Is there a clopen subset of $\beta \mathbb N \setminus \mathbb N$ homeomorphic to $\beta \mathbb N$? If so, is there any plausible description of any such a subspace?
Does $\beta \mathbb N$ embed into $\beta \mathbb N \setminus \mathbb N$?
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general-topology
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1You should include the definitions if you want a wider audience to be able to help you. This is also helpful for others when searching for questions that were already posed. – 2012-11-28
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1@Julian: I really don’t see anything here that needs to be defined. – 2012-11-28
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0@Brian: I agree with Julian. A search for "Stone-Cech compactification of the naturals" will not show this question. – 2012-11-28
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0So it would be better to create a tag stone-cech-compactification because there are lots of questions here concerning this matter. – 2012-11-28
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0@Martin: That’s a completely separate issue that has nothing to do with whether any of the terms needs to be defined. Adding the term *Čech-Stone compactification* may help searchers, but it’s most unlikely to help someone who doesn’t already know what $\beta\Bbb N$ is. – 2012-11-28
1 Answers
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There is not: if there were, $\beta\Bbb N\setminus\Bbb N$ would contain isolated points, and it does not. Specifically, if $h:\beta\Bbb N\to\beta\Bbb N\setminus\Bbb N$ were homeomorphism onto a clopen subset $X$ of $\beta\Bbb N\setminus\Bbb N$, the sets $\{h(n)\}$ for $n\in\Bbb N$ would be open in $X$ and therefore in $\beta\Bbb N\setminus\Bbb N$.
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0Thank you. How about containment of other extremally disconnected clopen subspaces? – 2012-11-28
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0@J.Lund: I don’t know; I never did much with extremally disconnected spaces, and my knowledge of $\beta\Bbb N$ is fairly modest. – 2012-11-28