I was reading the paper of Kando about charachterization of topological spaces by some continuous functions. The following generalization came to my mind.

Let $(X,T)$ be a topological space, and $C(X)$ the space of all continuous fuctions from $X$ to $X$ (with respect to $T$). Which properties of $C(X)$ imply properties in $X$?

Does anybody know a reference or any idea to answer this question?