Exercise 3, § 4

Let $(X_i)_{i \in I}$ be an infinite family of topological spaces such that each $X_i$ contains (at least) two distinct points $a_i, b_i$ such that there is a neighbourhood of $b_i$ which does not contain $a_i$. a) For each index $x \in I$, let $c_x$ be the point of the product space

for which $\operatorname{pr}_x c_x = b_x$ and $\operatorname{pr}_i c_x = a_i$ whenever $i \neq x$. Show that every point of the set $C$ of points $c_x$ is isolated. b) Deduce from a) that the topology of $X$ has a countable base if and only if $I$ is countable and the topology of each $X_i$ has a countable base (use Exercise 7 of § 1).

c) Show that if the index set $I$ is not countable then the point $b = (b_i)$ of $X$ has no countable fundamental system of neighbourhoods.