Let $B$ be a bounded subset of $\mathbb{R}^n$ and $X$ be a finite union of simply connected subsets of $B$ which are pairwise disjoint. Let $X=\bigcup X_i$. Let $\omega$ be a path in $B$ such that
- $\omega\cap X=\emptyset$
- $\omega\cap\partial B\neq\emptyset$ where $\partial B$ is the boundary of $B$
Proposition: There exists a continuous function $T:[0,1]\times[0,1]\rightarrow\mathbb{R}^n$, denoting $T_s(t):=T(s,t)$ and $T_s:=\{T(s,t)\mid t\in[0,1]\}$, such that
- $T_0=\omega$
- $T_1\cap B=\emptyset$
and for all $s\in[0,1]$
- $T_s\cap X=\emptyset$ and
- the arc length of $T_s$ is that of $\omega$
Proof: Imagine pulling the string $\omega$ out of a bag $B$ full of object(s) $X$. If it is on the surface and visible upon opening the bag, just pinch it there and pull out. It should weave in reverse around the objects and you can remove it from the bag.
I think it's false. Take as $B$ two tangent open disks, as $X$ two smaller disks also tangent and as $\omega $ the boundary of $B$.