Timeline for Is every locally connected subset of Euclidean space R^n locally path connected ?
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| Oct 7, 2020 at 10:48 | comment | added | Arnaud Chéritat | Old topic, but let me point out that C is compact hence cannot be locally connected without being path connected according to Vít Tuček's answer. In fact, the example C (which is nice once you understand the description) is not locally connected. It comes from the part of the first few $S_z$ that is close to $y\in$ the y-axis. | |
| Oct 30, 2011 at 21:40 | comment | added | Sergey Melikhov | OK, now I see. You probably want a bit more than that, because you don't want $D$ to be concentrated, say, only in increasing intervals of the graph and miss the decreasing ones. But this is easy to arrange. So the problem finally makes sense to me! I wonder if your technique applies also to mathoverflow.net/questions/25171 ? | |
| Oct 30, 2011 at 19:09 | comment | added | Jerzy Dydak | Sorry for not responding earlier. The problem is that I want the boundary of $D$ to be the edge of $0\times [-1,1]$ of the sine curve, not to be dense in all of it. | |
| Oct 30, 2011 at 18:59 | history | edited | Jerzy Dydak | CC BY-SA 3.0 |
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| Oct 22, 2011 at 20:06 | comment | added | Sergey Melikhov | There is an interesting example here ams.org/journals/bull/1942-48-02/S0002-9904-1942-07615-4 (see Property 7). | |
| Dec 5, 2010 at 23:38 | comment | added | Sergey Melikhov | Jurek, your example almost made me think I understand the problem, but... I don't think $C$ is locally connected at all points of the graph $G\subset S$ of $\sin\frac1x$. Let $p\in G$, and let $p_n$ be a sequence of distinct points in $D$ converging to $p$. Then $S_{p_n}$ Hasdorff-converge to a scaled copy $h(S)$ of $S$ lying in the horizontal plane. Its intersection with $S$ surely contains a $q\in G$ that is far from $p$ and from the $y$ axis. Now $q=h(r)$ for some $r\in G$, and the images $r_n$ of $r$ in $S_{p_n}$ converge to $q$. Any small connected sets containing almost all of the $r_n$? | |
| Nov 22, 2010 at 21:40 | history | edited | Jerzy Dydak | CC BY-SA 2.5 |
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| Nov 22, 2010 at 21:29 | history | edited | Jerzy Dydak | CC BY-SA 2.5 |
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| Nov 22, 2010 at 21:15 | history | edited | Jerzy Dydak | CC BY-SA 2.5 |
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| Nov 22, 2010 at 20:14 | comment | added | George Lowther | I assume that the idea is to scale and translate S to join z to the y-axis, then bend it a bit in the third dimension to avoid intersections. I don't know why this results in a "clearly locally connected" set though. | |
| Nov 22, 2010 at 19:00 | comment | added | Nate Eldredge | Also, what do you mean by a copy $S_z$ of $S$ in $R^3$? | |
| Nov 22, 2010 at 18:59 | history | edited | Nate Eldredge | CC BY-SA 2.5 |
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| Nov 22, 2010 at 18:56 | comment | added | Nate Eldredge | It's hard for me to see how you keep $S_z$ and $S_w$ from intersecting outside the $y$-axis. If it is convenient to include a picture, that would be a big help... | |
| Nov 22, 2010 at 18:20 | history | answered | Jerzy Dydak | CC BY-SA 2.5 |