Timeline for Subspaces of metric spaces having prescribed dimension
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 10, 2019 at 6:25 | vote | accept | Nicola Arcozzi | ||
| Jun 9, 2019 at 19:19 | comment | added | Andrea Marino | You are right. What about increasing sequences, in the sense of $\gamma(t) \subset \gamma(s)$ for $t<s$, and limita from the left? I am trying to thinking to limsup of compact spaces. But probably you are right, no hope. | |
| Jun 9, 2019 at 2:54 | comment | added | Nate Eldredge | @AndreaMarino: We'd need to look at the Hausdorff dimension, right? And that is certainly not continuous with respect to Hausdorff distance. Consider $X = [0,1]$ and $\gamma(t) = [0,t]$, so $\gamma(t)$ has Hausdorff dimension $1$ for every $t>0$ but dimension zero for $t=0$. | |
| Jun 9, 2019 at 2:45 | answer | added | Yuval Peres | timeline score: 2 | |
| Jun 7, 2019 at 9:56 | comment | added | Nicola Arcozzi | Hi Pietro. Skeeve: the question is about general metric spaces. Locally compact and complete could be a reasonable restriction. Euclidean space is restricting too much. Andrea: thanks for the hint. | |
| Jun 7, 2019 at 7:28 | comment | added | Skeeve | For $X=\mathbb R^n$ the answer is positive, see e.g. this question. | |
| Jun 6, 2019 at 18:55 | comment | added | Pietro Majer | Welcome Nicola, nice to see you | |
| Jun 6, 2019 at 15:32 | comment | added | Andrea Marino | Thinking about a fancy way that I don't know if it could work. Take a continous $\gamma$ from [0,1] to the space of compact subsets of $X$ with the hausdorff distance. 1. Is the hausdorff measure of $\gamma(t)$ a continous function of $t$? 2. Is there a compact subset K of X such that it ha s the same hausdorff measure? 3. Is $K$ connected to a point in the space of compact subsets? Of course, this would yield the answer, but I have no idea if it is effective.. | |
| Jun 6, 2019 at 14:55 | review | First posts | |||
| Jun 6, 2019 at 16:13 | |||||
| Jun 6, 2019 at 14:52 | history | asked | Nicola Arcozzi | CC BY-SA 4.0 |