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Looking over the treatment of the Eilenberg-Steenrod axioms in a few of my favorite introductory algebraic topology texts, I see that some include an "axiom of compact support", while others do not. Whether or not one needs such an axiom for the homology theory to be uniquely defined (assuming it satisfies the dimension axiom) seems to depend on the category of pairs for which one is defining the homology theory. For example, if one is only looking at the homology of compact pairs, the axiom of compact supports is certainly unnecessary. Spanier invokes the axiom to handle arbitrary polyhedral pairs, and this also seems to be the situation in Munkres. But then Hatcher (if I'm reading it correctly) proves uniqueness (assuming the dimension axiom) on CW pairs, without any mention of the compact support axiom, which I find somewhat surprising without limiting to compact pairs.

So for which categories containing possibly non-compact spaces does one need the compact support axioms (along with the dimension axiom) in order to know that the homology theory is unique? In particular, what is the status for polyhedral pairs, CW pairs, and arbitrary topological pairs? And if I'm reading Hatcher correctly, is there an intuitive reason why the compact support axiom isn't necessary for CW pairs but is for polyhedral pairs?

It's amazing the things one is forced to think about when teaching a graduate algebraic topology class!

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Maybe something to do with local compactness? –  David Roberts Jan 12 '11 at 0:49
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up vote 6 down vote accepted

The axioms that Hatcher uses (on page 160 of the current online version of his book) include an axiom about arbitrary wedges going to direct sums. This has the same effect (in the presence of his other axioms and in the CW setting) as an axiom about compact support (by which I assume you mean a statement that the homology of a space or pair coincides with the direct limit of homology of compact spaces or pairs mapping to it).

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Thanks, Tom. I think I see the idea, but it's still not completely clear to me why these are equivalent. For example, suppose we just have a single noncompact CW complex. Is it necessarily homotopy equivalent to a wedge of compact complexes? –  Greg Friedman Jan 12 '11 at 19:36
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