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Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces?

Everyone quotes the theorem of Atiyah-Janich in Atiyah's book - for example in the discussion Homotopy groups of Fredholm operators.

However, this theorem that $[X,\mathcal{F}] = K(X)$ is stated for compact spaces $X.$ Does the same hold for paracompact $X?$ Any $X?$

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  • $\begingroup$ What is your definition of $K$-theory for a noncompact space? $\endgroup$
    – Denis Nardin
    Commented Oct 2, 2014 at 23:21
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    $\begingroup$ Usually $K$-theory is defined for a CW-complex as the inverse limit of $K$-theories of its finite subcomplexes. This makes what you ask essentially a tautology. $\endgroup$
    – Denis Nardin
    Commented Oct 2, 2014 at 23:26
  • $\begingroup$ What if I take the definition to be the one in Atiyah's book? Simply the Grothendieck group of $Vect(X)$ with addition. $\endgroup$
    – user58951
    Commented Oct 3, 2014 at 0:04
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    $\begingroup$ That definition doesn't give you a cohomology theory (it fails to satisfy the Eilenberg-Steenrod axioms), so it cannot be representable. In particular it fails to satisfy the product axiom. $\endgroup$
    – Denis Nardin
    Commented Oct 3, 2014 at 1:32
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    $\begingroup$ @user58951: The correct definition of K-theory for paracompact spaces requires you to sheafify the presheaf of formal differences of vector bundles. The point here is that as you move to infinity you might need formal differences of vector bundles whose dimension increases indefinitely. For compact spaces this is unnecessary because you can choose a finite cover such that for each element of this cover you have an honest difference of vector bundles and because there are only finitely many of these, you can glue them together in a single difference. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 4, 2014 at 10:56

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