Let $S$ be a simplex in $\mathbf R^n$ and let $\{S_i\}$ be a triangulation of $S$.
The boundary of $S$ is defined as the union of the boundary faces of $S$. Is this union equal to the topological boundary of $S$ (with respect to the Euclidean norm topology)? The answer seems to be an obvious yes but I'm having trouble proving it. Any tips or references?
It seems to be a well established fact, although I have never seen it proved anywhere, that each boundary face of a subsimplex $S_i$ of the triangulation is either contained in a boundary face of $S$ or is shared by exactly one other subsimplex $S_j$. I have been trying to prove this for a while now and it seems highly nontrivial. The proof I am constructing relies on the answer to 1 being yes. Any tips or references are welcome.