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The region with trapped surfaces in spherical symmetry, its core, and their boundaries

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.