Union of two simply connected open subsets with path-connected intersection is simply connected

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Statement

Suppose X is a topological space with two non-empty open subsets U and V such that:

Then, X is a Simply connected space (?) (and in particular, a path-connected space).

Facts used

  1. Seifert-van Kampen theorem

Applications

Proof

The statement follows directly from the Seifert-van Kampen theorem. Both \pi_1(U) and \pi_1(V) are trivial, so we get \pi_1(X) is an amalgamated free product of two trivial groups, hence it must be trivial.