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1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
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3 \section{Adapting families of maps to open covers} \label{sec:localising} |
3 \section{Adapting families of maps to open covers} \label{sec:localising} |
4 |
4 |
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5 In this appendix we prove some results about adapting families of maps to open covers. |
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6 These results are used in Lemma \ref{small-top-blobs} and Theorem \ref{thm:map-recon}. |
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6 Let $X$ and $T$ be topological spaces, with $X$ compact. |
8 Let $X$ and $T$ be topological spaces, with $X$ compact. |
7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
9 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
8 unity $\{r_\alpha\}$. |
10 unity $\{r_\alpha\}$. |
9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
11 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |