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In the book of "Counterexample in Topology", the authors define a scattered space as : "A space is called scattered if it contains no nonempty dense-in-itself subsets", which is different from definition in this post A question on isolated points. Which one is correct? In my understanding, the former does not exclude the case of perfect set and dense-in-itself does not imply perfect.

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The definitions are equivalent. Saying that a non-empty set is dense in itself is saying precisely that it has no isolated points. A perfect set is dense in itself, hence not scattered, though you are correct in thinking that a set that is dense in itself need not be perfect.

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  • $\begingroup$ Thank you Brian. I prefer the latter since it is closer to "what really mean" :). $\endgroup$
    – yushang
    Apr 23, 2020 at 4:41
  • $\begingroup$ @yushang: You're welcome. (So do I.) $\endgroup$ Apr 23, 2020 at 5:44

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