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Kuratowski's Closure-Complement (Topology)

  • Thread starter tylerc1991
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Homework Statement


Let (X,T) be a topological space, and let A be a subset of this space.
Prove that there are at most 14 subsets of X that can be obtained from A by applying closures and complements successively.

The Attempt at a Solution


I understand the concept behind the theorem, that is, starting with an arbitrary subset of a topological space, and performing closures and complements successively you can only make 14 sets. In doing the proof however, I am not even sure where to begin.

I have read on wikipedia that to prove Kuratowski you must prove the following:
(1) kkS = kS
(2) ccS = S
(3) kckckckS = kckS.

where k = closure and c = complement of a subset S.

Is this true? and can someone give me some insight to #3? Thank you very much!!!
 

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