TITLE: Which of the following sets are compact:
QUESTION [5 upvotes]: Which of the following sets are compact:

$\{(x,y,z)\in \Bbb R^3:x^2+y^2+z^2=1\}$ in the Euclidean topology.
$\{(z_1,z_2,z_3)\in \Bbb C^3:{z_1}^2+{z_2}^2+{z_3}^2=1\}$ in the Euclidean topology.
$\prod_{n=1}^\infty A_n$ with the product topology where $A_n=\{0,1\}$ has discrete topology.
$\{z\in \Bbb C:|\operatorname{Re} z |\leq a \}$ for some fixed positive real number $a$ in the Euclidean topology.

$1$ is closed and bounded and hence compact,$2$ is closed but not bounded and hence not compact.
$3$ is compact by Tychonoff Theorem and $4$ is not bounded and hence not compact.
Are these correct?

REPLY [3 votes]: That's correct. However, 1, 2 and 4 need a proof.
All three sets are closed, being inverse images of a closed set under a continuous function.
The set in 1 is bounded, because it is contained in $[-1,1]^3$.
The sets in 2 and 4 are not bounded, because they contain element with arbitrarily large norm; can you show them?
Set 4:

 Easy: you can take $z=a+bi$ with arbitrary $b$.

Set 2:

 Consider $z_3=1$. Then you can take $z_2=iz_1$, for arbitrary $z_1$.