Spaces for which all transpositions are homeomorphisms The discrete, indiscrete, cofinite, cocountable, co-(any downwards closed collection of cardinals) topological spaces are part of the unique class of spaces such that all permutations of the space (bijections from the space to itself) are autohomeomorphisms. This is because their topologies may be defined purely in terms of cardinality, so everything is bijection-invariant. If we relax the requirement to just saying that all transpositions (i.e. permutations that just swap two elements) are homeomorphisms, do we get a broader class of spaces, or is this property equivalent to bijection-invariance?

The example in Ulli's answer uses a non-principal ultrafilter on $\mathbb{N}$. I'll supplement that nice answer with a more elementary (and choice-free!) example.
Consider the following topology on $\mathbb{Z}$: $U\subseteq \mathbb{Z}$ is open if and only if $U = \varnothing$ or there exists $N\in \mathbb{Z}$ such that $m\in U$ for all $m\geq N$. Any transposition is an autohomeomorphism, but the permutation $n\mapsto -n$ is not, since the preimage of the open set $\{n\in \mathbb{Z}\mid n\geq 0\}$ is the non-open set $\{n\in \mathbb{Z}\mid n \leq 0\}$.
