Does the Union of finite simply connected open subspaces of a space, with the condition that for each three of these sub spaces the intersection is also simply connected, also have to be simply connected?
I mean if i take the union of these simply connected spaces does it have to be also simply connected?
If the sub spaces does't have to be finite then the i think i could take the space of circles of radius $1/n$ and a line to make the space… But it's not possible now…
Take $X$ a circle and cover it by three open arcs that only overlap pairwise. Then the intersection of the three is empty, and therefore simply connected. The arcs are simply connected, but the circle is not.
This shows that we also need some condition on pairwise intersections.