Here is a way to construct an adjoint $F:\textbf{Top}\rightarrow \textbf{Haus}$ of the forgetful functor $U:\textbf{Haus}\rightarrow \textbf{Top}$ in my textbook.
Let $X\in \textbf{Top}$. Then take the quotient of $X$ by the closure of the diagonal $\Delta_X\subset X\times X$, which is indeed an equivalence relation on $X$, and call it $F(X)$.
Well, I think the above construction is wrong, since the closure of $\Delta_X$ is not necessarily an equivalence relation on $X$, unless $X$ has some good properties. How do I fix this? What would be the best Hausdorff space with respect to $X$? I tried to repace the closure of $\Delta_X$ by its equivalence closure, but it does not seem work right.