TITLE: can another topology be given to $\mathbb R$ so it has the same continuous maps $\mathbb R\rightarrow \mathbb R$?
QUESTION [33 upvotes]: 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.

REPLY [15 votes]: This is a special case of much more general results surveyed in the book
MR0393330 Magill, K. D., Jr. A survey of semigroups of continuous selfmaps. Semigroup Forum 11 (1975/76), no. 3, 189–282. 
For example, Theorem 2.3 of this book says that only the abstract semigroup structure
(with respect to composition) of the set of continuous maps is sufficient to
recover the topology.