I have been stuck for quite some time on a problem concerning the regularity of CW complexes. More precisely I am unable to prove (or disprove for that matter) that regularity is preserved under suspension. My question summarizes to seeing if the following result is true.
Lemma. Let $X$ be a regular CW complex. Then its suspension $SX$ and reduced suspension $\Sigma X$ also admit a regular CW complex structure.
For now, I have only tried proving the suspended case. My idea was to consider the natural CW complex structure of the suspension and show that the cells of this suspension have characteristic maps which are simply the suspension of the characteristic maps in the original CW complex. More explicitly, if $\sigma$ denotes a cell in $X$, I consider its induced cell (i.e. the equivalence class in the suspension of the cell $\sigma \times ]0,1[$ in $X \times I$) in the suspension, which I denote by $S \sigma$. If $\Phi:D^n \to X^{n-1}$ denotes $\sigma$'s characteristic map, then I thought of showing that the characteristic map of the cell $S \sigma$ in $SX$ is $S \Phi$. If this were the case, then I think I could be able to conclude: regularity of $X$ implies that $\Phi$ is a homeomorphism onto its image. From the functoriality of the suspension, one deduces that $S \Phi$ is a homeomorphism onto its image, which would allow one to conclude that the suspension is regular with this CW complex structure.
This argument could probably be transferred to the reduced suspension. But I don't know how to prove the key step of this argument.
I am open to any solutions and suggestions in order to prove (or disprove) this result. Any references to this are also welcome.
Thanks in advance!
PS: this is the first time I post a question on stackexchange, so I am not yet comfortable with its numerous functionalities. I therefore apologize for the bad presentation of this question.