In https://www.maths.ed.ac.uk/~v1ranick/papers/doylemoran.pdf, there is a short proof that compact 2 manifolds can be triangulated.
In it, they list two facts that I never heard before but they claim to be familiar. It's copied below.
The first one is intuitive, but since we're not assuming smooth structures we may not have a tubular neighborhood. Would someone have a proof for it?
The second one I think is straight-forward, but the way he writes makes me second-guess if I understand the statement right. A cellular set seems to be just any 2-cell, or a point. Am I missing something? Why this whole gadget of infinite intersections?
The definition of 2-cell is any topological space that is homeomorphic to the closed unit disk.