# Equivalent condition for deformation retraction

I am currently reading a paper where the author needs to show that a map $$i: X \hookrightarrow Y$$ between 2 $$CW$$- complexes $$X$$,$$Y$$ (and $$X$$ is contained in $$Y$$) is a deformation retraction.

He claims it suffice to show that for every $$\psi : D^n \hookrightarrow Y$$ such that $$\psi(\partial D) \subset X$$, we can produce a deformation retraction of $$\psi(D)$$ to a disk contained in $$X$$ while fixing $$\psi(\partial D)$$.

Why is this equivalent to showing that there is a deformation retraction from $$Y$$ to $$X$$ ?

• Is that all the paper assumes about the map $i : X \hookrightarrow Y$? There are no further conditions on $i$? For example, I presume $i$ should be continuous, but there need to be more conditions as well or else this statement is false. – Lee Mosher Jan 30 at 12:44
• $i$ is the inclusion map which is continuous. But other than that there are no other additional conditions. – cr1t1cal Jan 30 at 14:23
• Also $\psi$ is continuous as well. These are the maps that give Y it’s CW complex structure. – cr1t1cal Jan 30 at 14:32
• Do you mean deformation retraction or strong deformation retraction? – Paul Frost Jan 30 at 17:33