Definition talk:Topology Induced by Metric
- Besides the subspace topology, I've also seen the quotient topology called that - the latter especially when "collapsing a subspace to a point". --Lord_Farin (talk) 21:23, 30 October 2012 (UTC)
- hmm ... so disambiguation or transclusion? I expect it's going to have to be disambiguation. I'm minded of Definition:Induced Operation.
Didactic approaches and missing pieces
There's a common way to use metric spaces as an introduction to topological ideas. The first part of this approach is fairly well covered on PW:
- Define metric space, and probably make some connections to familiar Euclidean $n$-space.
- Define "topological" terms such as "open set", "closed set", "convergent sequence", and "continuous function" as specialized to metric spaces.
- Develop some topological theorems within the context of metric spaces.
The next part of the approach is reasonable well covered on PW:
- Define abstract notions of "topological space", "open set", "basis", and "subbasis".
- Define topological notions such as "convergent sequence" and "continuous function" based on topological spaces.
- Prove some basic theorems about bases, subbases, continuous functions, etc.
The third part is currently poorly covered on PW, and I personally am struggling to figure out how to do it in a structurally pleasant way.
- Prove that the $\epsilon$-balls of a metric space form a basis for a topology.
- Prove that open sets (as defined for metric spaces) comprise a valid topology.
- Prove that the definitions of convergence, continuous function, etc. that were given for metric spaces are compatible with the way those were previously defined for metric spaces.
Once that didactic sequence has been completed, further proofs about metric spaces are typically considered free to apply the general topological theorems without having to go into nitpicky details of justifying themselves. It's quite awkward, for instance, if a proof about metric spaces must continually invoke the theorems that convergence and continuity in a metric space match those in the induced topology before applying general topological theorems. This issue comes up in, for example, the proof that isometry preserves (pseudo)metric space completeness. While this is a theorem specifically about (pseudo)metric spaces, it makes heavy use of topological theorems. As it is, I was forced to bend everything to a metric space-specific approach, which just looks bizarre. I would expect most people would be willing/able to learn about the general topological notions before getting to a theorem of that level of sophistication.
While the approach above is very common and didactically strong, it is not the only approach taken in the literature. Some writers take a possibly-Bourbakist approach of defining and developing topological notions before introducing metric spaces. These authors prove that the $\epsilon$-balls form a basis for a topological space (or otherwise prove the induced topology is a topology), and then prove theorems specific to metric spaces in the context of the fact that they are topological spaces, never proving the general theorems in a purely metric-space context.
The reason I call that approach possibly-Bourbakist is because there is another common way to approach topological spaces, sometimes used alongside the first approach I described using metric spaces, is to begin the development with the real line, define open intervals, etc., define open sets from open intervals, etc., etc. Then they define topologies, and then work back with order topologies and so on. I'm sure we want to support all these approaches, but I'm not sure how to do it without inducing confusion as someone traveling in one direction of the development finds, elsewhere on the same page, a development going in the opposite direction. --Dfeuer (talk) 18:40, 13 January 2013 (UTC)
- We are pleased to welcome you in the realm of the decisions that we struggle with regularly as well... Usually, the course of action is to wait until someone brings up a brilliant approach. This person is likely the same as the one initiating the discussion, but not necessarily. --Lord_Farin (talk) 10:09, 14 January 2013 (UTC)