# What is the correct definition for a Lifschitz transition?

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?