I read the following statement in the old question "Intersection of Simply-Connected Sets" (Intersection of Simply-Connected Sets):
If $U$ and $V$ are simply connected and $U \cap V$ is path connected, then $U \cap V$ is simply connected.
The author said, that this follows by Mayer-Vietoris, but I dont see why this is the case since this is a statement about homology. Can anyone give me a hint?
I thought about using the fact, that the first homology group is the abelization of the fundamental group, but I couldnt derive a proper argumentation out of this.
If this statement is false, is there any general condition?
Thanks a lot !