Timeline for Connecting a compact subset by a simple curve
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 20, 2021 at 20:35 | comment | added | Robert Frost | It's likely I completely misuderstand this question and I certainly don't understand all of the terms and intricacies but it appears a circle can be constantly deformed into Sierpinski's triangle - in the unlikely event that observation is of help! | |
| Jan 13, 2021 at 16:55 | answer | added | Moishe Kohan | timeline score: 2 | |
| Jan 12, 2021 at 1:08 | vote | accept | Pietro Majer | ||
| Jan 10, 2021 at 22:49 | comment | added | YCor | Maybe the case of $K$ totally disconnected would make a good follow-up question. | |
| Jan 10, 2021 at 19:55 | history | became hot network question | |||
| Jan 10, 2021 at 17:37 | comment | added | erz | related (but obviously different) mathoverflow.net/questions/359390/… | |
| Jan 10, 2021 at 15:25 | comment | added | Pietro Majer | Then we proceed filling the gaps and connecting the components of $N_\epsilon(K)$ for $\epsilon=1/2,1/4,1/8\dots.$ Since $K$ is totally disconnect, the diameters of the components tend to $0$, which should prevent the lost of continuity and injectivity as would happen in your counterexample. | |
| Jan 10, 2021 at 15:25 | comment | added | Pietro Majer | I don't know the answer in the case of a totally disconnected $K$, but I would believe it is affirmative, by a construction that mimics the curve described above in the OP for the squared Cantor set. We may define a curve iteratively this way. First we join the finitely many connected components (if more than one) of the uniform neighborhood $N_\epsilon(K):=\{x: \rm{dist}(x,K)\le \epsilon\}$, where $\epsilon=1$, parametrizing the arcs on disjoint closed intervals of $[0,1]$. | |
| Jan 10, 2021 at 14:45 | comment | added | YCor | Do you know the answer when $K$ is totally disconnected? Then for $n=2$ the answer is yes (some self homeomorphism of the plane maps $K$ into a fixed line). However there are wild Cantor subsets in higher dimension. | |
| Jan 10, 2021 at 12:43 | answer | added | YCor | timeline score: 11 | |
| Jan 10, 2021 at 12:27 | history | edited | YCor | CC BY-SA 4.0 |
added obvious implicit assumption (false for n<2)
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| Jan 10, 2021 at 12:06 | comment | added | Moishe Kohan | In the case of $n=2$ one possible approach is to show that there exists a totally disconnected compact subset $C$ of $K$ intersecting every component of $K$. Such $C$ is necessarily contained in a Jordan curve since there is a self-homeomorphism of $R^2$ sending $C$ to a subset of the x-axis. | |
| Jan 10, 2021 at 11:54 | history | asked | Pietro Majer | CC BY-SA 4.0 |