This is a problem of my midterm test...
Denote invertible $2\times 2$ matrices on $\mathbb{C}$ by ${\rm GL}(2,\mathbb{C})$, and define the conjugation action of ${\rm GL}(2,\mathbb{C})$ on itself by what it usually means $(A,B)\mapsto ABA^{-1}$, then we get the orbit space of this action equipped with quotient topology. The problem is to prove that this space is not Hausdorff.
I heard that for any 2 dissimilar matrices $A,B$, we can always find a matrix similar to $A$ in any, arbitrarily small neighborhood of $B$.
Is that true? If is, how does it help with the proof? Thanks in advance.