Kuratowski’s closure-complement theorem

One of my favorite theorems in elementary topology is Kuratowski’s closure-complement theorem.

First some notation.  For any set A\subset\mathbf{R} let A^c denote the complement of A and A^- denote the closure of A.  (Recall that A^c=\mathbf{R}-A and A^- is the union of A and all the limit points of A).

Here’s the problem.  Find a set A so that we can construct as many distinct sets as possible from A using only the closure and complement operations.

For example, if A=[0,1), then we have:

  1. A=[0,1)
  2. A^c=(-\infty,0)\cup[1,\infty)
  3. A^{c,-}=(-\infty,0]\cup[1,\infty)
  4. A^{c,-,c}=(0,1)
  5. A^{c,-,c,-}=[0,1]
  6. A^{c,-,c,-,c}=(-\infty,0)\cup(1,\infty)

Note that A^{c,-,c,-,c,-}=A^{c,-} and that A^{-}=A^{c,-,c,-}, so we’re done.  We were able to construct six sets in this way (including A itself).

What’s the best you can do?

In 1922 Kuratowski proved that the largest number of sets that can be obtained in this way is XX. I’ll tell you what XX is tomorrow, and I will give an example of a set that achieves this number. (In fact, the proof is for any topological space, not just \mathbf{R}.)

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