Generalized Jordan Curve Theorem,

[James]

Let A,B be closed subsets of S^n, where n >= 2, where A and B are both 
homeomorphic to S^1.  Suppose A /\ B consists of 2 points.

If n >= 3, how do I show that if C denotes S^(n) - (A \/ B), then C has one 
component and 

H_n(C) = H_n(S^(n-2) one-point union S^(n-2) one-point union S^(n-2)) ?

I am hoping that I use the Generalized Jordan Curve theorem which says that if 
f : S^r ----> S^n is an embedding, then 

H_i(S^n - f(S^r)) = H_i(S^(n-r-1)). (Here these are reduced homology groups)

In this case, r = 1, so H_i(S^n - f(S^1)) = H_i(S^(n-2)). (reduced homology)

Thank you,

James