What is special about the homology of the compact spaces, what are the most elementary properties that are verified by homology of compact spaces that are not true for the homology of non compact spaces ? I’m asking because i read that Euler characteristic is not easily defined for non compact spaces while it is straightforward for compact spaces and i also read that we have a homology theory that is better suited for noncompact spaces rather than singular homology, what is making this difference between compact and noncompact spaces regarding homology and cohomology ? Thank you for your help !
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