# Is this conjecture about the boundary of a surface correct?

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 $$\bf f$$, $$\bf g$$.