this is a problem in my topology class:
True or false: the product of two simply connected spaces also simply connected.
What I have right now:
Let X1 and X2 be two simply connected spaces. Let f: $S^1$ --> X1 $\times$ X2. Define f1: $S^1$ --> X1, and use $\pi_1$(X1) = 0 to get a homotopy h1: $S^1 \times$ I --> X1 from f1 to a constant. Then because of the universal property of the product, h: $S^1 \times I$ --> X1 $\times$ X2, which projects to h1. And we can prove that h is a homotopy from f to a constant, so the product X1 $\times$ X2 is simply connected.
Is this correct? If not, please point out where I got wrong.
Thank you so much!