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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 $p\in P$. |
261 \item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on $U_i^{i-1}$ |
261 \item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on |
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262 $U_i^{i-1} \setmin (U_0^i \cup \cdots \cup U_{i-1}^i)$ |
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 |