  Function (linear) spaces on which an arbitrary function operates (i.e. the
space is stable w.r.t. the pointwise unary operation defined by the function)
were investigated, for continuous real or complex operations, by
deLeeuw-Katznelson, Sternfeld and Weit. They showed that, with suitable
assumptions, a real or complex function space on which a non-affine continuous
function operates is like an algebra. In this note, however, the point of view
is somewhat different: the scalars are any field K, and from the outset no
topology or continuity are assumed. Rather, a function space operated by a
non-additively-affine (multivariate) function induces a topology on the set on
which it is defined and the family of functions which operate on it induces
topologies on K^n. These are used to derive a density property and to
investigate "homomorphisms" of such spaces w.r.t. the "operations".
