Which of the following sets are compact:

Question

Which of the following sets are compact:

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

Answer 1

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 $z_1$.