Example · High school
Terms
| Term | This example | Why |
|---|
| diameter |
$\infty$
|
The diameter of $\reals$ is $\infty$, because $\reals$ contains points arbitrarily far apart.
|
| cardinality |
$\mathfrak{c}$
|
The cardinality of $\reals$ is $\mathfrak{c}$, the continuum, which is uncountable.
|
| measure |
$\infty$
|
The Lebesgue measure of $\reals$ is $\infty$, because $\reals$ contains intervals of every length.
|
| Hausdorff dimension |
$1$
|
The Hausdorff dimension of $\reals$ is $1$.
|
| open set |
✓ |
The set $\reals$ is open as a subset of itself, because every interval around a real number is inside $\reals$.
|
| closed set |
✓ |
The set $\reals$ is closed as a subset of itself, because its complement, $\emptyset$, is open.
|
| locally compact |
✓ |
$\reals$ is locally compact, because every point $x$ has the compact neighborhood $[x - 1, x + 1]$.
|
| paracompact |
✓ |
$\reals$ is paracompact, because it is a metric space.
|
| uncountable |
✓ |
|
| compact |
✗ |
The set of real numbers, $\reals$, is not compact, because the sequence $1, 2, 3, \dots$ has no convergent subsequence.
|
| sequentially compact |
✗ |
The set of real numbers, $\reals$, is not sequentially compact, because the sequence $1, 2, 3, \dots$ has no convergent subsequence.
|
| countable |
✗ |
|
| finite |
✗ |
|
| countably infinite |
✗ |
|
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