Let $X$ be a set of continuum cardinality. The group of permutations of $X$ acts on the set of topologies on $X$.


What can be said about the fixed points of this action? Are manifolds fixed points?


What can be said about the finite orbits?



Action of the permutation group on the set of topologies on a continuum
Let $S(X)$ be the group of permutations of $X$, $X$ infinite. It is a classical consequence of the Baer theorem (Onofri for $X$ countable) that $S(X)$ has no nontrivial finite quotient [actually every nontrivial quotient has cardinal $2^{|X|}$]. Hence every finite orbit of $S(X)$ on $T(X)$, the set of topologies on $X$, is a singleton. [Edit: actually $S(X)$ has no proper subgroup of index $<|X|$ and hence this even applies to orbits of cardinal $<|X|$.]


The orbits of $S(X)$ on $2^X$ are indexed by pairs $(u,v)$ of cardinals such that $\max(u,v)=|X|$, namely $C(u,v)$ the set of subsets of cardinal $u$ with complement of cardinal $v$. Let $W$ be the set of such pairs $(u,v)$; it has an obvious total ordering.


Hence an $S(X)$-invariant subset $\tau$ of $2^X$ is determined by a subset $E\_\tau$ of $W$. The condition that $\tau$ is stable under taking arbitrary unions means that $(u,v)\le (u',v')$ and $(0,|X|)\neq (u,v)\in E\_\tau$ implies $(u',v')\in E\_\tau$. In addition, the condition of being stable under taking finite intersection means that $(u,v)\in E\_\tau$, $v=|X|$ implies $(u',v')\in E\_\tau$ whenever $(u',v')\le (u,v)$, and $(|X|,n)\in E\_\tau$ for $0<n<\omega$ implies $(|X|,n')\in E\_\tau$ for all $n'\ge n$.


We deduce that every $S(X)$-invariant topology on $X$ is one of the following


1. the indiscrete topology $\{\emptyset,X\}$;
2. for some infinite $\alpha\le |X|$, the topology $\tau\_\alpha$ for which open subsets are $\emptyset$ and subsets with complement of cardinal $<\alpha$;
3. the discrete topology (this is the only Hausdorff one).


Note: this is equivalent to ask what are the topologies on $X$ for which the self-homeomorphism group is the whole group of permutations.


