Definition talk:Embedding (Topology)

What's the context of this? Topology, presumably? Does it have anything to do with the Embedding Theorem? I suspect not, and it's just another case where abstract algebra and topology just happen to have developed parallel definitions. --Matt Westwood 18:00, 10 April 2009 (UTC)
 * In so many of those kinds of cases, the parallel definitions are equivalent when discussing Lie groups with manifold structure. I'm not familiar enough with the subject of Lie groups to make a guess as to any equivalency here, but I wouldn't be surprised if an embedding of a manifold into another corresponded to an embedding of the associated Lie group into the other. Zelmerszoetrop 19:31, 10 April 2009 (UTC)
 * At my rate of mathematical development, I may get as far as Lie groups in about 2020 ... --Matt Westwood 19:46, 10 April 2009 (UTC)

Redefinition
"I checked sources. What was here before was just plain wrong, presumably based on a false generalization from properties of algebraic embeddings." (edit by Dfeuer)

In the interest of verifiability, it is a very good idea (actually, no, it's essential) to specify what those sources are for this new definition (wikipedia not accepted, of course, it goes without saying).

It is also a very bad idea (as has already been established ad nauseam) to state categorically that a particular definition is "wrong" unless one is prepared to set oneself up to be shot down very messily, at serious risk to one's ego. It is on occasion necessary to accept that one does not know all the details of every single usage of a given term, and that there may indeed be different conventions which may be used in different contexts. At the very least it is worth adding a section "also defined as" so as to specify what the previous definition is. --prime mover (talk) 08:23, 24 January 2013 (UTC)


 * In fact it appears that the original definition was correct after all. --prime mover (talk) 08:24, 24 January 2013 (UTC)

I checked both print sources I had available, as well as several online sources.I could not find anything substantiating the definition that was originally here. I have noidea where it could have come from, and it makes no sense from the perspective of the concept of an "embedding". I have to think that either someone wrote down something they only rememberedhalf of, or that they incorrectly generalized from algebraic embeddings (where in some very notable cases an injective homomorphism is necessary an isomorphism with a substructure) to the topological setting, where that does not hold at all.

"was correct after all" — and your source is....?--Dfeuer (talk) 08:31, 24 January 2013 (UTC)
 * FYI, mine were Munkres, Kelley, Joshi, PlanetMath, and Wikipedia. --Dfeuer (talk) 08:39, 24 January 2013 (UTC)


 * As we have established, BTW, Wikipedia is totally unacceptable as a source for glaringly obvious reasons. --prime mover (talk) 09:20, 24 January 2013 (UTC)


 * A homeomorphism is a continuous bijection. A continuous injection is what you get when you restrict the domain of such a bijection. Therefore the definitions are the same. Therefore the original definition was not "wrong".


 * Your replacement definition is clumsy. Best to state all your assumptions up front, rather than bury what is arguably the most important one in an "if" condition. It's one of the niceties of house style.


 * I didn't have any sources because, as you may glean from studying the history of this page, it wasn't mine to start with. And it turns out that I may not have any anyway, as all my sources that are directly to hand are point-set topology (in which this concept appears not to be used) not algebraic topology Now, what's your excuse for not posting up your sources on the definition page?


 * A little more discipline from you would go a long way. --prime mover (talk) 09:18, 24 January 2013 (UTC)


 * You are clearly not reading carefully. A homeomorphism is a bicontinuous bijection. Or a continuous and open bijection. Or a continuous bijection whose inverse is continuous. --Dfeuer (talk) 09:26, 24 January 2013 (UTC)


 * So add an "also defined as" section to the effect that it can be defined alternatively. It won't break your back. --prime mover (talk) 09:29, 24 January 2013 (UTC)


 * If you, or anyone else, can find a single source, however old or obscure, to support that definition, I will be more than happy to. Otherwise, we might as well include an alternative definition of $\pi$ as $3.14$. --Dfeuer (talk) 09:33, 24 January 2013 (UTC)

Add your sources to the page, though, particularly Kelley and Munkres. This is not optional, you have no excuse - they have both been set up in the "Books" section and it's just a simple matter of entering them in the usual templated format. --prime mover (talk) 09:44, 24 January 2013 (UTC)


 * Cursory search reveals that in the realm of Banach space theory, the previous definition is the common one. Cf Demengel-Demengel, "Functional Spaces for the Theory of Elliptic Partial Differential Equations", p.20. There's also a MathOverflow reference but that's of course less reliable. --Lord_Farin (talk) 10:05, 24 January 2013 (UTC)


 * I don't have that book, but Banach spaces have a lot more structure than general topological spaces, so I'm not sure how that's relevant. --Dfeuer (talk) 10:25, 24 January 2013 (UTC)

Expansion template
I could understand wanting to split this into two definitions, and perhaps add a better explanation, but I don't think there's anything in the template that's not already in the definition. $f$ can only be considered a homeomorphism by restricting its codomain, as is done already. --Dfeuer (talk) 11:34, 24 January 2013 (UTC)