Let $X_1$ and $X_2$ be connected topological spaces. I want to show that the product $X_1 \times X_2$ is connected.
By definition the base from which the topology $\mathcal J_{X_1 \times X_2} = \{\cup_{\alpha \in I} \mathcal B_{\alpha}, \mathcal B_{\alpha} \in \mathcal B \}$ is induced is given as $\mathcal B = \{ U_1 \times U_2 \mid U_i \in \mathcal J_i \}$.
I've tried proving connectedness by assuming $X_1 \times X_2 = U_1 \cup U_2$ with $U_1 \cap U_2 = \emptyset$, $U_1, U_2 \in \mathcal B_{X_1 \times X_2}$, and then use the fact that $X_i$ is connected.
If $U_i = \emptyset$ we are done, so one can assume $U_i \neq \emptyset, X_1 \times X_2$.
Can someone help me to show that $X_1 \times X_2$ is connected ?
