In this presentation we treat linear parabolic problems posed on an evolving domain Ω(t) that may undergo a topological change. Clearly, numerous scenarios can lead to topological changes in two- or three-dimensional evolving domains. We restrict to a specific subset of such scenarios in which the domain evolution is C2-smooth away from a single critical point in time, where a topological change occurs instantaneously. We introduce a class of level set domains that allows for instantaneous topological changes such as domain merging or splitting and the creation or vanishing of a hole in the domain. We discuss well-posedness issues of a space-time weak formulation of parabolic problems in such singular space-time domains. For discretization of the problem we use a Eulerian unfitted finite element, known from the literature, that combines a standard time-stepping scheme with an implicit extension of the finite element solution to a narrow band surrounding the physical domain. For this method we present a stability and error analysis.