0-cells in CW complexes

If X is a CW complex, then for each fixed point x, is it possible adapt the cellular decomposution of X such that x be a 0-cell?

Actulally, my real interest: is any point in X nondegenerated?

• The answer to you first question is yes and a proof can be found in every book treating CW complexes (you just need to prove it for the interior points of $D^n$, where it is obvious). What is a nondegenerate point? – Denis Nardin Jul 26 '17 at 13:04
• A point x of X called nondegenerated if the inclusion x in X is a (Hurewicz) cofibrarion, that is, have the homotopy extension property. The first question implies the second one, thank you. – Izael do Nascimento Jul 26 '17 at 13:08