An interval $(a,b)$ is totally bounded in $\Bbb R$.
Would $\bigcup\limits _{n\ge1}(1/2^n,1/{2^{n-1}})$ or $\bigcup\limits _{n\ge1}(1/\alpha^n,1/{\alpha^{n-1}})$ for $\alpha\in\Bbb R_{>2}$ be good examples?
2026-03-28 06:24:00.1774679040
Example of open and bounded subset of $\Bbb R$ that is not totally bounded
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Actually, in $\mathbb{R}^n$, every bounded set (open or not) is a subset of some compact set, and therefore it is totally bounded.