Let a group $G$ act on a (not necessarily irreducible) algebraic variety over ${\bf C}$.

It seems to be well-known that the quotient in the sense of geometric invariant theory (i.e., the categorical quotient in the category of algebraic varieties) agrees with the largest Hausdorff quotient (i.e., the categorical quotient in the category of Hausdorff spaces).

The reference which I find cited for this fact in the literature are two papers of Luna from the 70s: paper 1, paper 2. For example, this reference says (on page 6): *It can be shown that the categorical quotient in the category of affine varieties (over ${\bf C}$ and with respect to a reductive group action) is also the categorical quotient for Hausdorff spaces or complex analytic varieties [Lun75, Lun 76].*

My question is just how the claim about the categorical Hausdorff quotient follows from Luna's papers.

I understand that Luna is proving that not only (by definition) the invariant polynomials correspond 1-1 to polynomial functions on the GIT-quotient, but that also the same holds true for smooth or analytic functions instead of polynomials. Does this imply by some known results that the GIT-quotient must be the largest Hausdorff quotient?