# Talk:Order Topology on Convex Subset is Subspace Topology

Any ideas on making this clearer? I think I'm struggling more to explain what I mean at each step than to actually prove the theorem. It's a simple theorem, but it's coming out complicated. My recollection is that Munkres defines the order topology using a basis, rather than a sub-basis, which is conceptually simpler but which I suspect would make this particular proof even harder to follow. --Dfeuer (talk) 19:29, 5 February 2013 (UTC)

You will probably need to use that the order topology is self-dual, among the Duality Principle (Order Theory) on the underscores in your proof. --Lord_Farin (talk) 19:38, 5 February 2013 (UTC)
Further of interest is referring to Strict Upper Closure in Restricted Ordering. Other than that, spreading the first and second lines over more lines and adding more on the last deduction (explicitly deriving that in the last case, $y \in A \implies q \le y$) should suffice to make this a good proof. --Lord_Farin (talk) 19:43, 5 February 2013 (UTC)
As a final remark, please note that for the general ordering, the PW symbol is $\preceq$ rather than $\le$. --Lord_Farin (talk) 19:44, 5 February 2013 (UTC)
It's a total ordering. --Dfeuer (talk) 20:02, 5 February 2013 (UTC)
That's immaterial. It does not pertain exclusively to numbers, hence the convention is to use $\preceq$ rather than $\le$. --Lord_Farin (talk) 18:15, 6 February 2013 (UTC)
Numbers have nothing to do with the nature of ordering. --Dfeuer (talk) 18:58, 6 February 2013 (UTC)
May have a point there (and with your now deleted comment as well) - must have internalised the conventions regarding that, not questioning them any more. Minor point in any case. --Lord_Farin (talk) 19:12, 6 February 2013 (UTC)
I would not object strenuously to restricting use of $\le$ and $<$ to total orderings in most cases (if multiple orderings are under discussion, it's good to have multiple symbols available). --Dfeuer (talk) 19:15, 6 February 2013 (UTC)
It's convention only, but it's convention for this site specifically. It's not between partial and total orderings, it's between orderings on numbers and on general ordered sets. The reason is that when $\le$ and $<$ etc. are used they "condition" the reader into taking for granted that the entities being worked on are numbers and therefore work like numbers. Therefore the decision. Total or not, the symbol to be used here (in order to ensure site consistency, and also with the definition as given on the various pages defining orderings) needs to be $preceq$ or $\preccurlyeq$ and $\prec$ etc.
If you can find a citation which says: "$\preceq$ for partial, $\le$ for total" then we can remark on this on the ordering page, so as to cover us in case anyone complains at our convention. --prime mover (talk) 19:30, 6 February 2013 (UTC)

It's obviously your personal convention. Your claim that it conditions people to think the elements being compared are numbers would seem to apply only to people unfamiliar with the general context, who will in any case stop reading the minute they see a term like "ordered set", "lattice", "linearly ordered space", "close packed set", etc. Can you provide citations for $\le$ being only for numbers? Because I could easily find as many texts as you'd like that use them more generally. --Dfeuer (talk) 19:47, 6 February 2013 (UTC)

No I can't - as I said, it's the convention that is used on this site. I explained why. As I say, if you could find citations to say that "$\le$ is supposed to be used for total orderings" then trot them out. Just stating that you "could" is not enough. --prime mover (talk) 21:13, 6 February 2013 (UTC)
I never claimed that $\le$ is used specifically for total orderings; I only said I wouldn't object strongly to such a rule. I did say that I could trot out multiple texts that use $\le$ for things other than numbers. --Dfeuer (talk) 21:22, 6 February 2013 (UTC)
What's your point, then? Mine is that this site (for better or worse) reserves $\le$ etc. for numbers. Yours appears to be that other conventions exist and can be cited, and (because those citations differ from what's on this site) we ought to change the convention used by this site. For the usual reasons I recommend that the currently implemented convention remains. --prime mover (talk) 22:31, 6 February 2013 (UTC)
FYI, Munkres uses $\le$ for total orders and $\preceq$ for partial ones, which is probably where I got that notion. And yes, I do think it much more helpful to distinguish total orderings from partial ones than to distinguish numerical orderings from non-numerical ones. As for other authors, Steen & Seebach don't seem to use partial orderings at all, but consistently use $\le$ for total orderings. MacLane & Birkhoff use $\leqslant$ for orderings of any sort. Kelley uses $<$ when he needs one transitive relation ($\leqq$ if he's insisting on it being reflexive), $\prec$ when he needs another one, and $\ll$ if he needs a third (no, I am not suggesting we follow Kelley in not requiring an ordering to be antisymmetric). For myself, I like the free-for-all approach. For a site designed to be usable by high school students, I think Munkres's approach the best. The mental tie between $\le$ and total ordering is considerably tighter than the one between $\le$ and numbers. --Dfeuer (talk) 07:59, 7 February 2013 (UTC)
Whatever. We stick with what we have. --prime mover (talk) 09:07, 7 February 2013 (UTC)