Let $U$ be a simply connected open set of $\mathbb{R}^2$, and $\phi:[0,1] \rightarrow U$ a cross-cut, that is an injective continuous map such that $\phi((0,1)) \subset U$, while $\phi(0) \in \mathbb{R}^2 \backslash U$ and $\phi(1) \in \mathbb{R}^2 \backslash U$.
Then $U \backslash \phi((0,1))$ has exactly two connected components which are both simply connected (see e.g. Newman, Elements of the Topology of Plane Sets of Points, Chapter VI, Theorem (5.1)), say $D_1$ and $D_2$.
Is $D_{i} \cup \phi((0,1))$ homeomorphic to the half plane $H= \{(x,y) \in \mathbb{R}^2 : x \geq 0 \}$?
Thank you very much for your attention.
NOTE. Since each $D_i$ is a simply connected open subset of $\mathbb{R}^2$, we know that it is homeomorphic to $\mathbb{R}^2$ (see e.g. Newman, Elements of the Topology of Plane Sets of Points, Chapter VI, Theorem (6.4)).
