  Affine ind-varieties are infinite dimensional generalizations of algebraic
varieties which appear naturally in many different contexts, in particular in
the study of automorphism groups of affine spaces. In this article we introduce
and develop the basic algebraic theory of topologically integrable derivations
of complete topological rings. We establish a bijective algebro-geometric
correspondence between additive group actions on affine ind-varieties and
topologically integrable derivations of their coordinate pro-rings which
extends the classical fruitful correspondence between additive group actions on
affine varieties and locally nilpotent derivations of their coordinate rings.
