Q: Topology of point-wise convergence Does there exists a topology on the product space $\Pi_{\alpha\in\Lambda}X_\alpha$ other than the product topology such that the convergence and the point-wise convergence are equivalent?

A: Consider the  product space $\mathbb{R}\times\mathbb{R}$ with the co-countable topology. Here the convergent sequences are eventually constant sequences. And hence $(x_{1,n},x_{2,n})\to (a,b)$ iff $x_{1,n}\to a$ and $x_{2,n}\to b$. But projections are not continuous as the complement of the $P_1^{-1}(\mathbb{R}\setminus\{1\})$ contains the uncountable set $\{1\}\times\mathbb{R}$ 
