Exercise 21, § 8

Let $X$ be a Hausdorff space with no isolated points, and let $\mathfrak{T}_0$ be the topology of $X$. Let $\mathfrak{M}$ denote the set of complements of countable subsets $A$ of $X$ such that $\overline{A}$ has only a finite number of non-isolated points. Show that if $\mathfrak{T}$ is the topology on $X$ generated by the union of $\mathfrak{M}$ and the set of subsets which are open in $\mathfrak{T}_0$, then $\mathfrak{T}^* = \mathfrak{T}_0$ (Exercise 20) and that every sequence which converges with respect to $\mathfrak{T}$ has only a finite number of distinct terms (which implies that, with respect to $\mathfrak{T}$, no point of $X$ has a countable fundamental system of neighbourhoods, although $X$ itself may be countable and therefore every point of $X$ is then the intersection of a countable family of neighbourhoods of this point).