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

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

   257 \begin{itemize}

   257 \begin{itemize}

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

   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 $g$,

   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 $p\in P$.

   260 for all $p\in P$.

   262 (and hence supported on $U_i$).

   263 (and hence supported on $U_i$).

   263 \end{itemize}

   264 \end{itemize}

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

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

   266 

   267