TITLE: Boundary triangulation induces triangulation
QUESTION [8 upvotes]: In $R^n$ (the real space) we have an open connected set $D$, such that $\partial D$ is triangulable. Can we prove the closure $\bar{D}$ is triangulable or any counterexample?
Furthermore, the $\partial D$ are piecewise algebraic in the question I am considering, I do not know whether this would be helpful for the above statement.
Thanks for any help.

REPLY [4 votes]: In many categories, the answer is known to be yes, see
Emil Saucan, MR 2184196 Note on a theorem of Munkres, Mediterr. J. Math. 2 (2005), no. 2, 215--229.