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](http://math.stackexchange.com/questions/1890383/is-there-another-topology-on-mathbbr-that-gives-the-same-continuous-functio/1890409#1890409) but all we could prove is that $\tau$ must be a refinement of the euclidean topology.

Regards.