It is known [\textit{R. Geroch}, \textit{E. H. Kronheimer} and \textit{R. Penrose}, Proc. R. Soc. Lond., Ser. A 327, 545-567 (1972; Zbl 0257.53059)] that space-times generally have a boundary, defined by certain past and future sets. A past set is one which coincides with its own past; it is indecomposable if it is not the union of two past sets. The author shows that the set of indecomposable past sets is partially ordered by inclusion and is inductive. Hence Zorn's lemma provides maximal indecomposable past sets. These are described as the essential part of the boundary, because every point of the manifold belongs to one of them. In fact, there is an intimate connection between absolute event horizons and maximal indecomposable past sets. The same cannot be said for black hole event horizons, but the paper proves a weaker result there.