Let X1,X2 be two spaces which are simple connected, and let their intersection be path connected, show that their union is simple connected.(adsbygoogle = window.adsbygoogle || []).push({});

I think I only need to show that their intersection is simply connected, cause then I think that the fundemental group of X1UX2 is isomorphic to to th union of the fundemental groups of X1,X2 and their intersection (correct?).

Well, if we look at a point x in the intersection, then there's a loop in X1, and a loop in X2, if we look at the path that is common to both loops in the intersection then it's also a loop (if we choose the direction of both loops to be counterclockwise or clockwise) in the intersection and its homotopic to the constant loop, cause this loop is both in X1 and X2, that way we get that the intersection is simply connected.

Is this way off?

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# Simpley Connectedness.

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