Relatively Compact Subspace/Examples/Open Unit Interval in Itself

Example of Relatively Compact Subspace

The open unit interval $\openint 0 1$ is not a relatively compact subspace of $\openint 0 1$ itself.

Proof

From Open Real Interval is not Compact, $\openint 0 1$ is not a compact space no matter what topological space it is embedded in.

From Underlying Set of Topological Space is Clopen, $\openint 0 1$ is both closed and open in $\openint 0 1$.

From Set is Closed iff Equals Topological Closure:

$\openint 0 1 = \cl {\openint 0 1}$

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