I am trying understanding product maps better. Suppose that we have open maps$$f_1:X_1\rightarrow Y_1, f_2:X_2\rightarrow Y_2$$ Then, if $U_i\subseteq X_i \Rightarrow f_i(U_i)\subseteq Y_i$ for $i=1,2$.
Now consider, $$f_1\times f_2:(X_1\times X_2)\rightarrow(Y_1\times Y_2)$$ I believe that this should also be an open map.
Let $U$ be an open subset of $X_1\times X_2$, then $U$ can be expressed as $U_1\times U_2$ where $U_1\subseteq X_1$ and $U_2\subseteq X_2$. I want to show that $(f_1\times f_2)(U)$ is an open subset of $Y_1\times Y_2$. If $x=(x_1,x_2)\in U$, then $(f(x_1),f(x_2))\in (f_1\times f_2)(U)$. Also, $f_1(U_1)$ is open in $Y_1$ and $f_2(U_2)$ is open in $Y_2$. Here is where I am stuck and having difficulty continuing.
Hints are welcome, please no full solutions. I want to work through this. Thank you for your help.