User:Dfeuer/To Do

Refactor and rework the definitions for the usual topology on the real line and real vector spaces.

The current arrangement defines the Euclidean metrics for real vector spaces and (as a special case, essentially) the complex numbers, which is a valuable thing to do. It also separately defines a Euclidean metric for powers of $\Q$, which seems silly. But then it goes and defines a "Euclidean topology" for $\R^n$ based on the Euclidean metrics, and this makes much less sense.

1. There's nothing particularly Euclidean about it: hyperbolic geometry, for example, induces the same topology.

2. Some texts (e.g., Kelley and Munkres) introduce the usual topology on the reals in order-theoretic terms (it is the order topology, and because the reals are unbounded above and below that gives the order topology a basis of open intervals). Then the usual topology on $\R^S$ is defined as the product topology, and the Euclidean metric is shown to induce it in the finite-dimensional case. --Dfeuer (talk) 18:48, 25 February 2013 (UTC)


 * The great importance of these definitions leads me to ask that you proceed on the basis of preferably multiple sources which you undoubtedly possess &mdash; otherwise, drop me a note; I can probably provide you with a few. Among these sources, those already mentioned on PW should be considered first, of course. &mdash; Lord_Farin (talk) 19:09, 25 February 2013 (UTC)


 * I don't have terribly many, but I can use them. We actually already have most of the information here on PW&mdash;it's just arranged in a very metric-centered way (presumably the influence of Sutherland). Most of what I'm talking about is rearranging it to make the order-based approach a coequal one rather than a secondary one. Some proofs (I don't remember which) will want alternative versions added to take advantage of the other perspective. --Dfeuer (talk) 19:14, 25 February 2013 (UTC)


 * Ok. In due time I'll monitor and comment; please give me a heads up when you're about to start (and I'd then also appreciate a more detailed outline of what you envisage). &mdash; Lord_Farin (talk) 19:18, 25 February 2013 (UTC)