Hi,

doing my research I found the following problem and I´ll be glad if someone could give a reference.

We say that a compact connected subset $K$ of the plane is psuedo laminated if the following hold:

$K$ has a partition by $C^1$ curves without boundary. For each $x$ in $K$ denote by $W(x)$ the leaf through $x.$

For any $x\in K$ and any $y\in W(x)$ there exist a neighborhood $U_x$ and a continuous map $\phi:U_x\times [0,1]\to\mathbb{R}^2$ such that: 1) $\phi(x,0)=x$ 2) $\phi(x,1)=y$ 3) for any $z\in K\cap U_x$ we have that $\phi(z,t)\in W(z)$ for any $t\in[0,1].$

I´m looking for the following results:

1) Let $K$ be a compact connected subset of $\mathbb{R}^2$ and pseudo laminated. Then $K$ separates the plane in at least two connected components

2) If the complement of $K$ has just two connected components, then $K$ is just a circle.