Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y) :1\leq x^2+y^2\leq 2\text{ or }3\leq x^2+y^2\leq 4\}$. Then which are true:
$1$.$X\setminus K$ has $3$ connected components.
$2$.$X\setminus K$ has no relatively compact connected components.
$3$.$X\setminus K$ has two relatively compact connected components.
$4$.All connected components of $X\setminus K$ are relatively compact.
I got that $X\setminus K$ is $\{(x,y):x^2+y^2<1\text{ or }2<x^2+y^2<3\text{ or }4<x^2+y^2<5\}$. Thus $X\setminus K$ has $3$ connected components.
Since neither components is closed hence not compact thus $X\setminus K$ has no relatively compact connected components. So $1,2$ are correct.
Are these options correct. I would be grateful if someone could check that.
I agree with your components. But relatively compact means their closure is compact. And the first two components have compact closures, while the last one has not (we can take sequences going to points with norm 5, that have no convergent subsequence because those points are not in $X$).