238 such that $g\circ f(P)$ is a small neighborhood of the 

   238 such that $g^{-1}\circ f(P)$ is a small neighborhood of the 

   243 We assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.

   243 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$.

   244 

   244 

   245 

   245 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite.

   246 

   246 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$.

   247 \nn{...}

   247 

   248 

   248 We will need some wiggle room, so for each $\alpha$ choose open sets

   249 \[

   250 	U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset \cdots \supset U_\alpha^N

   251 \]

   252 so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that

   253 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$.

   254 

   255 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us

   256 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties:

   257 \begin{itemize}

   258 \item $h(p, 0) = f(p)$ for all $p\in P$.

   259 \item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$,

   260 for all $p\in P$.

   261 \item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on $U_i^{i-1}$

   262 (and hence supported on $U_i$).

   263 \end{itemize}

   264 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small,

   265 and the subdivision mentioned above is chosen fine enough to insure this.

   266 

   267 By reparametrizing, we can view $h$ as a homotopy (rel boundary) from $h(\cdot,0) = f: P\to\Homeo(X)$

   268 to the family

   269 \[

   270 	h(\bd(P\times [0,N]) \setmin P\times \{0\}) = h(\bd P \times [0,N]) \cup h(P \times \{N\}) .

   271 \]

   272 We claim that the latter family of homeomorphisms is adapted to $\cU$.

   273 By the second bullet above, $h(P\times \{N\})$ is the constant family $g$, and is therefore supported on the empty set.

   274 Via an inductive assumption and a preliminary homotopy, we have arranged above that $f(\bd P)$ is

   275 adapted to $\cU$, which means that $\bd P = \cup_j Q_j$ with $f(Q_j)$ supported on the union of $k-1$

   276 of the $U_\alpha$'s for each $j$.

   277 It follows (using the third bullet above) that $h(Q_j \times [i-1,i])$ is supported on the union of $k$ 

   278 of the $U_\alpha$'s, specifically, the support of $f(Q_j)$ union $U_i$.