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It is possible construct a connected compact metric space $X$ and a continuous decomposition $\mathcal{G}$ of $X$ that satisfies: 1)$X/\mathcal{G}$ is locally connected. 2)If $M$ is a compact connected subset of $X$ and $M\cap G \neq \emptyset$ for some $G\in \mathcal{G}$, then $G\subset M$. 3)There exist $G\in\mathcal{G}$ and a compact connected subset $A$ of $X$ such that $A\not\subset D$ and $D\not\subset A$.

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  • $\begingroup$ Something is wrong with your question. Your condition 2) implies that $\mathcal{G}$ contains only singleton sets, contradicting 3). 1) does not enter into it. What did you really mean? What is your motivation? $\endgroup$
    – Lasse Rempe
    Commented May 4, 2016 at 14:39

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