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.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.4$: Properties of compact spaces