In section 3, pg 8 of String theory on Elliptic curves they claim that for $X$ compact with $x_0$ an involution fixed point, the map $i:x_o\rightarrow X$ is equivariant and **equivariantly split**, which according to them implies

$KR^{j}(X) \approx KR^{j}(X - \{x_0 \}) \oplus KO^{j}(x_0)$

I don't understand what equivariantly split means here relative to the functor $KR^j( - )$ other than you can simply do this separation nor why this is so.

1) I would appreciate if someone could put in more steps into the isomorphism $KR^{j}(X) \approx KR^{j}(X - \{x_0 \}) \oplus KO^{j}(x_0)$

2) Is there a reference were they explain in more detail what is meant by equivariantly split? I haven't seen any mention of the term the way is being used here for any kind of equivariant cohomology theory

Note that it seems to be $KR$-theory with **compact supports** but I don't know if this is essential for the above argument.

Thank you for your time