Given $X$, when is it possible to find $Y$ such that $X \approx C_p(Y)$

Question

Given a topological space $X$. Is there a way to construct a space $Y$ such that $C_p(Y)$ is homeomorphic to $X$?

Answer 1

Not necessarily. A space of the form $C_p(Y)$ (where $Y$ is Tychonoff, as we will assume all spaces to be) is quite special. E.g. it is connected, even path-connected, and contains lots of copies of $\mathbb{R}$ as a subspace. For finite $Y$ we get spaces $C_p(Y) \cong \mathbb{R}^n$, for infinite $Y$ it will be infinite dimensional. And so on.

So we certainly cannot see all spaces $X$ in such a way...