Union of two simply connected open subsets with path-connected intersection is simply connected
Suppose is a topological space with two non-empty open subsets and such that:
- Both and are Simply connected space (?)s and in particular Path-connected space (?)s.
- The intersection is a non-empty path-connected space.
The statement follows directly from the Seifert-van Kampen theorem. Both and are trivial, so we get is an amalgamated free product of two trivial groups, hence it must be trivial.