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Let be $\gamma = \{C_1,\ldots, C_k\}$ a partition of a compact metric space $X$ such that $diam(C_j)<\delta$ for all $j$. Suppose that there exist compact sets $L_i\subset C_i$ for all $i\in\{1,\ldots,k\}$ and pairwise disjoint. I want to prove that there exist $\gamma' = \{C_1',\ldots, C_k'\}$ with $diam(C_j')<\delta$ and $L_j\subset int(C_j')$ for all $j$. How do i do? Does $\gamma'$ is a partition? I need a new partition satisfying the above conditions.

Thanks.

Teo Viddoto
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1 Answers1

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Let $\varepsilon:=\min_i \,(\delta-{\rm diam}(C_i))\ >0$. Then set $$C_i':=\{x\,\mid\,d(x,C_i)<\varepsilon/2\}\,. $$ We still have ${\rm diam}(C_i') < {\rm diam}(C_i)+\varepsilon\le \delta$, and obviously ${\rm int}(C_i')\supseteq C_i\supseteq L_i$.

Berci
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