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I am trying to get my head around the Van Kampen Theorem, and how this could be applied to find the fundamental group of X = the union of the unit sphere S^{2}in R^{3}and the unit disk in x-y plane? I was thinking of splitting the sphere into 3 regions - two spherical caps each having open boundary 'disk', and a spherical cap (representing an open extension of the disk in the x-y plane through the middle of the sphere).

I think that these regions would all then be open, and the fundamental group of each is just trivial, so the the fundamental group of the whole object X is just trivial. Is this actually the case? Or is this argument somehow flawed?

Thanks!

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# Using the V-K Thm to find fundamental grp of sphere union disk in R3

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