Timeline for Unbounded countable subset
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2016 at 3:03 | review | Close votes | |||
| Jun 15, 2016 at 6:58 | |||||
| Jun 8, 2016 at 4:14 | review | Close votes | |||
| Jun 8, 2016 at 9:03 | |||||
| Jun 7, 2016 at 20:20 | comment | added | thejoshwolfe | Is the OP missing a claim about the size of the set? Isn't the empty set a counterexample for the first sentence? | |
| Dec 7, 2009 at 20:15 | vote | accept | Sune Jakobsen | ||
| Nov 27, 2009 at 19:57 | comment | added | Alicia Garcia-Raboso | After Sune's explanation I am happy to withdraw my downvote. | |
| Nov 27, 2009 at 16:37 | answer | added | Pete L. Clark | timeline score: 4 | |
| Nov 27, 2009 at 16:21 | comment | added | Sam Nead | I think that your generalization still holds. Perhaps transfinite induction will be useful? | |
| Nov 27, 2009 at 7:04 | comment | added | Sune Jakobsen | I haven't solved it. But if the answer to my question was yes, you could find a countable subset $J\subset I$, such that for every $i\in I$ there is a $j\in J: K_j\subset K_i$. If there is a j that works for every i, the intersection would be $K_j$ and thus non-empty. Otherwise you could find a decresing sequence $K_1\supset K_2\supset \dots$ and reduce the problem to the homework problem. | |
| Nov 27, 2009 at 0:40 | comment | added | Sam Nead | Well, I apologize in all directions, in that case. <p>Sune: would you like to explain the resolution of your refined intersection problem? | |
| Nov 27, 2009 at 0:16 | comment | added | David E Speyer | For the record, I remember inventing and thinking about this question when I was first learning set theory. My motivation went as follows: An equivalent formulation of Zorn's lemma is "In a nonempty poset where every totally ordered subset has an upper bound, there is a maximal element." At the time I found it hard to think about arbitrary totally ordered sets, so I wondered if I could replace this by "In a nonempty poset where every ascending sequence $(a_i)_{i \in Z}$ has an upper bound, there is a maximal element." | |
| Nov 26, 2009 at 18:26 | comment | added | Sune Jakobsen | It was not a homework problem, but it was inspired by a homework problem. The problem was: Show that if $K_1\supset K_2\supset \dots$ is a decreasing sequence of non-empty compact sets, the intersection $\cap_{i=1}^{\infty} K_i$ is non-empty. I was wondering if this could be generalized: Let $(K_i)_{i\in I}$ be a system of non-empty compact sets, such that for for all $i,j\in I: K_i\subset K_j \vee K_i\supset K_j$. Is the intersection $\cap_{i\in I} K_i$ non-empty? This is why thought of the problem, but I got interested in the problem for its own rights. | |
| Nov 26, 2009 at 17:54 | comment | added | Sam Nead | David - it is presented as a straight problem. Nothing about "I need this for..." Or "I was reading X and thought...". No motivation... However, I will be happy to be corrected by M. Jakobsen. | |
| Nov 26, 2009 at 17:47 | comment | added | David E Speyer | I am baffled as to why people think this is a homework problem. It could be assigned in a set theory class, but it is a very natural question and the counter-examples are not elementary. I'll bow to peer pressure and not give an explicit construction, but the basic hint here is to read up on ordinal numbers. | |
| Nov 26, 2009 at 17:46 | history | edited | Sune Jakobsen | CC BY-SA 2.5 |
added 83 characters in body; added 1 characters in body
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| Nov 26, 2009 at 17:24 | comment | added | Jose Brox | You should require $a>s$ (strictly greater than) for some $s$, otherwise you could have S bounded and $a$ a maximum. | |
| Nov 26, 2009 at 17:22 | vote | accept | Sune Jakobsen | ||
| Dec 7, 2009 at 20:15 | |||||
| Nov 26, 2009 at 17:13 | answer | added | Kristal Cantwell | timeline score: 4 | |
| Nov 26, 2009 at 17:06 | comment | added | Alicia Garcia-Raboso | I agree, this is a homework problem: -1. | |
| Nov 26, 2009 at 16:36 | comment | added | Sam Nead | No. This is a homework problem. | |
| Nov 26, 2009 at 16:31 | history | asked | Sune Jakobsen | CC BY-SA 2.5 |