Talk:Compact Subspace of Linearly Ordered Space/Lemma 1

't Appears to me that the Nec and Suf have been mixed up. Suf means: to prove from the two conditions that the thing is compact. --Lord_Farin (talk) 22:31, 7 February 2013 (UTC)


 * This sort of thing is why I don't approve of mixing "iff" with "necessary and sufficient". There is apparently an understanding on PW of which way it goes by PW convention, but that is not memorable. --Dfeuer (talk) 22:38, 7 February 2013 (UTC)


 * First mentioned goes first. So, if $p \iff q$ is to be proved, $p \implies q$ is necessary condition, $q \implies p$ sufficient. I am working on an alternative, hopefully shorter proof of the necessary condition - so as to comfort you I'm not only thrashing against minor presentational things. --Lord_Farin (talk) 22:40, 7 February 2013 (UTC)


 * We read:


 * $p \implies q$ as $p$ is a sufficient condition for $q$.


 * $p \implies q$ as $q$ is a necessary condition for $p$.


 * IMHO, it is more natural to do it the other way round LF based on how we read it (or at the very least how this wiki says you should read it).


 * (I am quite fond of books that write $\implies$ direction and $\Longleftarrow$ direction but perhaps I have no taste :P) --Jshflynn (talk) 22:57, 7 February 2013 (UTC)


 * Convinced now that there may arise ambiguity. I've been prone to this confusion myself. A solution should be crafted. I also think to have an alternative proof. I'll post it to my sandbox. --Lord_Farin (talk) 23:01, 7 February 2013 (UTC)


 * Proof is up. Please read it and check it for flaws. It's likely to be essentially the same argument but I think I've removed a few subproofs by contradiction. --Lord_Farin (talk) 23:19, 7 February 2013 (UTC)


 * I'm not a fan of $\implies$ direction and $\Longleftarrow$ direction because in heading titles they are unwieldy and ugly. We might want to use "forward direction" and "reverse direction", but I would caution against making too much fuss over this because (a) if we were to impose a change of standard now, there are lots of pages to be amended, and I'll be the one who ends up doing it and I don't want to, (b) someone somewhere will throw his teddy out of the lecture theatre because it's not how he likes it, and (c) I like "necessary condition" and "sufficient condition". --prime mover (talk) 23:25, 7 February 2013 (UTC)


 * Something could be put as an acceptable new alternative so as to not make the "problem" not worse (at least not by the ones considering it a problem). Forward and reverse direction seems a fine solution. We won't be crucified for doing that (at least, no more than we are already anyway). --Lord_Farin (talk) 23:29, 7 February 2013 (UTC)

Converse
I've been banging my head against the converse for a bit, but I haven't managed to get very far. I'm starting to think it's probably necessary to use Zorn's lemma to magick up a minimal subcover and then prove that under these conditions a minimal subcover must be finite. --Dfeuer (talk) 02:57, 8 February 2013 (UTC)


 * I think I have a proof, but as of now it seems to require $\text{AC}_{|Y|}$ (which is an improvement over Zorn nonetheless). --Lord_Farin (talk) 09:04, 8 February 2013 (UTC)


 * The problem lies in that e.g. we can't avoid choice in picking a cover element for $\inf Y$, because there may not exist a greatest element. --Lord_Farin (talk) 09:13, 8 February 2013 (UTC)