| interior |
$\operatorname{int} (0, 1) = (0, 1)$
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The interior of $(0, 1)$ is $(0, 1)$, because every point of $(0, 1)$ has a small interval around it inside $(0, 1)$; that is, $(0, 1)$ is open.
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| boundary |
$\partial (0, 1) = \setdef{0, 1}$
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The boundary of $(0, 1)$ is $\setdef{0, 1}$, because every interval around $0$ or $1$ contains points inside and outside $(0, 1)$.
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| exterior |
$\operatorname{ext} (0, 1) = (-\infty, 0) \cup (1, \infty)$
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The exterior of $(0, 1)$ is everything outside $[0, 1]$, because each of those points has a small interval around it that misses $(0, 1)$.
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| closure |
$\overline{(0, 1)} = [0, 1]$
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The closure of $(0, 1)$ adds its boundary points $0$ and $1$.
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| open set |
✓ |
The interval $(0, 1)$ is open in $\reals$, because every point of $(0, 1)$ has a small interval around it inside $(0, 1)$.
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| bounded |
✓ |
$(0, 1)$ is bounded.
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| closed set |
✗ |
The interval $(0, 1)$ is not closed in $\reals$, because the sequence $1/n$ in $(0, 1)$ converges to $0$, which is not in $(0, 1)$.
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| finite |
✗ |
$(0, 1)$ is not finite, because it contains $1/n$ for every $n \geq 2$.
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| compact |
✗ |
$(0, 1)$ is not compact, because it is not closed: the sequence $1/n$ has no limit in $(0, 1)$.
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