  In 1979 Valiant introduced the complexity class VNP of p-definable families
of polynomials, he defined the reduction notion known as p-projection and he
proved that the permanent polynomial and the Hamiltonian cycle polynomial are
VNP-complete under p-projections.
  In 2001 Mulmuley and Sohoni (and independently B\"urgisser) introduced the
notion of border complexity to the study of the algebraic complexity of
polynomials. In this algebraic machine model, instead of insisting on exact
computation, approximations are allowed. This gives VNP the structure of a
topological space. In this short note we study the set VNPC of VNP-complete
polynomials. We show that the complement VNP \ VNPC lies dense in VNP. Quite
surprisingly, we also prove that VNPC lies dense in VNP. We prove analogous
statements for the complexity classes VF, VBP, and VP.
  The density of VNP \ VNPC holds for several different reduction notions:
p-projections, border p-projections, c-reductions, and border c-reductions. We
compare the relationship of the completeness notions under these reductions and
separate most of the corresponding sets. Border reduction notions were
introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Our paper is the
first structured study of border reduction notions.
