TITLE: Mayer-Vietoris sequence for topological K-theory
QUESTION [7 upvotes]: I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman.  They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence.  
I'm a bit confused with some of their computations, for instance $S^3$ (their example 1.4 in the first section).  They take subsets $U_+ = S^3 \backslash(0,0,0,-1)$ and $U_- = S^3 \backslash(0,0,0,1)$ and then they say that $K^0(U_\pm) \simeq \mathbb Z$.  I don't understand where this comes from since $U_\pm$ are non-compact so I believe $K^0(U_\pm)$ should be the reduced $K^0$ of a 1-point compactification.  The compactifications of these spaces are $S^3$ so shouldn't $K^0(U_\pm) = \tilde {K^0}(S^3) = 0$?  However, it seems like if you replace $U_\pm$ by shrinking it a bit to make it closed, these computations work out. 
So I'm wondering what the exact statement of Mayer-Vietoris is for $K$-theory (specifically, what type of covers you can take) or if Freed, Hopkins, and Teleman are using a different definition of $K^0$ for which $K^0(U_\pm)$ is indeed $\mathbb Z$.  Any references would also be appreciated since I couldn't find much in the literature about a Mayer-Vietoris sequence for $K$-theory.

REPLY [9 votes]: "since $U_\pm$ are non-compact so I believe $K_0(U_\pm)$ should be the reduced $K_0$ of a 1-point compactification"
This is a convention often used in $K$-theory of $C^*$-algebras: by default, people take "$K$-theory" to mean compactly supported $K$-theory.
But that this not the convention used for that particular Mayer-Vietoris computation in FHT.
There, the version of $K$-theory that is being used is homotopy invariant (so that the $K$-theory of $\mathbb R^n$ is the same as the $K$-theory of a point).