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I encounter with a problem in Topological Manifold written by Lee:

How many different topological structure of $\{1,2,3\}$?

It is easy to make a list of the question, and the answer is $9$.

However I am interested in the more general case: $\{1,2,\cdots,n\}$. My question is obviously that how many structure on it classified by homeomorphism. Is there any general formula?

Thanks in any help.

gaoxinge
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1 Answers1

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This is an open problem with a lot of devolopment. You should take a look here

http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ8_02%2FS1446788700005231a.pdf&code=ad4ac623d5c04ebc83c2e95bce65491e

http://www.emis.de/journals/JIS/VOL9/Benoumhani/benoumhani11.pdf

EQJ
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