Another implememntation of van Kampen thoerem.

[MathematicalPhysicist]

There this question that someone gave me an answer to, in the ask topology forum, but I feel it's an immediate conclusion of this theorem mentioned in the title, it goes like this:

Let X=X1UX2 and X1,X2 are open and simply connected in X, show that if the intersection of both sets X1 and X2 is composed of two path components, then \pi_1(X) is isomorphic to Z the additive group of integers.

Now I think, that what I need to find is that \pi_1(X1) and \pi_1(X2) are isomorphic to \pi_1(S^1) which is isomorphic to Z, and then just use the above theorem, the question which isomorphism will do the job?

any hints?

thanks in advance.

 

[cogito²]

If X_1,X_2 are simply-connected, then by definition \pi_1(X_1) = \pi_1(X_2) = 0 which certainly will cause difficulties if you're trying to prove they are isomorphic to Z.