Toward a topological classification of convex sets in infinite-dimensional Fréchet spaces

Main Theorem: Let C be an infinite-dimensional convex set in a Fréchet space. If C has an almost internal point, then for every compact subset K of the completion C such that K∩C=K∩aff(C) the pair (C,C) is strongly (K,K∩C)-universal and the space C is strongly K∩C-universal. rollaries are derived from this theorem. Among them there are (1) conditions under which two convex sets with almost internal points are homeomorphic, (2) conditions under which a convex set with an almost internal point is homeomorphic to a convex set in l2, (3) a characterization of convex sets with almost internal points, homeomorphic to Σ, (4) a characterization of ∞-convex sets homeomorphic to Σω.