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It is written in this article that "a non-scattered Čech-complete space, contains a compact subspace which can be continuously mapped onto the Cantor-set".

Can anyone explain this? I mean, if, $X$ is not scattered, then it contains a non-empty subset which is dense-in-itself. So, how can it be continuously mapped onto the Cantor set? Isn't the Cantor set scattered?

Thank you!

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topsi
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    The Cantor space is not scattered. It's just nowhere dense (if you consider it as a subspace of $\Bbb R$). – Asaf Karagila May 29 '14 at 12:41
  • Now that you mentioned it, I am not sure that I can tell the difference. Scattered = does not contain a dense-in-itself subset. So, what is the difference between this and a nowhere dense space? – topsi May 30 '14 at 09:34
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    Shir, I'm not sure how to answer that. But note that dense-in-itself means without isolated points. The Cantor space doesn't have any isolated points, and many of its subsets don't have isolated points either. Nowhere dense set just means that its complement is dense; but I don't recall this being used as a "standalone property" of spaces. – Asaf Karagila May 30 '14 at 09:41
  • I see. ok I get the general idea. will continue thinking about it. Thanks! – topsi May 30 '14 at 09:50

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