Example of a CW complex that is not a $\Delta$-complex?

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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?

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Take the unique cell complex which has one $0$-cell and one $2$-cell.