We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X\rho)\rightarrow (X,\rho)$.

Does there exist a topology $\tau$ that is similar to the euclidean topology on $\mathbb R$?

This was asked here but all we could prove is that $\tau$ must be a refinement of the euclidean topology.

Regards.