@Tyrone I don't understand your argument.

I was not (yet) aware. The euclidian n-cube is the "superposition" of all globular realizations of the n-cube in the sense that the execution paths in the euclidian order are the union of all possible execution paths coming from globes. And I don't know what to do with this geometric observation. The euclidian n-cubes are not cofibrant for the "globular" model structure, that I call the q-model structure now, because they contain too many execution paths to be q-cofibrant.

@DavidRoberts You are absolutely right ! I had not realized that either. I'm going to modify my question.

@DavidRoberts In fact in my situation, the continuous bijection is also a homotopy equivalence. I realized that by reading Gabriel's answer.

@DavidRoberts Yes a half-open interval. Morally speaking, the idea is to dig a hole by adding open subsets. Anyway, that cannot happen in my situation so this answer is very helpful for me.