I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states:
$\eta_t(u)$ is a homeomorphism for each $t\in [0,1]$.
I could see how if we restricted $\eta_t$ to $\overline{X\setminus A_2}$ then it would be a homeomorphism of $X$ because $\eta_t(u)=u$ for all $u\in \overline{X\setminus A_2}$. How about $A_2$?
There seems to be probably a thought failure it says "$\eta(t,\cdot)$ is a homeomorphism for every $t\in [0,1]$". This is crucial, beginning with the fact that $(4.2)$ is not stated for $t$ but $u$ and so forth.
Further the following error in the text may have brought confusion: $\psi$ shall be defined for $A_2$ and not $A_1$. Then the next equation for the corresponding case follows for $\phi$ on $A_2$ followed by the next equation (Cauchy) where follows for $\dot{w}$ the boundary $t\geq 0$. Herefrom the solution for $w$ that is defined on $[0,\infty[$.
Hence in your argumentation you are then right but keep in mind that only $\phi\equiv0$ not else. So due my understanding there is no other case.