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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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    @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

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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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    @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