User:Mathmensch/Disc1wDE

Hello,

the lemma Graph (topology) has been transformed to a redirect to Topological graph theory, although it seems as though a graph in topology is merely the object of study of the latter. I am confused on when to construct new lemmata; certainly Linear Algebra and Vector space are two different articles, and even Group (mathematics) and Group theory (although this I find perhaps not right since group theory is only about groups, whereas linear algebra also studies matrices and so on, which is why one could argue for a merger between the two).

See also Wikipedia:Merging#Reasons_for_merger at "Merging should be avoided if:" 2. I assume the reason for the merger was 2. in the first list; I don't know whether context is required there; when reading Hatcher's book, this concept was presented to me without this particular context, and it was understandable.

Namely, I am currently reading Hatcher's book, and there are a couple theorems on graphs, like for instance that every graph contains a maximal tree, from that one can compute the fundamental group, covering spaces of graphs are again graphs, and there is also an application to group theory (subgroups of free groups are again free).

I would be particularly pleased if apart from the rules, which seem ambiguous here, I would gain knowledge on the day-to-day practice of merging lemmata and articles, so that I will be able to avoid creating new lemmata in vain by checking some more-or-less well-defined criteria on when not to do so.

Since you mentioned the possibility at the top of your talk page, I would invite you to reply at my own talk page, so that I am notified (furthermore my talk page is much less busy, and since your reply may shed some light on the issue, I may want to come back to it in the future, which is easier when the archive is somewhat less crowded). --Mathmensch (talk) 06:01, 5 September 2016 (UTC)

It's a fuzzy boundary, but to me the topics are similar enough that a new article would need to demonstrate the need for a new article by providing significantly more content about whatever it is about. In the case of Graph (topology), on the other hand, the new article was significantly shorter than the paragraph about the same topic within topological graph theory. —David Eppstein (talk) 06:12, 5 September 2016 (UTC)

Hello there,

1) May I move this section to my talk page?

2) If, say, I were to expand Graph (topology) so that it would contain significantly more content on the subject than topological graph theory, would you then think it is better to have two lemmas? --Mathmensch (talk) 12:37, 5 September 2016 (UTC)

I'd rather keep a copy here for my records, but if you want to copy it to your talk page I have no problem with that. I'm not sure why you call them lemmas; they're generally called articles here. But I'd have to see the expansion before formulating an opinion on it. —David Eppstein (talk) 20:10, 9 September 2016 (UTC)

I created a page (User:Mathmensch/Disc1wDE) which can be embedded in both our talk pages as a template. --Mathmensch (talk) 06:46, 10 September 2016 (UTC)

I just created a draft of Graph (topology) in my user namespace. I would ask for comment; it would be best if you could use the page User:Mathmensch/Disc1wDE for commenting, since this is embedded in both our talk pages. --Mathmensch (talk) 10:59, 10 September 2016 (UTC)

Ok, some comments:

• I think it's too WP:TECHNICAL. Here's my attempt at a more accessible start: "A topological graph is a topological space defined from an undirected graph by replacing each vertex by a point and each edge by a curve." You can see, I hope, that not requiring readers to already understand CW complexes and "the usual way" lowers the amount of background needed to understand this material. The same goes for all the rest of the material. You need to pay much more attention to whether the jargon and notation you are introducing is really necessary, or whether it is just there to make things look more mathy.
• In the same vein of reducing technicality, If you are going to discuss complexes, what is the point of using CW complexes? These are all just simplicial complexes, the added generality of CW complexes is pointless here.
• In general, many concepts that are used here are neither linked to pages describing them nor explained within the article, leaving the article readable only to people who are already familiar with those concepts. These include graphs (the normal kind of graph), skeleton, quotient topology, quotient map, gluing, closed, one-dimensional, unit ball, etc., and that's just in the lead section. The rest is if anything worse.
• In the bullet about "usual category of graphs", graphs points to a disambiguation page, and there is no actual link to the category of graphs. Also I thought the usual category of graphs was over directed graphs (and directed graph homomorphisms) but here the graphs are undirected.
• "The usual category of graphs and graph homomorphisms is naturally contained within the category of graphs" reads like a tautology. I think that the second instance of "category of graphs" should be replaced by something more like "category of topological graphs and continuous maps". Also, there is no entry for inclusion in Glossary of category theory so although I can guess I am a bit unsure what the right meaning of an inclusion of categories is supposed to be, especially in this case because it's not clear what the objects are — do two isomorphic combinatorial graphs lead to a single object in the category, or two different objects? What about two different but homeomorphic embedded topological graphs? And if they do all lead to different objects, which of multiple homeomorphic topological graphs is the one that is supposed to represent a given combinatorial graph?
• The applications section is missing a link to the main article Nielsen–Schreier theorem on this application.
• There are some closely related concepts that should be included,
• Are you assuming that the starting combinatorial graph is finite? Because otherwise the existence of a maximal tree would seem to need the axiom of choice or some similar assumption.
• It is not really even true that a topological graph is just a topological space of this form, right? Because the vertices are distinguished as special points of the graph. If you subdivide an edge of a graph, you get a different graph with the same topological space. So it is also not formally correct to toss around concepts like covering space, fundamental group, homotopy equivalence, etc., as if they applied to graphs. What they actually apply to is the underlying topological space, forgetting the distinction of some points as vertices.

—David Eppstein (talk) 21:21, 20 September 2016 (UTC)

Yes, these are all true. Gonna include them ASAP.

Could you give some info or a reference on the closely related concepts that are to be included? --Mathmensch (talk) 06:01, 21 September 2016 (UTC)

Although Hatcher's book defines them without point distinction. Let's see what we do here. --Mathmensch (talk) 06:05, 21 September 2016 (UTC)

Aha, perhaps the points are distinguished by being the 0-skeleton. This would mean that CW complexes have a "memory" for how they were formed, or the associated topological space does not bear all the information about CW complexes. Interesting. --Mathmensch (talk) 06:08, 21 September 2016 (UTC)

Some more thoughts:

• You really should be using more than one source.
• This does not line up well with graph homomorphisms because its continuous functions allow both edge subdivisions and edge contractions while graph homomorphisms don't.
• There's another kind of topological space derived from graphs (I think this idea is from Thurston but I don't have a source): make a non-Hausdorff space with a closed point for each vertex, an open point for each edge, and the closure of each edge being the set of it and its endpoints. For this one it actually is true that homeomorphism of spaces is the same as isomorphism of graphs, and it has much of the other properties you would want. Its continuous maps are almost like graph homomorphisms — no edge subdivision are possible — but homomorphisms after you augment the graph to have a self-loop on each vertex. So its categorical product becomes the strong product of graphs, etc.

—David Eppstein (talk) 18:44, 23 September 2016 (UTC)

Hmm, so there is a different structure (which seems still a fairly discrete one). As usual I agree on what has been said, and I will also look for more sources ASAP (for now I'm "a bit" busy). --Mathmensch (talk) 17:10, 25 September 2016 (UTC)

@David Eppstein: The article User:Mathmensch/sandbox/Graph_(topology) has now been submitted to be reviewed, after all your concerns were addressed. --Mathmensch (talk) 10:22, 1 February 2017 (UTC)