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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?

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    $\begingroup$ I can’t suggest a specific reference, but I think you’ll find something relevant if you Google “endomorphisms of a topological space.” $\endgroup$
    – Martin Skilleter
    Commented May 20, 2021 at 11:11

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