Talk:Embedding

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Needs references and a proper lead. Geometry guy 23:30, 22 June 2007 (UTC)

1. What do you think of removing the fist section, I never saw use of embedding for groups...? Maybe it is used I simply do not know...

2. Can anybody use TeX makeup for last section? I know nothing about comuters, so please help...

Tosha

It seems that if you are going to discuss the equivalence between injective immersion and embedding in the compact case, then the definition of embedding as an injective proper immersion in the general case could be useful. Also, perhaps you could make sure the reader knows that only M needs to be compact.--Orthografer 05:16, August 16, 2005 (UTC)

1 Proper embeddings
2 abstract embedding
3 Problem in section "Differential topology"
4 Hooked Arrow

[edit] Proper embeddings

I fixed the definition of a proper embedding. Cthulhu.mythos 08:47, 2 June 2006 (UTC)

In my opinion this notion of properness (involving boundaries) does not make sense in the general topolgical case but only for (differetiable) manifolds. --Trigamma 20:00, 14 January 2007 (UTC)

There is a different (but -- in the case of manifolds -- related) notion of "proper" in general topology: A map is proper if the preimages of compact sets are compact. In this sense, an embedding $f \colon X \to Y$ is proper iff

f - 1 (K)

is compact whenever $K \subset Y$ is compact.--Trigamma 20:00, 14 January 2007 (UTC)

I agree that the given definition of proper embedding makes no sense at all in the context of general topology: "An embedding is proper if it behaves well w.r.t. boundaries: . . . transversal . . .". Trigamma's definition is the correct one and should replace what is currently in the article.

Furthermore: Neither the concept of a boundary, nor transversality, makes sense for an abstract space per se. To the best of my knowledge the concept of a boundary requires that the space be a manifold (or possibly some kind of generalized manifold). Likewise, transversality only makes sense in similar contexts.70.20.161.83 15:21, 4 May 2007 (UTC)

Visibly the problem remained so I fixed it. There is a confusion between the two notions of boundaries too, the general topology one (also known as frontier) and the other in topological (not necessarily differentiable) manifold (with boundaries) theory. Cenarium (talk) 22:13, 22 December 2007 (UTC)

[edit] abstract embedding

Happily I comment to you that abstract embedding had been accepted in PlanetMath [1], i.e. my friends there agree --kiddo 16:19, 24 April 2007 (UTC)

"injective" does not make any sense for arbitrary morphisms in categories. or what do you understand under this term? even if you replace it by monomorphism, this is not the correct meaning for topological spaces e.g.. apart from that, it is clear that it is impossible to summarize all the meanings in the given examples in the "naked setting" of category theory. I suggest to delete this general remark in the article, also in planetmath. ---oo- 01:46, 26 July 2007 (UTC)

I've modified this section which was poor and misleading in my opinion : the concept of embedding is much stronger than the concept of monomorphism or injectivity and may be not related. Moreover, as pointed out in the reference, there is no appropriate way to define embeddings in a purely categorical context. I see that this section has been moved to the bottom of the page, which seems to be appropriate since this is not a categorical concept. Cenarium (talk) 19:16, 22 December 2007 (UTC)

[edit] Problem in section "Differential topology"

In the "Differential topology" section of the article it is stated "When the manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion." This is ambiguous, since there are two manifolds involved in an embedding: the domain and the codomain. In fact, if only the codomain is compact, the quoted statement is false (consider the image of the line y = sqrt(2)x in the square torus T = R²/Z²).Daqu 15:35, 4 May 2007 (UTC)

[edit] Hooked Arrow

I think something should be put in about $\hookrightarrow$ as a symbol often used for embeddings/injective maps SetaLyas (talk) 11:28, 4 April 2008 (UTC)