Definition talk:Order Topology/Definition 1

Yep. Brief. I intend on three definitions: this very brief one, something like the original one (possibly stripping out the $\Uparrow$ and $\Downarrow$ notation which is introduced here for no obvious reason), and one as the topology generated by the basis consisting of open rays and open intervals, which is the one Munkres uses and is the one that relates most directly to the usual order-based definition of the usual topology on the reals. --Dfeuer (talk) 20:10, 25 February 2013 (UTC)

Please do away with the arrows indeed; I introduced them for no apparent reason. As an aside, weren't you refactoring order theory things? Things can get chaotic (at least for other users) if you hop around so often. — Lord_Farin (talk) 20:14, 25 February 2013 (UTC)

Yes, I was. It just gets a bit dull to do just that. Unfortunately, what I hopped to was the usual topology on the reals, which got me into the godawful mess of our product space definition pages. Yes, I know it's a judgement call to call it a mess, but:

• When introducing finite (non-binary) product topologies, basing them on a definition of (potentially infinite) box products is turning things upside down, didactically.
• The box topology is typically introduced in intro texts because it is what a student's intuition might expect the definition of the product topology to be. Texts then explain why the product topology is not defined this way. I imagine working topologists probably use the box topology to a certain extent in the construction of counterexamples, but that's really what it is: a counterexample, rather than part of the main line of development of the field.
• People really do extend the word "product topology" to the infinite case, rather than switching to "Tychonoff topology".
• While the product space is indeed a special case of "initial topology", it's conceptually simpler (as well as much better-behaved), so introductory texts don't always/usually define "initial topology" before defining "product topology". I think it makes sense to include a more elementary definition as well. --Dfeuer (talk) 20:38, 25 February 2013 (UTC)

I know what you mean with "dull" :). Hopefully you can understand where I'm coming from as well. I can't stress enough how much more value such a clear exposition has over just two lines in a refactor template. Forcing yourself to write a clear and reasonable exposition will automatically make it less judgemental and more argument-driven, which is a good thing for wiki efforts.

I'm not currently prepared to go into a discussion as to the direction of product topologies because I'm attending to other things. I'd like to defer it to a moment when I'm less deeply nested in my long-term plans to avoid getting lost and leaving loose ends. — Lord_Farin (talk) 21:44, 25 February 2013 (UTC)

Fair. --Dfeuer (talk) 21:52, 25 February 2013 (UTC)