  Let V be a variety of not necessarily associative algebras, and A an inverse
limit of nilpotent algebras A_i\in V, such that some finitely generated
subalgebra S \subseteq A is dense in A under the inverse limit of the discrete
topologies on the A_i.
  A sufficient condition on V is obtained for all algebra homomorphisms from A
to finite-dimensional algebras B to be continuous; in other words, for the
kernels of all such homomorphisms to be open ideals. This condition is
satisfied, in particular, if V is the variety of associative, Lie, or Jordan
algebras.
  Examples are given showing the need for our hypotheses, and some open
questions are noted.
