Exercise 5, § 8

a) Let $X$ be an infinite set. Show that all the topologies $\mathfrak{T}_c$ on $X$ (§ 2, Exercise 8) such that $1 < c \leq \mathrm{card}(X)$ are topologies of an accessible space but are not Hausdorff.

b) Show that the intersection of all the Hausdorff topologies on $X$ is the non-Hausdorff topology $\mathfrak{T}_{\mathrm{card}(\mathbf{N})}$, which is also the coarsest topology of an accessible space on $X$. (Use Exercise 8 of § 2, and remark that there exist Hausdorff topologies on $X$ in which there are countably infinite subsets of $X$ which are not closed).