3
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This problem was motivated by the MO problems: "Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?".

Let $p$ be a positive real number. A metric space $(X,d)$ is called $\ell_p$-chain connected if for every points $x,y\in X$ and $\varepsilon>0$ there exists a sequence $C_0,\dots,C_n$ of connected subsets of $X$ of diameter $<\varepsilon$ such that $x\in C_0$, $y\in C_n$ and $\sum_{i=1}^n d(C_{i-1},C_i)^p<\varepsilon$. Here $d(A,B)=\inf\{d(a,b):a\in A,\;b\in B\}$.

Problem 1. Is each connected compact subset of a Euclidean space $\mathbb R^n$ $\ell_1$-chain connected? What is the answer for $n=2$?

Remark 1. By the method of the proof of Lemma from the answer to this MO-problem, it can be shown that each connected compact subspace of $\mathbb R^n$ is $\ell_p$-chain connected for any $p>n-1$.

Problem 3. Is each connected compact metric space $\ell_1$-chain connected? $\ell_p$-connected for some $p$?

Remark 2. Each $\ell_1$-chain connected compact metric space is almost path connected in the sense of this MO problem.


Added in Edit. It seems that the the answer of Anton Petrunin to this problem also yields a counterexample to Problem 3.

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  • $\begingroup$ Taras, I like your $\ell_p$ problems (perhaps mathoverflow.net/questions/308239/… even more than this one; the connected subsets $C_k$ can be in practice simply 1-points sets--the continua can be this weird, as you know). Along your line, I have cruder notions (less metric, more combinatorial). And an exmaple too (I hope with a good probability :)). I feel that these combinatorial notions would provide a framework for your more subtle metric direction. Would you be iinterested? (It's hm-MO hence I tread carefully :-) ). $\endgroup$ – Wlod AA Aug 16 '18 at 2:35
  • $\begingroup$ In addition to $\ell_p$-pth connectedness, you may also address a respective variation of Hahn-Mazurkiewicz theorem about continuous surjections. $\endgroup$ – Wlod AA Aug 16 '18 at 2:40
  • $\begingroup$ @WlodAA Thank you for your interest in these questions. Of course the countinua can be singletons. In this case the $\ell_p$-connectedness became related to the Hausdorff dimension. It seems that each metric contiunuum of Hausdorff dimension $<p$ is $\ell_p$-connected. $\endgroup$ – Taras Banakh Aug 16 '18 at 5:19
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    $\begingroup$ @WlodAA The solenoid is not weird enough: its path components are dense, so it is $\ell_p$-chain conneceted for every $p>0$ (with just one jump). $\endgroup$ – Taras Banakh Aug 16 '18 at 6:37
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    $\begingroup$ @WlodAA Concerning the question of 40+ old, it is immediately related to the Hausdorff dimension (more precisely, if it is $\le 2$). $\endgroup$ – Taras Banakh Aug 16 '18 at 6:39

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