A contradictory result from a corollary with the fact that the quotient space of a Hausdorff space is not necessary Hausdorff

Question

It is known that a quotient space of a Hausdorff space is not necessarily Hausdorff; however, in the book of Topology by Munkres, at page 140, it is given that

But, we can always choose $Z = X$ and $g=i$ so that $g = i$ is a surjective continuous map. Hence, if $Z=X$ is Hausdorff, then by part $b$, $X^*$ must be Hausdorff, which is not true, as there are lots of counterexamples, by what is wrong with the above argument ?

Answer 1

If $Z=X$ and $g=id$ then $$X^*=\{ g^{-1}(z) | z \in Z \}=X$$

I think that you are confusing $X^*$ with some quotient of $X$, which is not the case in this corollary.