Split faces and projective sets in a metrizable compact convex set

Abstract

If H is a subset of a metr izable compac t convex set K in a locally convex Hausdor f f topological vector space then its complementary set H' is defined to be the union of all the faces of K disjoint f rom H. It is shown that i f H has projective class 2n or 2 n + t, then H ' has projective class 2n + 2. (Recall that a subset of a complete, separable metric space is defined to be of projective class 0 if it is a Borel set, of projective class 2n + 1 if it is the cont inuous image of a subset of projective class 2n and of projective class 2n if it is the complemen t of a subset of projective class 2 n 1.) The case where H is a split face of K with complemen ta ry face H ' is then considered and it is shown that if H and H ' are both projective sets of class 1 (usually known as analytic sets) then H and H ' are automat ica l ly Borel sets. Further, any pair of affine Borel functions on H and H ' have a c o m m o n affine Borel extension. Finally it is shown that if H is a countably generated split face then its c o m p l e m e n t a r y face is a Borel set. Thus H is a set of projective class 2 (usually known as a co-analytic set).