N-sphere is simply connected for n greater than 1

Statement
Suppose $$n$$ is a natural number greater than 1. Then, the $$n$$-sphere $$S^n$$ is a fact about::simply connected space. In other words, it is a fact about::path-connected space and its fact about::fundamental group is the trivial group.

Stronger facts

 * n-sphere is (n-1)-connected

Facts used

 * 1) uses::Union of two simply connected open subsets with path-connected intersection is simply connected, which is a corollary of the uses::Seifert-van Kampen theorem.
 * 2) uses::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 $$n$$-sphere $$S^n$$ satisfies the conditions for fact (1). Denote by $$p$$ and $$q$$ any two antipodal (diametrically opposite) points of $$S^n$$. Then, define $$U = S^n \setminus \{ p \}$$ and $$V$$ as $$S^n \setminus \{ q \}$$. We obtain $$U \cap V = S^n \setminus \{ p,q \}$$. We find that:

Thus, we see that the conditions to apply fact (1) are met, and we obtain that $$S^n$$ has trivial fundamental group.

Proof using fact (2)
For this, we note that $$S^n$$ is the suspension of $$S^{n-1}$$. For $$n > 1$$, $$n - 1 \ge 1$$, so $$S^{n-1}$$ is a path-connected space. Hence, by fact (2), $$S^n$$ is simply connected.

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.