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A subset of a topological space is called the Kuratowski set if we can get 14 different sets by applying closures and complementation successively.

I want to find a set which is uncountable and is a Kuratowski set.(I got this idea from Munkres). Is there a set which is uncountable and is a Kuratowski-14 set?

  • I can't think of any 14-set on $\mathbb{R}$ that is not uncountable. Also, related: http://math.stackexchange.com/questions/186017/what-is-the-smallest-cardinality-of-a-kuratowski-14-set – anak Jul 06 '15 at 13:36
  • See here. – Dejan Govc Jul 06 '15 at 13:43
  • From Characterization of Kuratowski 14-sets by Eric Langford, Amer. Math. Monthly, p. 362, 1971:

    "Suppose that $X$ is a subset of the real line under its usual topology. Then $X$ is a 14-set iff the following: there exists a nonempty open interval $I$ such that $X$ and $Xk$ (the complement of $X$) are both dense in $I$, and both $X$ and $Xk$ contain nonempty, relatively open, nowhere dense subsets of $R$."

     – mathematrucker Jul 07 '15 at 03:03

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