TITLE: The fundamental group of a product is the product of the fundamental groups of the factors
QUESTION [22 upvotes]: Hello :) i want to prove the following statement:

$\pi_1(X\times Y,(x_0,y_0))\equiv\pi_1(X,x_0)\times\pi_1(Y,y_0)$

But how to do that? Is this just the projection and the use of the product topology?
Thank you for help :)
I also want to prove that the fundamental group of a n-sphere is trivial for $n>1$, but i have no idea. From my point of view homotopy is very difficult...

REPLY [5 votes]: The proof of Owen for the first question is correct.
Of course van Kampen works for the second question, but it is a slight overkill. I would prefer the following: 
$\pi_1(S^n)=0$ if and only if every loop based at $p$ is contractible through loops based at $p$. Given an arbitrary path based at $p$, we can easily find a contraction by stereographically projecting (from a point not lying on the path) it to $\mathbb R^n$, contracting it in $\mathbb R^n$ and composing the contraction with the inverse of the stereographic projection.