Furthermore, let C∞(M,N) denote the space of smooth mappings between M and N. The notation C∞ means that the mappings are infinitely differentiable, i.e. partial derivatives of all orders exist and are continuous.
For a fixed integer k ≥ 0 consider an open subset U ⊂ Jk(M,N), and denote by Sk(U) the following: The sets Sk(U) form a basis for the Whitney Ck-topology on C∞(M,N).
Let us denote by Wk the set of open subsets of C∞(M,N) with respect to the Whitney Ck-topology.
To see this, let ℝk[x1,...,xm] denote the space of polynomials, with real coefficients, in m variables of order at most k and with zero as the constant term.
This is a real vector space with dimension Writing a = dim{ℝk[x1,...,xm]} then, by the standard theory of vector spaces ℝk[x1,...,xm] ≅ ℝa, and so is a real, finite-dimensional manifold.