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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?

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You 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. –  Julian Kuelshammer Nov 28 '12 at 10:23
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@Julian: I really don’t see anything here that needs to be defined. –  Brian M. Scott Nov 28 '12 at 10:24
    
@Brian: I agree with Julian. A search for "Stone-Cech compactification of the naturals" will not show this question. –  Martin Argerami Nov 28 '12 at 12:48
    
So it would be better to create a tag stone-cech-compactification because there are lots of questions here concerning this matter. –  J. Lund Nov 28 '12 at 14:42
    
@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. –  Brian M. Scott Nov 28 '12 at 20:05

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up vote 4 down vote accepted

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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Thank you. How about containment of other extremally disconnected clopen subspaces? –  J. Lund Nov 28 '12 at 10:05
    
@J.Lund: I don’t know; I never did much with extremally disconnected spaces, and my knowledge of $\beta\Bbb N$ is fairly modest. –  Brian M. Scott Nov 28 '12 at 10:09

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