I am trying to prove the following statement, but I do not see how to proceed:
Let $(X_1,\tau_1),(X_2,\tau_2)$ be two topological spaces. Let the no. of connected comp. of $X_1$ and $X_2$ be finite. Then the no. of connected comp. of $X_1 \times X_2$ (equipped with the product topology) is equal to the no. of connected comp. of $X_1$ multiplied with the no. of connected comp. of $X_2$.
If $\{C_i\}_i$ and $\{D_i\}_j$ are the two sets of connected components of $X_1$ and $X_2$ respectively then I claim that $\{C_i \times D_j\}$ is the set of connected components of $X_1 \times X_2$.
We can reduce this problem to showing that if $X$ and $Y$ are two connected topological spaces then $X \times Y$ is connected since a topological space is the disjoint union of all of its connected components (if the set of connected components is finite) and $\times$ distributes over $\bigsqcup$. I.e $(A \bigsqcup B) \times C \approx (A \times C) \bigsqcup (B \times C)$
Now, to prove that $X \times Y$ is connected for $X$ and $Y$ connected I refer you to this math.se post:
