Quiz, week 9

Which of these metric spaces are compact?

$([0, 1], d_2)$
$((0, 1), d_2)$
$(\mathbb{R}_+, d_2)$
$([0, 1] × [2, 3], d_2)$
$([0, 1], d_1)$
$(\{1, 2, 3\}, d_2)$
$(\{\sin, \log\}, d_\infty)$ , where $d_\infty(f, g) = \sup_{x \in [1, 2]} |f(x)-g(x)|$ .
$(\{(x, y) : x² + y² = 1\}, d_2)$
$(\{(\cos x, \sin x) : x \in [0, 2π]\}, d_2)$

Find a sequence that does not have a convergent subsequence in

$([0, 1), d_2)$ .
$A = B_1(1)$ inside the space $(\mathbb{R}, d_2)$ .
$(B[0, 1], d_\infty)$ (harder)

What is wrong with these nonsensical questions?

Is $(\sin, d_\infty)$ a compact metric space?
What is the domain of $(\mathbb{R}, d_2)$ ?
What is the first element of $\{\sin, \cos, \log\}$ ?
What is the contrapositive of “The set $[0, 1)$ has no maximum.”