The definition that was given to me by my professor said that a Lifschitz transition is when the topology of the Fermi surface changes. For example, if we have a one-dimensional Fermi surface, we would have some curve $E_\lambda(k)$ where $k$ ranges over the Brillouin zone and $\lambda$ is some real-valued parameter. If we vary $\lambda$, wouldn't a change in the topology imply the change in the number of singularities since this would actually change the topology of the curve/surface? Nevertheless, I have also been told that the transition occurs when the number of roots of $E_\lambda(k)$ changes. In this case, the topology won't necessarily change. For example, we can homeomorphically transform a curve with no singularities into another curve with no singularities but a different number of roots. As a concrete example, let $E_\lambda(k)=k^2+\lambda$. We have a different number of roots depending on if $\lambda$ is positive, negative or zero, but all of the curves are simply connected and closed, and therefore, have the same topology. What is the correct definition of Lifschitz transition?