Let $K \subset L \subset S^2$, where $S^2$ is the 2-sphere and $K$ and $L$ are compact subsets with empty interior and $L$ is connected (I don't think all of those are relevant hypotheses though, but well).
Suppose $S^2 - L$ is connected. Is it true then that $S^2-K$ is connected as well ?
This is not a formal proof, but I would think you could justify these statements formally to get one.
Since $L$ and $K$ have empty interiors, they must be curves. There are only two ways for $S^2 - L$ to not be connected:
Either $1$) the curve (or set) $L$ begins and ends on a boundary of $S^2$ or $2$) $L$ is a closed curve. Now $S^2 - L$ will not be connected because it is the union of two disjoint open sets.
Since $K$ is a subset of $L$, if $L$ does not form a closed curve and does not connect two boundary points of $S^2$, then neither can $K$, therefore $S^2 - K$ must also be connected.
On the other hand, if it had not stated that $L$ and $K$ had empty interiors, then this would not necessarily be the case: