newer things

That sounds like a question we should know how to answer. I have a domain $V$ and a diffeotopy $h_t : V \to V$, $t \in [0, 1]$. Call $U_n = h_{1/n}(V)$. Assume $d(U_n, V) < 1/n$. Say $f$ is a function on $V$ and $f_n$ are functions on $U_n$ such that $\|f_n - f|_{U_n}\|_{C^0} < 1/n$. Can I say $\int_{U_n} f_n \to \int_V f$?

So, just to write it out for myself, I need to prove that the homotopy fiber of the inclusion map $i : Y \to CX \cup_f Y$ is $X$, and the inclusion of the fiber $X$ in $E_i \cong Y$ is homotopy equivalent to $f : X \to Y$.