Is it always possible to make both pieces connected? Inside a path-connected set, there is a path-connected subset ( **G** reen) whose complement ( **P** ink) is disconnected: ![enter image description here]( We would like to move a small piece from G to P, such that both of them are path-connected: ![enter image description here]( Is it always possible? Formally, given two sets $G,P$ such that $G$ and $G\cup P$ are path-connected, and given $\epsilon>0$, we would like to find a subset $H\subseteq G$, such that: * The area of $H$ is at most $\epsilon$; * $G\setminus H$ is path-connected; * $P\cup H$ is path-connected. Is this always possible? If not, what conditions on $G,P$ are required to make it possible?

For a counterexample, let $$ G = \\{0\\}\times[0,1] \cup (0,1) \times ([0,1]\setminus \mathbb Q) \\\ P = \\{1\\}\times([0,1]\setminus \mathbb Q) $$

$G\cup P$ is path-connected, but the only way to connect two different points of $P$ is to go all the way down to the $x$-axis, then horizontally and then up again. So $H$ has to be all of $G$, which has area (that is, Lebesgue measure) $1$.

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Or, for a less pathological example, consider

$$ G = (\mathbb R\setminus \\{0\\})\times (0,\infty) \cup \\{(0,0)\\} \\\ P = (\mathbb R\setminus \\{0\\})\times (-\infty,0) $$

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Or, nicer yet,

$$ G = \mathbb R \times (0,1) \\\ P = \mathbb R^2 \setminus G $$