Is the inclusion of the wedge sum into the reduced cylinder a relative cell complex?

Question

For a CW-complex $X$ the map $$X\coprod X \hookrightarrow X \times I$$ is a relative cell complex. What I want to know is if this still holds in the pointed case. That is, if $X$ is a pointed CW-complex with non-degenerated base point and $I_+ = [0,1] \coprod *$, then is it true that $$X\vee X \hookrightarrow X \wedge I_+$$ is a relative cell complex?

Answer 1

Yes, it is a CW pair since $X \wedge I_+$ is the quotient of $X \times I $ by $\{*\} \times I$. It is easy to show that any point in a CW complex can be a 0-cell of a CW structure on it, so this quotient has a CW structure, and the image of $X \sqcup X$ under the quotient map is the subcomplex $X \vee X$.