Talk:Compactification (mathematics)

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As far as i can tell, compactification of extra dimensions in string theory has nothing to do with Stone-Čech compactification in point set topology. In the first case, we simply choose a compact manifold (e.g. CY₃) to replace a noncompact one, turning the spacetime M₄xR⁶ into M₄xCY₃ by fiat. We do not require that there exist an isometry (indeed, there will not be one), or even a continuous embedding from R⁶ into CY₃, nor do we require any uniqueness. In the second case, given a noncompact topological space, we look for a (in some instances unique) compact space with a continuous embedding.

I think they should be in separate articles. The are simply not related. Comments?

Well, no. The Stone-Cech is the most complicated compactification, as the one-point is the simplest. Compactification with no continuity condition would be a strange thing (one can Stone-Cech a discrete space if one really has to ...); I suspect that the string theory usage is driven by certain motivations that could be explained, but if you are talking about non-compact to compact manifolds, there must be quite an amount of topology to carry over.

So, some examples of compactifications would add to the article.

Charles Matthews 08:33, 27 May 2004 (UTC)Reply

You know, for some reason, i thought i was looking at an article entitled "Stone-Cech Compactification". Now i see that the article is titled simply "Compactification". This is fine then. Clearly stringy compactification doesn't belong in an article on stone-cech, but i was mistaken. I think i must have seen the Redirect notice and thought it was the title Lethe

I propose to add to the list of comparctifications also Nagata compactification: any separated scheme of finite tye over a noetherian basis can be embedded as an open subscheme of a proper scheme over the same base. There's no article on Nagata compactification theorem, I think that it should be one but I'm not expert enough to write it; however stating it here and create a red link can be useful, right? -- F4wk3s (talk) 10:40, 24 February 2011 (UTC)Reply

Degree of a subvariety in topological cohomology?

Latest comment: 4 days ago1 comment1 person in discussion

Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems. For example, any two different lines in RP2 intersect in precisely one point, a statement that is not true in R2. More generally, Bézout's theorem, which is fundamental in intersection theory, holds in projective space but not affine space. This distinct behavior of intersections in affine space and projective space is reflected in algebraic topology in the cohomology rings – the cohomology of affine space is trivial, while the cohomology of projective space is non-trivial and reflects the key features of intersection theory (dimension and degree of a subvariety, with intersection being Poincaré dual to the cup product).

This entire paragraph in this form is either misleading or just not true. Not only Bezout theorem does not work over non-algebraically closed fields, but it's a theorem from algebraic geometry not topology, about Chow rings not cohomology, and indeed translates to singular cohomology rings, but only for smooth varieties over C and with not that obvious reason. In particular, this is complete nonsense over R. ~SpectralFlux 01:58, 24 August 2025 (UTC)Reply