  We consider the nonlocal Cahn-Hilliard equation with singular (logarithmic)
potential and constant mobility in three-dimensional bounded domains and we
establish the validity of the instantaneous strict separation property. This
means that any weak solution, which is not a pure phase initially, stays
uniformly away from the pure phases $\pm1$ from any positive time on. This work
extends the result in dimension two for the same equation and gives a positive
answer to the long standing open problem of the validity of the strict
separation property in dimensions higher than two. In conclusion, we show how
this property plays an essential role to achieve higher-order regularity for
the solutions and to prove that any weak solution converges to a single
equilibrium.
