Relatively Compact Subspace/Examples/Open Unit Interval in Reals
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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