I have to prove the following:
We have two metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ and their product-space $X=X_1 \times X_2$ with metric $d=d_1 \times d_2$ (so $d(x)=d(x_1,x_2)=d_1(x_1)+d_2(x_2)$ ). We have a projection $\pi_i$ which projects $X$ on $X_i$ for $i=1,2$.
First question was proofing the projection is continuous, which was fairly simple. The next question was where I got stuck upon: $\\$
Let $K=K_1 \times K_2$ with $K_i \subset X_i$. Proof:
$K$ is compact in $X$ $\iff$ $K_i$ is compact in $X_i$ for $i=1,2$.
"$\Rightarrow$" gave no problems: Let $K$ compact in $X$, $\pi_i$ is continuous, which implies that $\pi_i(K)=K_i$ is then compact. (Weierstrass)
"$\Leftarrow$" This is the problem. I tried with collection of open sets, but this went wrong. I hope someone can give a nice proof for this.