I am really short of time, so let me summarize. I have joined the Institute of Mathematical Sciences for my PhD. Right now, I am desperately trying to avoid being succumbed to the pressure of grueling coursework here. But a mathematician is a masochist, so I am enjoying the mathematical agony!

A famous theorem in point-set topology due to Kuratowski was stated as an assignment problem in our topology class. With a hint from our Topology professor, I solved the problem which I regard as the best problem in point-set topology I ever solved yet. The problem seems too fascinating to be even true. It appeared ﬁrst in a paper due to Kuratowski and was made popular by Kelley in his book *General Topology* and can be found as a “starred” problem in Munkres’ *Topology*. I could solve it, not mainly because of the hint given by my professor but with the comfort that someone has already solved the problem. It does seem daunting to begin working on an open problem. Without further ado, I state the problem:

**Let be a topological space and . By iteratively applying operations of closure and complemention, one cannot produce more than 14 disjoint sets.**

The proof of this theorem is found in the adjoining section “Notes” or by clicking here.

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October 9, 2010 at 19:48

Raghu TejaGreat work Abhisheik! Extraodinary intution it was.

October 9, 2010 at 20:41

abhishekparabThere is a mistake I came to know recently. I will update it soon.

November 5, 2010 at 17:52

chandrasekharHey sounds a very interesting problem. Is your solution the only way of proving the problem.

November 5, 2010 at 17:54

chandrasekharA similar post which might happen to interest you may be this. http://mathoverflow.net/questions/5635/does-autaut-autg-stabilize

I say similar because, in Spaces Closures and complements are the famous operations, where as in groups, the homomorphisms (namely Automorphisms) are very important

November 5, 2010 at 18:33

chandrasekharHey don’t mistake, i see the book ” Proofs from the book” in the photo which is above. Do you have an Indian edition or is it a foreign one.

November 5, 2010 at 20:24

abhishekparab@chandrasekhar:

1. I have no idea if it is the only way of solving it. For me, it was a natural way to arrive at the solution. There may be different proofs on the internet.

2. I have seen the classic MO problem earlier, the results mentioned are highly nontrivial.

3. Proofs from THE BOOK is available in the Indian edition by Springer within 400 rupees.

December 22, 2010 at 11:20

nishantchandgotiaHi,

This had been one of my favourite problems too. Though I must say Munkres has plenty of problems like this.

I have not gone through your solution but your approach seems similar to what I know.

Its quite amazing you managed this as an assignment problem since I took quite a while. The question that remains is how did Kuratowski Reach this?

December 22, 2010 at 17:24

abhishekparabThough I don’t have an idea as to how Kuratowski might have reached this result, there is an interesting book in Topology by him. It contains succinct proofs and is a pleasant reading.