Is it true that quotient space of a Hausdorff space is necessarily Hausdorff?
In the book "Algebraic Curves and Riemann Surfaces", by Miranda, the author writes:
"$\mathbb{}P^2$ can be viewed as the quotient space of $\mathbb{C}^3-\{0\}$ by the multiplicative action of $\mathbb{C}^*$. In this way, $\mathbb{}P^2$ inherits a Hausdorff topology, which is the quotient topology from the natural map from $\mathbb{C}^3-\{0\}$ to $\mathbb{}P^2$"
It is true that the complex projective plane $\mathbb{}P^2$ is Hausdorff, but the above reasoning by Miranda will be true if the statement in the question is true.