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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.
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)