A plane domain homeomorphic to the plane or to the disc is contractible

Both $\mathbb{C}\cong {\mathbb{R}}^2$ and the unit disc $\mathbb{D}$ are convex subsets of ${\mathbb{R}}^2$, so [L1] makes $E$ contractible. In particular, the identity map ${\mathrm{id}}_E$ is homotopic to a constant map $c_e$ for some $e\in E$.

Postcompose the homotopy from step 1.1 by $h^{-1}$ and precompose it by $h$. By [L2], this yields a homotopy from [step 1.1, L2, algebra] $h^{-1}\circ {\mathrm{id}}_E\circ h={\mathrm{id}}_{\mathrm{\Omega }}$ to the constant map at $h^{-1}(e)$. Thus ${\mathrm{id}}_{\mathrm{\Omega }}$ is nullhomotopic.

Let $f:\mathrm{\Omega }\to Y$ be any continuous map into any topological space $Y$. Postcomposing the nullhomotopy from step 2.1 by $f$ and using [L2] again shows that [step 2.1, L2, L3] $f=f\circ {\mathrm{id}}_{\mathrm{\Omega }}$ is homotopic to the constant map at $f(h^{-1}(e))$. Hence every continuous map out of $\mathrm{\Omega }$ is nullhomotopic, so [L3] makes $\mathrm{\Omega }$ contractible. ∎