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219 |
219 |
220 \medskip |
220 \medskip |
221 |
221 |
222 \nn{need to clean up references from the main text to the lemmas of this section} |
222 \nn{need to clean up references from the main text to the lemmas of this section} |
223 |
223 |
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224 \noop{ |
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225 |
224 \medskip |
226 \medskip |
225 |
227 |
226 \nn{do we want to keep the following?} |
228 \nn{do we want to keep the following?} |
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229 |
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230 \nn{ack! not easy to adapt (pun) this old text to continuous maps (instead of homeos, as |
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231 in the old version); just delete (\\noop) it all for now} |
227 |
232 |
228 The above lemmas remain true if we replace ``adapted" with ``strongly adapted", as defined below. |
233 The above lemmas remain true if we replace ``adapted" with ``strongly adapted", as defined below. |
229 The proof of Lemma \ref{basic_adaptation_lemma} is modified by |
234 The proof of Lemma \ref{basic_adaptation_lemma} is modified by |
230 choosing the common refinement $L$ and interpolating maps $\eta$ |
235 choosing the common refinement $L$ and interpolating maps $\eta$ |
231 slightly more carefully. |
236 slightly more carefully. |
247 \item each $f_i$ is supported on some connected $V_i \sub X$; |
252 \item each $f_i$ is supported on some connected $V_i \sub X$; |
248 \item the sets $V_i$ are mutually disjoint; |
253 \item the sets $V_i$ are mutually disjoint; |
249 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
254 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
250 where $k_i = \dim(P_i)$; and |
255 where $k_i = \dim(P_i)$; and |
251 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
256 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
252 for all $p = (p_1, \ldots, p_m)$, for some fixed $gX\to T$. |
257 for all $p = (p_1, \ldots, p_m)$, for some fixed $g:X\to T$. |
253 \end{itemize} |
258 \end{itemize} |
254 |
259 |
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260 } |
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261 % end \noop |
255 |
262 |
256 \medskip |
263 \medskip |
257 \hrule |
264 \hrule |
258 \medskip |
265 \medskip |
259 |
266 |