Let {$ X_i $: i $ \in $ I} be a family of topological spaces. Show that $ \prod_{i \in I} X_i $ is path-connected if, and only if, $ X_n $ is path-connected for each n $ \in $ I.
Let x and y be two points in the product. Let t $\in$ [0,1] and z(t) be a point such that $z_i$(t)=$f_i$(t) where $f_i$:[0,1] $\to x_i$ is a path connecting $x_i$ with $y_i$ in $X_i$ . Since any open subbasis set U in the product is an inverse projection $\pi^{−1}_i$($U_i$) of a proper open set $U_i \subset X_i$ for some i , the preimage $z^{−1}$(U) is just $f^{−1}_i$($U_i$) which is open in [0,1].
I was only able to think this. Any help will be appreciated. Thanks in advance.