Kuratowski's Closure-Complement (Topology)

[tylerc1991]

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!

 

[mbowron]

The first two properties above are not specific to the Kuratowski problem, but the third one is.

For further reading and exploration, check out my note and set generator here:

http://mathdl.maa.org/mathDL/60/?pa=content&sa=viewDocument&nodeId=3343