Timeline for Is there a "universal" connected compact metric space?
Current License: CC BY-SA 3.0
13 events
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| Nov 26, 2017 at 17:58 | comment | added | John Baez | Okay, good - I see this encyclopedia defines snake-like to mean the same thing as what I'd call chainable. The pseudo-arc is chainable but also what Wikipedia calls "crooked", and I mistakenly assumed snake-like was a synonym for crooked. | |
| Nov 24, 2017 at 18:43 | comment | added | Jeremy Brazas | "chainable," "linearly chainable," and "snake-like" are often used to mean the same thing. | |
| Nov 20, 2017 at 17:45 | comment | added | John Baez | At least, so say Irwin and Solewecki. But looking at Mioduszewski's paper linked to above, I see he claims that pseudo-arc maps onto every 'snake-like' continuum. He doesn't define this concept, but elsewhere I read that a continuum is snake-like if it's chainable and has an extra property too. So, I don't get why Irwin and Solewecki claim what they do. | |
| Nov 20, 2017 at 17:16 | comment | added | John Baez | Another versal property: the pseudo-arc maps onto every chainable continuum, meaning one for which every open cover refines to a cover $U_1, \dots , U_n$ where $U_i \cap U_j \ne \emptyset$ iff $|i - j| \le 1$. This was proved by Mioduszewski in 1962. | |
| Nov 20, 2017 at 8:54 | comment | added | John Baez | Yes, we can say "metrizable". | |
| Nov 12, 2017 at 22:40 | answer | added | D.S. Lipham | timeline score: 9 | |
| Nov 12, 2017 at 17:53 | comment | added | Harry Richman | This is nit-picky, but wouldn't it be more appropriate to call these universal metrizable spaces, rather than metric spaces? Since we are asking for existence of continuous maps, not isometries, and as a result there is no preferred metric to choose on $K$ or $I$, just any that is compatible with the topology | |
| Nov 12, 2017 at 7:11 | history | edited | John Baez | CC BY-SA 3.0 |
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| Nov 12, 2017 at 6:46 | history | edited | Martin Sleziak |
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| Nov 12, 2017 at 6:45 | history | edited | John Baez | CC BY-SA 3.0 |
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| Nov 12, 2017 at 4:29 | answer | added | Jeremy Brazas | timeline score: 37 | |
| Nov 12, 2017 at 3:27 | history | edited | John Baez | CC BY-SA 3.0 |
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| Nov 12, 2017 at 3:08 | history | asked | John Baez | CC BY-SA 3.0 |