This seems like something that is probably obvious to geometric topologists but I'm not so familiar with the theory of triangulation. Internet searches have not proved as fruitful as I had imagined. Open subsets of $\mathbb{R}^n$ are really ``nice" manifolds so surely the answer to the following question is yes, but I'd like to see a procedure for how to do it without appealing to much heavier machinery.
Q: Can all connected open subsets of $\mathbb{R}^n$ be triangulated? Can they be PL-triangulated?
I know that arbitrary 2-manifolds and 3-manifolds can be triangulated (implying a positive answer to my question in these dimensions) but in higher dimensions things get more subtle. I'm hoping to hear that I can PL-triangulate any connected open subset $M$ of $\mathbb{R}^n$, $n\geq 2$ so that as points in $M$ approach $\partial M$, the diameter of the simplices containing these points tends to $0$.
A reference, specific to this fundamental case would be ideal.
Yes, every open subset $\Omega$ of $\mathbb{R}^n$ can be triangulated. Indeed, it is always possible to subdivide $\Omega$ into closed cubes with disjoint interiors and where each cube intersects at most finitely other cubes. Then everything drops down to triangulate cubes.
On can take a look at the Wikipedia page on the Whitney covering lemma and to the references therein:
https://en.wikipedia.org/wiki/Whitney_covering_lemma