Please, explain me the Stone–Čech compactification of reals without using topological language. I want to understand the concept.
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Do you know stereographic projection? – pmun May 09 '21 at 01:53
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@srm99 yes..... – Anixx May 09 '21 at 01:55
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Stereoraphic projection is an example of one point compactfication of $\mathbb{R}^2$, meaning you are adding one point to make it compact. Similarly, how many points you can add to $\mathbb{R}$ or $\mathbb{R}^2$ and still it is compact? The compactness obtained by adding maximum quantity of points is known as Stone-Cech compactification (Largest compactification), in very layman terminology. Find https://math.stackexchange.com/questions/1790222/stone-%C4%8Cech-compactification-of-real-line, in your context. Hwvr, I need to think a bit to spell it out completely wthout topology and ultrafltrs. – pmun May 09 '21 at 02:07
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@srm99 one-point compactification is a simple thing. – Anixx May 09 '21 at 02:09
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@srm99 particularly, I am interested how it corresponds with the compactification that we get if we add all divergent (to infinity, strongly divergent) sequences to the reals like this: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 – Anixx May 09 '21 at 02:30
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3I would say: first understand the Stone-Cech compactification of the integers. – GEdgar May 11 '21 at 11:54