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I ask this as a logic question. Can every result in algebraic topology be rewritten without the reference to algebraic structures and only fleshed out in terms of open sets and continuous maps?

For example, the line ${\mathbb{R}}$ and the circle $S^1$ are not homeomorphic, since $\pi_1({\mathbb{R}})$ is trivial while $\pi_1(S^1)={\mathbb{Z}}$. Can this be put totally without reference to any groups? (I don't allow proofs like if you take away one point from ${\mathbb{R}}$ it becomes disconnected, but if you take away one point from $S^1$ it remains connected.)

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6 
 Can you elaborate on why you don't allow such proofs? – Artie Prendergast-Smith May 6 2010 at 12:39
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 I removed the lo.logic tag. This has nothing to do with logic, other than in the sense that all mathematics does. Further, in a trivial sense, the answer to your question is yes, if you only work with locally compact Hausdorff spaces – since such a space $X$ can be recovered as the spectrum $C_0(X). This is of course not true of topological spaces in general, however. But I expect you knew that, which makes me wonder what you really want. – Harald Hanche-Olsen May 6 2010 at 12:49
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 ? The definition of the fundamental group is in terms of continuous maps, mod an equivalence relation defined in terms of other continuous maps. I'm unclear how to do what you suggest doing, other than not referring to it as a group. – Tyler Lawson May 6 2010 at 12:51
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 How about "S^1 is compact and R isn't"? That just uses open sets and no groups. But beyond that, I have to admit that I've no idea what you're asking here. Algebraic topology works by simplifying horrible topological stuff. So I would be surprised if there was a result from algebraic topology that really couldn't be proven without it, but why would you want to go through all that complication? Also, by simplifying stuff algebraic topology allows us to see things more clearly and see relationships that aren't obvious otherwise. – Andrew Stacey May 6 2010 at 14:25
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 Sorry, I'm voting to close. As far as I can tell, the use of logic is not concerned too much with whether we use certain words to package concepts we can already define, and many of the algebraic structures, such as fundamental group, arise very naturally. As stated the question seems a lot like asking someone to develop finite group theory without using the specific word "prime". One question that might be more suitable is how little set theory we need to still make sense of universal constructions that appear in algebraic topology like adjoint functors, Kan extensions, and so on. – S. Carnahan♦ May 6 2010 at 16:38
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closed as not a real question by Andrew Stacey, S. Carnahan♦, Pete L. Clark, Qiaochu Yuan, Harald Hanche-Olsen May 7 2010 at 4:11

1 Answer

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Of course yes, as homotopy (resp. singular homology) groups are defined in terms of standard topological spaces -spheres (resp. simplices)- and continuous maps from those to your target space. You would just have to spell out your algebraic topology proof in terms of the basic definitions. It would be cumbersome and useless. That's why algebraic topology is useful, at least as a "language".

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(I didn't downvote the question) – Qfwfq May 6 2010 at 12:46

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