TITLE: Is a CW complex, homeomorphic to a regular CW complex?
QUESTION [9 upvotes]: Is a CW complex, homeomorphic to a regular CW complex ?

Regular means that the attaching maps are homeomorphisms (1-1).  In particular, an open (resp. closed) $n$-cell, is homeomorphic to an open (resp. closed) $n$-ball.

REPLY [8 votes]: Something that you could employ in answering this is to note that every regular CW-complex is in fact triangulable.
It turns out that every closed topological manifold of dimension other than four is homeomorphic to a CW-complex [Kirby-Siebenmann, On the triangulation of manifolds and the Hauptvermutung]. On the other hand Ciprian Manolescu has recently proved an important result that there exist non-triangulable manifolds in any dimension above four (before this work, the only known examples of non-triangulable manifolds were of dimension four).
Ok, the above employs some extremely high-powered (and modern) machinery, but the idea take a non-triangulable manifold which is a CW-complex is pretty simple, at least! And I guess this also shows that the space can still be "nice", in a particular sense.