TITLE: Example of a CW complex that is not a $\Delta$-complex?
QUESTION [6 upvotes]: Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $\Delta$-complexes can be realized as CW complexes with some added restrictions on the characteristic maps. Is there a simple example of a CW complex that is not a $\Delta$-complex?

REPLY [6 votes]: Take the unique cell complex which has one $0$-cell and one $2$-cell.