So I'm having a little trouble with the part of Van Kampen's theorem my professor presented to us. He called this the easy 1/2 of Van Kampen's theorem.(adsbygoogle = window.adsbygoogle || []).push({});

Theorem (1/2 of Van Kampen's) - Let X_{,x}=A_{,x}U B_{,x}(sets with basepoint x) where A and B are open in X and A[itex]\bigcap[/itex]B is path-connected. Then [itex]\pi[/itex]_{1}(X) is generated by [itex]\pi[/itex]_{1}(A) and [itex]\pi[/itex]_{1}(B).

[itex]\pi[/itex]_{1}(A) and [itex]\pi[/itex]_{1}(B) are not necessarily subsets of [itex]\pi[/itex]_{1}(X), at least in general. So if anyone can enlighten me on exactly what the Professor meant. I would think he just means the embedding of [itex]\pi[/itex]_{1}(A) and [itex]\pi[/itex]_{1}(B) in [itex]\pi[/itex]_{1}(X) but I don't think, at least in general, the homomorphism induced by the inclusion is injective.

Thanks very much.

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# Van Kampen's Theorem

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