If X is a locally compact Hausdorff space and $X=X_1 \cup X_2$ where $X_1, X_2$ are disjoint open and closed subsets X, I want to show that $C_0(X)$ isomorphic to $C_0(X_1) \oplus C_0(X_2)$
I have that $K_0(A \oplus B) \cong K_0(A) \oplus K_o(B)$ but I am having some trouble seeing exactly how this helps me with $C_0$ as I am quite new in this area but this is what's "closest" to what I want to end up with. Is there a nice way to say that :
$$C_0(X) = C_0(X_1 \cup X_2 ) \cong C_0(X_1 \oplus X_2 ) $$
And is this even true? Also if $K_0(X) \cong C_0(X)$ somehow, such that it simply follows from my proposition in the book, this would be nice, but I can't seem to find an answer. Or is there another approach to this type of problem?