A property of ideals of jets of functions vanishing on a set.

For a set E ⊂ Rⁿ that contains the origin, we consider I^m(E) - the set of all m^th degree Taylor approximations (at the origin) of C^m functions on Rⁿ that vanish on E. This set is a proper ideal in P^m(Rⁿ) - the ring of all m^th degree Taylor approximations of C^m functions on Rⁿ. Which ideals in P^m(Rⁿ) arise as I^m(E) for some E? In this paper we introduce the notion of a closed ideal in P^m(Rⁿ), and prove that any ideal of the form I^m(E) is closed. We do not know whether in general any closed proper ideal is of the form I^m(E) for some E, however we prove in a subsequent paper that all closed proper ideals in P^m(Rⁿ) arise as I^m(E) when m + n ≤ 5. [ABSTRACT FROM AUTHOR]