Removable sets for classes of functions

Abstract

In a general context it is shown that a closed countable union of closed sets of removable is again a of removable singularities. 0. Introduction. Dolzenko noted in [1], that a closed countable union of closed sets of removable for bounded analytic functions in the plane is also removable for such functions. The sets of removable for bounded analytic functions are the null sets of inner analytic capacity, y, and the result is more striking since y is not known to be subadditive. It turns out that the special character of the functions considered by Dolzenko has little to do with the result. In this paper we formulate the notion of set of removable singularities (F-null set) in a general context, and we show that countable unions of such sets are again sets of the same type. This fact emerges as essentially an equiva~lent of the Baire Category Theorem. The argument is most naturally presented in the language of categories [3]. The sheaf case, which covers Dolzenko's theorem, is dealt with in ?1. Other cases are sketched in ?3, while ??2 and 4 are devoted to examples. The author is grateful to the referee for many helpful suggestions. 1. Removability in Baire spaces. Suppose X is a topological space, Y is a Hausdorff topological space, ( is the category of open subsets of X, with inclusions as morphisms, and C(U, Y), for U E ?, is the class of continuous functions mapping U to Y. Let y= Uu,6 2C(u ). An element of Fy is a of continuous maps from some U to Y. Fy forms a category, with restrictions as morphisms. Denote the restriction C(U, Y)-#C(V, Y), where Vc U, by p(U, V). Let F: (p.y be a contravariant functor. Then for each U E 0, F(U) is some of continuous maps from U to Y. Also p(U, V)f E F( Y) whenever Vc U andf E F(U). We say F is local if f belongs to F(U) whenever f E C(U, Y), Ic Secondary 30A 14, 30A44, 31C15, 32D20, 35A20, 54D05.