Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of diameter $<\varepsilon$ in $X$ such that $x\in C_1$, $y\in C_n$ and $C_i\cap C_{i+1}\ne\emptyset$ for all $i<n$. It is clear that $\sim_\varepsilon$ is an equivalence relation on $X$. The equivalence class $[x]_\varepsilon:=\{y\in X:y\sim_\varepsilon x\}$ will be called the *$\varepsilon$-connected component* of $x$.

**Problem.** What can be said about the $\varepsilon$-connected components of $X$? In particular, is each $\varepsilon$-connected component $[x]_\varepsilon$ dense in $X$? Is each $\varepsilon$-connected component $\sigma$-compact?

**Added in Edit.** It can be shown that each $\varepsilon$-connected component $[x]_\varepsilon$ is $\sigma$-compact. To prove this fact, fix a countable base $\mathcal B$ of the topology of $X$, which is closed under finite unions and let $\mathcal B_\varepsilon$ be the subfamily of $\mathcal B$ consisting of basic sets of diameger $<\varepsilon$. It can be shown that each connected subset of diameter $<\varepsilon$ is contained in the connected component of some set $\bar B$ with $B\in\mathcal B_\varepsilon$. Now fix an enumeration $\{B_n\}_{n\in\omega}$ if the family $\mathcal B_\varepsilon$. Given any point $x\in X$ let $C_0=\{x\}$ and for every $n\in\mathbb N$ let $C_n$ be the union of $C_{n-1}$ with all connected components of $\bar B_n$ that intersect $C_{n-1}$. It can be shown that the sets $C_n$ are compact and $[x]_\varepsilon=\bigcup_{n\in\omega}C_n$.