.