Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| interior |
$\operatorname{int} \mathbb{Q} = \emptyset$
|
Every interval contains rational and irrational numbers, so $\mathbb{Q}$ has an empty interior and an empty exterior.
|
| boundary |
$\partial \mathbb{Q} = \reals$
|
Every interval contains both rational and irrational numbers, so every real number is a boundary point.
|
| exterior |
$\operatorname{ext} \mathbb{Q} = \emptyset$
|
|
| closure |
$\overline{\mathbb{Q}} = \reals$
|
The closure of $\mathbb{Q}$ is $\reals$, because every real number is a limit of rational numbers.
|
| dense set |
✓ |
$\mathbb{Q}$ is dense in $\reals$ because every interval contains a rational number.
|
| open set |
✗ |
The set $\mathbb{Q}$ is not open in $\reals$, because every interval around a rational number contains irrational numbers.
|
| closed set |
✗ |
The set $\mathbb{Q}$ is not closed in $\reals$, because rational numbers converge to irrational ones, such as $1, 1.4, 1.41, 1.414, \dots \to \sqrt{2}$.
|
| almost everywhere |
✗ |
$\mathbb{Q}$ does not hold almost everywhere, because $\mathbb{Q}$ has measure zero.
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