  When de Sitter first introduced his celebrated spacetime, he claimed,
following Schwarzschild, that its spatial sections have the topology of the
real projective space RP^3 (that is, the topology of the group manifold SO(3))
rather than, as is almost universally assumed today, that of the sphere S^3.
(In modern language, Schwarzschild was disturbed by the non-local correlations
enforced by S^3 geometry.) Thus, what we today call "de Sitter space" would not
have been accepted as such by de Sitter. There is no real basis within
classical cosmology for preferring S^3 to RP^3, but the general feeling appears
to be that the distinction is in any case of little importance. We wish to
argue that, in the light of current concerns about the nature of de Sitter
space, this is a mistake. In particular, we argue that the difference between
"dS(S^3)" and "dS(RP^3)" may be very important in attacking the problem of
understanding horizon entropies. In the approach to de Sitter entropy via
Schwarzschild-de Sitter spacetime, we find that the apparently trivial
difference between RP^3 and S^3 actually leads to very different perspectives
on this major question of quantum cosmology.
