Exercise 8, § 2

Let $X$ be a set, and $c$ a cardinal number which is either infinite or equal to 1. The set of subsets of $X$ consisting of $X$ and those $M \subset X$ such that $\operatorname{card}(M) < c$ is the set of closed subsets of $X$ for a topology $\mathfrak{T}_c$; $\mathfrak{T}_1$ is the coarsest topology on $X$ and $\mathfrak{T}_c$ is the discrete topology if $c > \operatorname{card}(X)$. Every permutation of $X$ is an automorphism of the topological space obtained by giving $X$ a topology $\mathfrak{T}_c$.

Show that conversely any topology $\mathfrak{T}$ on $X$, which has the property that every permutation of $X$ is continuous with respect to $\mathfrak{T}$, is necessarily one of the topologies $\mathfrak{T}_c$. [Let $c$ be the smallest cardinal such that $\operatorname{card}(F) < c$ for each closed subset $F \neq X$ in the topology $\mathfrak{T}$; observe that when $X$ is infinite and $c > \mathrm{card}(X)$, $\mathfrak{T}$ is necessarily the discrete topology.]