If we remove a countable family of simple non-closed curves , which are mutually disjoint , from the plane , then is the resulting space connected?

Question

Let $\mathcal F:=\{f:[0,1]\to \mathbb R^2 : f \mbox{ is injective }\}$ be a countable family . If $Im f \cap Im g=\emptyset , \forall f,g \in \mathcal F$ , then is it true that

$\mathbb R^2 \setminus \cup_{f \in \mathcal F} Im f$ is connected ?

Answer 1

Let $f_x(t) = (x,t)$ for $x\in\mathbb{R}$ and $t\in[0,1]$. Then $f_x$ is an injective curve and $\operatorname{Im} f_x\cap \operatorname{Im} f_y \neq \emptyset$ if and only if $x=y$. The union $\bigcup_{x\in\mathbb{R}}f_x(t) = \mathbb{R} \times [0,1]$ separates the plane into two half planes.