Different Hausdorff topologies with the same continuous mappings

Question

Fix a set $X$. For a topology $\tau$ on $X$, let $F_\tau = \{f: X \to X \mid f \text{ continuous w.r.t. } \tau\}$. What would be an example of a set $X$ and two Hausdorff topologies $\tau$ and $\sigma$ on $X$ such that $\tau \neq \sigma$ but $F_\tau = F_\sigma$?

Answer 1

One way to find an example is via this fact.

In more detail: suppose $(X, \tau)$ is a nontrivial Hausdorff space with the property that any $\tau$-continuous function $X \to X$ is either the identity or constant. Let $f: X \to X$ be any non-identity bijection of $X$ (eg one that swaps two elements). Let $\sigma$ be the topology on $X$ that makes $f: (X, \tau) \to (X, \sigma)$ into a homeomorphism (ie "transport the structure along $f$"). Then $F_\tau$ and $F_\sigma$ both consist only of the identity function and the constant functions. But $\tau$ and $\sigma$ are not equal, because otherwise $f$ would be a continuous function from $(X, \tau) \to (X, \tau)$.