According to the Penrose-Hawking singularity theorems [22-24], a trapped surface inevitably leads to a geodesically incomplete spacetime manifold, implying the imminent formation of a singularity.

Like the singularity theorems, the principle of topological censorship [14] assumes the presence of a trapped surface.

Furthermore, since the formation of trapped surfaces would violate spacetime coherency (5), crucial assumptions underpinning the singularity theorems and the principle of topological censorship may not apply.

The highlight was the cosmological

singularity theorems, developing from Roger Penrose's ideas about black holes, showing that (under reasonable assumptions) classical general relativity necessarily implies there was a start to the universe: a space-time singularity that is the boundary to where normal physics applies.

It is considered as a cornerstone for the achievement of the

singularity theorems, the analysis of gravitational collapse, the cosmic censorship hypothesis, the Penrose inequality, etc.

Moreover, the unconditional universal coupling is a crucial assumption (1) for the singularity theorems of Hawking and Penrose (Hawking and Ellis 1973).

A hidden agenda of the implicit assumption of universal equivalence of energy and mass is to justify the universal coupling that is a vital assumption of the singularity theorems (Wald 1984).

Chandra had a deep appreciation of Penrose's work; in fact he once remarked to me, "The

singularity theorems of Hawking and Penrose are the most important results in general relativity since Einstein

Indeed, in the next section we shall call attention to a generalization of the Extreme Value Theorem which is used crucially in the proofs of spacetime singularity theorems.

A full answer to this question is beyond the scope of this paper, but let us call attention to one topic whose standard treatment is essentially non-constructive, but which arrives at some of the most striking results in general relativity and cosmology, namely the spacetime singularity theorems of Hawking, and Hawking and Penrose.

If Friedmann's solutions are supplemented by the Hawking-Penrose singularity theorems, and the theorems are satisfied, it follows that our Friedmann univers began to exist with a big bang singularity.

D3: U is the universe = U is a cosmological model C that is described by Friedmann's solutions to the field equations and that satisfies the Hawking-Penrose singularity theorems.