239 We will need some wiggle room, so for each $\alpha$ choose a large finite number of open sets

   239 We will need some wiggle room, so for each $\alpha$ choose $2N$ additional open sets

   240 \[

   240 \[

   241 	U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset U_\alpha^2 \supset \cdots

   241 	U_\alpha = U_\alpha^0 \supset U_\alpha^\frac12 \supset U_\alpha^1 \supset U_\alpha^\frac32 \supset \cdots \supset U_\alpha^N

   242 \]

   242 \]

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

   243 so that for each fixed $i$ the set $\cU^i = \{U_\alpha^i\}$ is an open cover of $X$, and also so that

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

   244 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-\frac12} \supset \ol{U_\alpha^i}$.

   245 \nn{say specifically how many we need?}

   245 %\nn{say specifically how many we need?}

   246 

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

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

   256 

   255 So $\bd P = \sum Q_\beta$, and the support of $f$ restricted to $Q_\beta$ is $V_\beta^N$, the union of $k-1$ of

   256 the $U_\alpha^N$'s.  Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$

   257 of the $U_\alpha^i$'s.

   258 

   259 Define

   260 \[

   261 	W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i .

   262 \]

   263 

   264 We will construct a sequence of maps $f_i : P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties:

   265 \begin{itemize}

   266 \item[(A)] $f_0 = f$;

   267 \item[(B)] $f_i = g$ on $W_i^i$;

   268 \item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and

   269 \item[(D)] there is a homotopy $F_i : P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the 

   270 support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$.

   271 \nn{check this when done writing}

   272 \end{itemize}

   273 

   274 Once we have the $F_i$'s as in (D), we can finish the argument as follows.

   275 Assemble the $F_i$'s into a map $F: P\times [0,N] \to \Homeo(X)$.

   276 View $F$ as a homotopy rel boundary from $F$ restricted to $P\times\{0\}$ (which is just $f$ by (A))

   277 to $F$ restricted to $\bd P \times [0,N] \cup P\times\{N\}$.

   278 $F$ restricted to $\bd P \times [0,N]$ is adapted to $\cU$ by (D).

   279 $F$ restricted to $P\times\{N\}$ is constant on $W_N^N = X$ by (B), and therefore is, {\it a fortiori}, also adapted

   280 to $\cU$.

   281 

   282 \nn{resume revising here}