Definition talk:Locally Euclidean Space

I wonder whether to separate out the definition of "dimension" of such a space into its own page so it can more easily be categorised and conceptually linked with other instances of "dimension". --prime mover (talk) 12:52, 1 December 2012 (UTC)

Hausdorff

I landed here because I was looking at this page and was looking for the Hausdorffness property. I found it here, but it is rather uncommon to assume Hausdorffness in the definition of locally Euclidean spaces. Should this be changed? --Geometry dude (talk) 09:56, 17 September 2014 (UTC)

I'd be careful about "just" changing it, because there are many pages which depend on it. If it were changed so as to say something different, then the pages that depend on its definition being as it is may then be rendered invalid. --prime mover (talk) 18:40, 17 September 2014 (UTC)

I understand that, but on the contrary, someone from the outside might land on the manifold page and take the wrong definition, because he or she didn't follow the link down to locally Euclidean. :-( --Geometry dude (talk) 22:50, 17 September 2014 (UTC)

Hence what I said somewhere else and needs to be repeated over: this is why it's best for any field of knowledge to be developed from the ground up, rather than dropping in at the top. --prime mover (talk) 05:05, 18 September 2014 (UTC)

And unfortunately, we haven't applied this philosophy consistently since the inception of this site. This page is one of the victims. — Lord_Farin (talk) 09:27, 18 September 2014 (UTC)

I still believe that omitting Hausdorffness in other pages and hiding it in locally Euclideaness is a mistake. They are two seperate concepts. That is there are locally Euclidean spaces that are not Hausdorff, e.g. the line with two origins, and there are Hausdorff spaces that are not locally Euclidean, e.g. a two point set equipped with the discrete topology. --Geometry dude (talk) 12:20, 18 September 2014 (UTC)

Okay, now we're talking about something different: your definition of Locally Euclidean Space does not match the definition given here. Are we going to have to say: let's throw away everything written by Linus44 and ARBowen (or whatever the username is of the other guy) because it's unreliable? I need to understand how important these differences are and whether they are actually "wrong" (as in: the person posting stuff up was incorrect) or whether there exist schools of thought which define the stuff in a completely different way with a view to approaching the subject from a different direction?

I'm afraid I'm going to have to call you out, Geometry dude, as I see no sources quoted for any of the stuff you've posted up. You've been contributing to this site for long enough now to have learned how it is structured, so you're familiar with how we provide citations: can you add the source work you are using as the basis for your information in the appropriate place in the Books section? Then we will at least have a verifiable source work which can then include all the background material as necessary, and we will then all understand the implications behind what it is you're sharing. All that will then be needed is for that background material to be added. --prime mover (talk) 12:32, 18 September 2014 (UTC)

You're right, I should add some sources. I will do that. What I was trying to say is that I have never encountered a definition of locally Euclidean where Hausdorffness is assumed except here and that I have found a fair amount of sloppiness and also errors in the differential geometry section here, which is why I became a contributor here to begin with. I do, however, understand that you are reluctant to change definitions as you can't tell directly which one is the "right definition" in the sense that it matches the idea behind the concept and whether it is the one agreed upon in the literature. --Geometry dude (talk) 15:11, 18 September 2014 (UTC)

A definition for local Euclideaness is given on page 2 in 2003: John M. Lee: Introduction to Smooth Manifolds. It does not include Hausdorffness. --Geometry dude (talk) 15:37, 18 September 2014 (UTC)

I've been led to understand that early work in topology assumed Hausdorffness as a basic topological property that all topological spaces possessed, so it's possible that some of the work on here is based on older works which have made an attempt to make sense of this, by specifically specifying this property so as to retain backward compatibility. Maybe. --prime mover (talk) 17:53, 18 September 2014 (UTC)

That would definitely make sense. But then the definition would be outdated. --Geometry dude (talk) 19:17, 18 September 2014 (UTC)

Outdated or not, it merits inclusion. --prime mover (talk) 21:52, 19 September 2014 (UTC)

One of the philosophical foundations of $\mathsf{Pr} \infty \mathsf{fWiki}$ is that people reading any book should be able to come here and find how the definitions in that book relate to the rest of the mathematical literature. Depending on, e.g., how many books treat manifolds in this way and what results are specific for Hausdorff manifolds, we could approach this in different ways.

An unfortunate consequence of the presentation of the existing material is that we will have to scrutinise the results for hidden Hausdorffness assumptions; this makes a new attempt at the field hard. I would first like to see listed exactly what and how many pages are affected by this discrepancy. If the ramifications are small, it won't be hard to rewrite the pages into a new form in a sandbox and publish them later. If there is a lot of material, it will be a difficult and time-consuming exercise to achieve meaningful progress. But we'll think about that as and when it's necessary. — Lord_Farin (talk) 10:14, 20 September 2014 (UTC)

As far as I know, Hausdorffness is almost always assumed when treating manifolds and if not, one usually talks about non-Hausdorff manifolds. Is there a way to see all the pages on $\mathsf{Pr} \infty \mathsf{fWiki}$ that link to this one? --Geometry dude (talk) 12:50, 22 September 2014 (UTC)

Tools > What links here. — Lord_Farin (talk) 16:18, 22 September 2014 (UTC)

Thank you very much. I had a look at all the links and the only pages there where Hausdorffness is needed, but not explicitly stated are the manifold definition pages. I suggest dropping the Hausdorffness condition here and adding it to the manifold pages, as it is common in the literature. --Geometry dude (talk) 18:59, 22 September 2014 (UTC)

Seeing as the definitions are not sourced, I see no harm in this approach. Worst-case we make an error or two, but these won't stand out on PW scale -- I can live with the thought of a few possible mistakes in a field as sparsely populated as this one. But let us hear what PM thinks. — Lord_Farin (talk) 19:03, 22 September 2014 (UTC)

What puzzles me is: if each point has a nbhd homeomorphic to a patch of $\R^n$, which is definitely Hausdorff itself, how can you have a non-Hausdorff L.E.S.? I confess to not having studied my Steen and Seebach deeply enough to have understood whether Hausdorffness follows in a L.E.S. in this context, but an intuitive glance at it suggests that Hausdorffness *ought* to follow from the condition given.

What would be *very* useful would be an example given of a non-Hausdorff locally Euclidean manifold, so as to get an intuitive feel for what is and what is not. --prime mover (talk) 20:30, 22 September 2014 (UTC)

see for example: http://en.wikipedia.org/wiki/Non-Hausdorff_manifold The first two are quite accessible. --Geometry dude (talk) 20:44, 22 September 2014 (UTC)

Okay, it seems plausible, as long as the relation $\sim$ is defined adequately. These spaces would do well to be added to $\mathsf{Pr} \infty \mathsf{fWiki}$ --

Dimension

Sorry to bother you Prime.mover, but there's another issue. You repeatedly asked me to reference a page along the lines of "dimension of a topological manifold", but this is defined implicitly on this page. You said that would be okay on Definition_talk:Dimension_(Linear_Algebra), but then you added the explain thingy on Definition:Coordinate_Function. I don't know how to proceed with respect to that. --Geometry dude (talk) 19:17, 18 September 2014 (UTC)

The question went "Should pages in the differential geometry referring to dimension link there?" I must have misunderstood what you meant by "there".

I have a simplistic view of the universe, because I'm a very simple man and have a brain which doesn't work very well. (This I know because I have had this constantly dinned into me all my life.) :-) A result of this is that websites, for example, that I have a say in the design of are simple. So when there is a link to "dimension" in a page, I expect it to go to a page which defines "dimension".

We already have a page defining "dimension". It's split into several subpages, because each instance of "dimension" is appropriate to a particular context.

So what we want is a page containing a definition of "dimension" in the context of wherever it is that needs a definition of "dimension" and when that page refers to "dimension" in that context, it should contain a link to that page.

I can't see what I can do to make it more complicated than that. --prime mover (talk) 19:34, 18 September 2014 (UTC)

The issue is resolved now. Just for anyone stumbling across this discussion. --Geometry dude (talk) 12:52, 22 September 2014 (UTC)

Full specification

I see you fixed my mistake about the dimension. Therein lies the caveat that every single definition needs to be made in as complete a context as possible. The problem is, when you're familiar with a topic, you take stuff for granted and forget that someone who's never been near it before may not share that knowledge.

This is a vexed point amongst contributors to $\mathsf{Pr} \infty \mathsf{fWiki}$: some of them believe that unless you already have plenty of background in a particular topic, there's no point even trying to understand it, so it's a waste of effort specifying everything in full detail; the people who understand the concept know the details, and the people who don't have no business reading it. The $\mathsf{Pr} \infty \mathsf{fWiki}$ philosophy does not intersect with this. --prime mover (talk) 20:39, 22 September 2014 (UTC)

I agree that one has to be careful when defining stuff. Of course, there's no point in having a wiki if it isn't of any use in understanding things. Don't forget that I did not create the page though. :P --Geometry dude (talk) 15:49, 23 September 2014 (UTC)