Let $C$ be the closed unit cube in $\mathbb R^3$, and let $A$ be one face of the cube $C$ (say the face above and parallel to $xy$-plane). Let $U\subset\mathbb R^2$ be open and path-connected such that $U\subseteq A$.
I claim that the space $X:=(C-A)\cup U$ is simply connected, by dragging any loop in $X$ into $int( C)$ which is convex subset of $\mathbb R^3$ i.e contactable.
My Question is: How to prove that formally? Can you find a retraction (homotopy map) which shows that $X$ is homotopy equivalent to $C-A$ (or equivalently that $X$ is contactable)?