  The faces of a convex set owe their relevance to an interplay between
convexity and topology that is systematically studied in the work of
Rockafellar. Infinite-dimensional convex sets are excluded from this theory as
their relative interiors may be empty. Shirokov and the present author answered
this issue by proving that every point in a convex set lies in the relative
algebraic interior of the face it generates. This theorem is proved here in a
simpler way, connecting ideas scattered throughout the literature. This article
summarizes and develops methods for faces and their relative algebraic
interiors and applies them to spaces of probability measures.
