  We consider the region $\mathscr{T}$ in spacetime containing future-trapped
closed surfaces and its boundary $\B$, and derive some of their general
properties. We then concentrate on the case of spherical symmetry, but the
methods we use are general and applicable to other situations. We argue that
closed trapped surfaces have a non-local property, "clairvoyance", which is
inherited by $\B$. We prove that $\B$ is not a marginally trapped tube in
general, and that it can have portions in regions whose whole past is flat. For
asymptotically flat black holes, we identify a general past barrier, well
inside the event horizon, to the location of $\B$ under physically reasonable
conditions. We also define the core $\mathscr{Z}$ of the trapped region as that
part of $\mathscr{T}$ which is indispensable to sustain closed trapped
surfaces. We prove that the unique spherically symmetric dynamical horizon is
the boundary of such a core, and we argue that this may serve to single it out.
To illustrate the results, some explicit examples are discussed, namely
Robertson-Walker geometries and the imploding Vaidya spacetime.
