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.