I was reading the paper of [Kando](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-6/issue-1/Characterization-of-topological-spaces-by-some-continuous-functions/10.2969/jmsj/00610045.pdf) 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 $C(X)$ implies properties in $X$?

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