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

From Topospaces

## Contents

## Statement

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.

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

## Facts used

## Applications

## Proof

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.