Exercise 6, § 7

Let $(X_i)_{i \in I}$ be an uncountable family of discrete spaces, each of which has at least two points. On the product set consider the topology

which is generated by all products $\prod_{i \in I} M_i$, where $M_i \subset X_i$ for all $i \in I$ and $M_i = X_i$ for all but a countable number of indices (cf. § 4, Exercise 9). Show that every countable intersection of open sets is open in $X$, and deduce that no point of $x$ can have a countable fundamental system of neighbourhoods.