Definition talk:Usual Topology

This is &hellip; awkward. This is currently a redirect to Definition:Euclidean Space/Euclidean Topology, which defines the usual topology on $\R^n$ as the topology induced by the Euclidean metric. However, the usual topology on $\R$ is very often defined as the order topology on $\R$ with the usual ordering. The latter (ultimately equivalent) definition is every bit as important as the former, but I'm not really sure how to structure that properly in this context. --Dfeuer (talk) 09:11, 11 February 2013 (UTC)


 * Never seen it defined that way. Sources? --Lord_Farin (talk) 09:56, 11 February 2013 (UTC)


 * It's not exactly so, but Kelley introduces the usual topology on the reals as the family of all sets which contain an open interval about each of their points. Since the reals are unbounded above and below, this is the order topology (rays are only needed for sets that are bounded in some direction). He doesn't introduce metric spaces for another bunch of pages. I believe Munkres does something similar, but it's bedtime now. --Dfeuer (talk) 10:05, 11 February 2013 (UTC)


 * Good catch. If there is genuine ambiguity in what "usual topology" is defined as then we need to do two things:
 * a) Replace this redirect with a disambig, or a page which otherwise explains the two different usages (with links)
 * b) Go through the existing links to this redirect and replace them with what is really meant. --prime mover (talk) 10:36, 11 February 2013 (UTC)


 * It's not a genuine ambiguity. It's two didactic approaches. leading to alternative definitions of the special case of the real line. --Dfeuer (talk) 13:57, 11 February 2013 (UTC)


 * If they ultimately lead to the same object ... where's the problem? --prime mover (talk) 14:23, 11 February 2013 (UTC)

Aside from the fact that some texts define it that way, it's pretty important because theorems about the reals very often generalize along two paths: to Euclidean spaces, metric spaces, and uniform spaces in one direction, and to linear continua, Dedekind complete or close packed spaces, and linearly ordered spaces in the other direction. So the order-theoretic view of the usual topology on the real line is plenty important enough to merit an alternative definition. --Dfeuer (talk) 04:16, 12 February 2013 (UTC)


 * So explain to me what your problem is. --prime mover (talk) 06:10, 12 February 2013 (UTC)


 * The usual topology on $\R$ can be described in two equivalent but conceptually very distinct ways, and Dfeuer wants to treat them on a more equal footing. --Lord_Farin (talk) 14:33, 12 February 2013 (UTC)