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.



Boundary triangulation induces triangulation
In many categories, the answer is known to be yes, [see](https://arxiv.org/pdf/math/0403055.pdf)


*Emil Saucan*, MR 2184196 [**Note on a theorem of Munkres**](http://dx.doi.org/10.1007/s00009-005-0040-z), *Mediterr. J. Math.* **2** (2005), no. 2, 215--229.


