I came up with an intuitive conjecture about boundaries of surfaces based on the idea that at a boundary point we can wrap a string across the edge, and the two halves of the string (on opposite sides of the surface) can be brought as close as we want:

Let $S \subseteq \mathbb{R}^n$ be a surface. A point $P \in S$ is a boundary point iff there is a point $Q \in S \setminus \{P\}$ such that for all $\epsilon>0$, there are functions $\mathbf{f}, \mathbf{g} : [0,1] \to \mathbb{R}^n$ such that:

  1. $\mathbf{f}(0) = \mathbf{g}(0)$ and $||\mathbf{f}(0) - P|| < \epsilon$ (string halves touch at a point close to $P$)
  2. $\forall t \in [0,1]: ||\mathbf{f}(t)-\mathbf{g}(t)|| < \epsilon$ (string halves are close to each other)
  3. $\forall t \in [0,1]: ||\mathbf{f}(t) - P|| > \epsilon \implies \frac{\mathbf{f}(t) + \mathbf{g}(t)}{2} \in S$ (string halves are separated by surface, except near $P$)
  4. $||\mathbf{f}(1) - Q|| < \epsilon$ and $||\mathbf{g}(1) - Q|| < \epsilon$ (string ends are close to $Q$)
  5. $(\mathbf{f}([0,1]) \cup \mathbf{g}([0,1])) \cap S = \emptyset$ (string doesn't pass through the surface)

Is this conjecture correct? Does it have any useful applications? Is this a known theorem?


I presume that by "surface" you mean a 2-dimensional submanifold, in which case this conjecture is incorrect when $n \ge 4$: Your property is true for all $P \in S$, not just for boundary points.

On the other hand, I think the conjecture is true when $n=3$. I also think the conjecture would be true if, rather than assuming that $S$ is 2-dimensional, you assume instead that $S$ is of dimension $n-1$.

A version of this property can be used to prove the theorem that if $S \subset \mathbb R^3$ is a 2-dimensional subsurface-with-boundary, if $S$ is connected, and if $\partial S \ne \emptyset$, then $\mathbb R^3 - S$ is path connected. But for this application you must prove the property for any $Q$ in $S$, instead of just for some $Q$. If you look up proofs of this theorem, you'll see objects similar to your two paths $\bf f$, $\bf g$.


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