Description of singularities In Hawking's (and Ellis') book the section 8.3 says:

The preceding theorems prove the occurrence of singularities in a large class of solutions but give little information as to their nature. To investigate this in more detail, one would need to define what one meantbythesize, shape,locationandsoonofa singularity.This would be fairly easy if the singular points were included in the space-time manifold. However it would be impossible to determine the manifold structure at such points by physical measurements. In fact there would be many manifold structures which agreed for the non-singular regions but which differed for the singular points. For example, the manifold at the $t = 0$ singularity in the Robertson-Walker solutions could be that described by the coordinates
  $\{t,r\cos\theta,r\sin\theta \cos\phi,r\sin\theta \sin\phi\}$ 
  or that described by
  $\{t,Sr\cos\theta,Sr\sin\theta \cos\phi,Sr\sin\theta \sin\phi\}.$ 
  In the first case the singularity would be a three-surface, in the second case a single point.

I don't understand very well the last sentence, (in fact I'm not sure what coordinates are the second ones), so if someone could explain me why in the first case it would be a surface and in the second a single point I would be very grateful. 
 A: Hawking and Ellis appear to confuse a map with a manifold. The spacetime manifold consists of coordinate systems which can be determined by measurement. It exists only in so far as it can be physically measured. It does not extend to singularities, and does not even extend beyond an event horizon.
Given a boundary on a manifold (e.g. an event horizon) there may be a number of different ways of extending the manifold mathematically, for example treating the event horizon itself as a singular point, or extending to the now familiar interior solutions. However, those solutions do not in themselves constitute physical science in the absence of further evidence. 
We can draw maps in different ways. Most familiar is

but after a change in coordinates such that the radial velocity of light is constant

In the first map the singularity is shown as a point, in the second it appears as a surface. However, these representations are properties of the maps, not the properties of a geometrical manifold. If spacetime were a topological manifold (without metric) either might be valid, and I think that is what Hawking and Ellis are suggesting. But that depends on the philosophical assumption that there exists a prior manifold independent of measurement. I would disagree fundamentally with such an assumption. The manifold in general relativity is not a substantive physical prior, it is constructed out of physical measurements, and consequently it has no meaning at points where measurement is not possible. The same thing is seen in quantum mechanics, the property of position does not always exist for a particle.
