Lebesgue measure of set of spheres with irrational radii

Question

For each $\alpha \in \mathbb R$ Let $$ S_{\alpha} =\{(x,y,z) \in \mathbb R^3/x^2+y^2+z^2=\alpha^2 \}$$ Let $E=\bigcup_{\alpha \in \mathbb R\vert \mathbb Q} S_\alpha$. Which of the following options are true?

1. The lebesgue measure of $E$ is infinite

2. $E$ contains a non-empty open set

3. $E$ is path connected

4. Every open set containing $E^c$ has infinite Lebesgue measure.

I think options 1 and 4 are true. Because Lebesgue measure of irrationals in $ \mathbb R$ is infinite. For option 2, if $E$ contain non empty open set than it would intersect some point of sphere with rational radius. I am confused about options 3 and 4. Thanks.

Answer 1

$1.$ is true because $\mu(E^c)=0$ since it's a countable sum of sets that are measure zero.

$2. $ is obviously false.

$ 3. $ is false, because you can't connect points lying on a different circle

$4.$ is also false, because $\mu(E^c)=0$