N-sphere is simply connected for n greater than 1
- Union of two simply connected open subsets with path-connected intersection is simply connected, which is a corollary of the Seifert-van Kampen theorem.
- Suspension of path-connected space is simply connected (this actually follows from the previous fact, but the explanation it offers is in some ways more direct and intuitive).
Proof using fact (1)
We try to show that the -sphere satisfies the conditions for fact (1). Denote by and any two antipodal (diametrically opposite) points of . Then, define and as . We obtain . We find that:
|1||Both and are open||No||Since is a manifold, points are closed, so their complements are open subsets.|
|2||Both and are simply connected||No||In fact, via the stereographic projection relative to the point , is homeomorphic to , which is contractible and hence simply connected. Similarly, via the stereographic projection relative to the point , is homeomorphic to , which is contractible, and hence simply connected.|
|3||The intersection is nonempty and path-connected||Yes||In fact, it is homeomorphic to . Since , , so is path-connected, hence is path-connected.|
Thus, we see that the conditions to apply fact (1) are met, and we obtain that has trivial fundamental group.
Proof using fact (2)
Note that this proof is actually the same as the proof using fact (1), but is stated in a different conceptual language that perhaps sheds more light.