I want to prove the continuity of a function $f: (X_1,\tau_1) \times (X_2,\tau_2) \rightarrow (X'_1,\tau'_1) \times (X'_2,\tau'_2)$ where $f(x,y) = (f_1(x),f_2(y))$ and my question is:
What is $f^{-1}(\bigcup\limits_{\lambda_1,\lambda_2 \in \Lambda} U_{\lambda_1} \times U_{\lambda_2})$? Is it $(f_1^{-1}(\bigcup\limits_{\lambda1 \in \Lambda}U_{\lambda_1}) \times (f_2^{-1}\bigcup\limits_{\lambda_2 \in \Lambda} U_{\lambda_2}))$ ?
EDIT: I believe that the answer to my question is $(f_1^{-1}(\bigcup\limits_ {\lambda_1\in\Lambda}U_{\lambda_1})\times X_2) \cap (X_1 \times f_2^{-1}(\bigcup\limits_{\lambda_2\in\Lambda}U_{\lambda_2}))$ Am i correct?
Bonus question: This is the first time I'm writting something in LaTeX, and it's honestly really cool. However, how can I write something like U_lambda_1, instead of having to write U_lambda'? Also, what is the code for uppercase Tau? \Tau doesn't work
$$f^{-1}\left(\bigcup\limits_{\lambda_1,\lambda_2 \in \Lambda} U_{\lambda_1} \times U_{\lambda_2}\right)= \bigcup\limits_{\lambda_1,\lambda_2 \in \Lambda} f^{-1}\left(U_{\lambda_1} \times U_{\lambda_2}\right)= \bigcup\limits_{\lambda_1,\lambda_2 \in \Lambda} f_1^{-1} \left(U_{\lambda_1}\right) \times f_2^{-1}\left(U_{\lambda_2}\right).$$