$\mathbb{Q} \cap [0, 1]$

Terms

Term	This example	Why
diameter	$1$	The diameter of $\mathbb{Q} \cap [0, 1]$ is $1$, because it contains $0$ and $1$.
cardinality	$\aleph_0$	The cardinality of $\mathbb{Q} \cap [0, 1]$ is $\aleph_0$, because its elements can be listed in order of their denominators.
measure	$0$	The Lebesgue measure of $\mathbb{Q} \cap [0, 1]$ is $0$, because it is countable: the $n$th rational number can be covered by an interval of length $\varepsilon / 2^n$, for a total length of at most $\varepsilon$.
Hausdorff dimension	$0$	The Hausdorff dimension of $\mathbb{Q} \cap [0, 1]$ is $0$, as for every countable set.
infinite	✓
countably infinite	✓	$\mathbb{Q} \cap [0, 1]$ is countably infinite, and so infinite, because its infinitely many fractions can be listed.
bounded	✓
countable	✓	$\mathbb{Q} \cap [0, 1]$ is countable because the fractions $p/q$ in $[0, 1]$ can be listed in order of their denominators $q$.
uncountable	✗	$\mathbb{Q} \cap [0, 1]$ is bounded, because all of its elements are between $0$ and $1$, and it is not uncountable.
finite	✗	$\mathbb{Q} \cap [0, 1]$ is not finite, because it contains $1/n$ for every $n \geq 1$.