In January 1965, Roger Penrose publishes "Gravitational collapse and space-time singularities", the first rigorous result since Einstein to show, without assuming perfect symmetry, that the gravitational collapse of a sufficiently massive star inevitably leads to a spacetime singularity. Penrose introduces the key concept of a "closed trapped surface" — a region from which neither light nor matter can escape — to prove the result in a purely topological way, without depending on exact solutions of Einstein's equations. Hawking, directly inspired by this work, extends the method between 1965 and 1970 to cosmological contexts, showing that reasonable physical conditions imply that the universe itself must have begun in a singularity: the Big Bang. The culmination of this joint program is the 1970 paper "The singularities of gravitational collapse and cosmology", signed by both, which establishes the general geodesic incompleteness theorem under four minimal physical conditions. These theorems do not prove that black holes exist in the real universe — that question is left open to the cosmic censorship conjectures Penrose would later develop — but they do establish that, given certain reasonable physical conditions, general relativity predicts the existence of singularities as an unavoidable mathematical consequence, transforming the singularity from a suspicious oddity of exact solutions into a generic feature of the theory.