Timeline for How complicated can the path component of a compact metric space be?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jun 3, 2021 at 19:55 | vote | accept | Jeremy Brazas | ||
| Jun 2, 2021 at 13:49 | history | edited | LSpice | CC BY-SA 4.0 |
x is not -> X is not
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| Jun 2, 2021 at 13:35 | answer | added | Benjamin Vejnar | timeline score: 4 | |
| Jun 2, 2021 at 11:53 | comment | added | Benjamin Vejnar | A path component of a compact metric space is in general an analytic set (=continuous image of a Polish space), but it need not to be a Borel set (see a paper by Becker, 1998: The number of path-components of a compact subset of $\mathbb R^n$, Corollary 4.2). For path components of compact subsets of $\mathbb R^2$, something more can be said. | |
| May 31, 2021 at 17:42 | comment | added | erz | I suspect that in general the answer is negative: cannot imagine a compact space having a path component $\mathbb{R}^2\backslash \mathbb{Q}^2$... | |
| May 31, 2021 at 13:19 | comment | added | Jeremy Brazas | @erz I like this construction. In fact, you can use the one-point compactification if $X$ is locally compact and separable. This does seem to give a partial answer: every path-connected, locally compact, separable metric space is the path component of some compact metric space. | |
| May 30, 2021 at 6:26 | comment | added | Alessandro Codenotti | @erz yes, embed it into the Hilbert cube and take closure | |
| May 29, 2021 at 22:50 | comment | added | erz | does every separable metric space even have a metrizable compactification? I am not sure if this is useful, but: if your space $X$ is locally compact and has a metrizable compactification $Y$, then $X$ is open in $Y$, and the closure in $Y\times [-1,1]$ of the graph of $\sin (\frac{1}{d(x, Y\backslash X)})$, $x\in X$ is what you need (it seems). | |
| May 29, 2021 at 19:26 | comment | added | Jeremy Brazas | Ok, these things have been added to the question in case it was confusing. Thank you. | |
| May 29, 2021 at 19:18 | history | edited | Jeremy Brazas | CC BY-SA 4.0 |
added 31 characters in body
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| May 29, 2021 at 19:11 | comment | added | Wojowu | Surely you need to assume that your space is path connected in order to hope for it to be a path component. | |
| May 29, 2021 at 19:07 | comment | added | Moishe Kohan | When you say "Is every separable..." you probably mean "Is every separable...homeomorphic to a path component..." | |
| May 29, 2021 at 19:01 | history | asked | Jeremy Brazas | CC BY-SA 4.0 |