Large sets moving through a ``narrow hole'' (Q2785020)

The paper is a contribution to the problem of moving a subset of \(\mathbb{R}^3\) through a given hole in a plane. A subset \(H\) of a plane \(P_0\) in \(\mathbb{R}^3\) is a hole if both \(H\) and \(P_0 \setminus H\) are arcwise connected. Let \(S \subset \mathbb{R}^3\) and \(\varepsilon > 0\). A homeomorphism \(h: S \rightarrow h(S) \subset \mathbb{R}^3\) is an \(\varepsilon\)-perturbation if \(d(x, h(x)) < \varepsilon\) for every \(x \in S\). A set \(S\) is simple if it is a finite union of segments. The author proves that, in a given plane \(P_0\), there exists a hole \(H\) of zero area which is universal in the following sense for all simple sets: for every simple set \(S\) and every \(\varepsilon >0\) there exists an \(\varepsilon\)-perturbation \(h: S \rightarrow S^* = h(S)\) and a homotopy \((f_t : S^* \rightarrow \mathbb{R}^3 \setminus H)_{t \in [0,1]}\) such that \(S^*\) is simple, all \(f_t\) are translations, \(f_0= \text{ id}\), and the sets \(S^*\) and \(f_1(S^*)\) are separated by \(P_0\). So, as the author intuitively says, there exists a ``universal'' hole of zero area such that one can perturb a given simple set a little so that a simple set obtained as the result of the perturbation can be moved through this hole.