In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler Whitney.
Construction Let M and N be two real, smooth manifolds. 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.
Whitney Ck-topology For some integer k ≥ 0, let Jk(M,N) denote the k-jet space of mappings between M and N. The jet space can be endowed with a smooth structure (i.e. a structure as a C∞ manifold) which make it into a topological space. This topology is used to define a topology on C∞(M,N). For a fixed integer k ≥ 0 consider an open subset U ⊂ Jk(M,N), and denote by Sk(U) the following:
S k ( U ) = { f ∈ C ∞ ( M , N ) : ( J k f ) ( M ) ⊆ U } . {\displaystyle S^{k}(U)=\{f\in C^{\infty }(M,N):(J^{k}f)(M)\subseteq U\}.}
The sets Sk(U) form a basis for the Whitney Ck-topology on C∞(M,N).
Whitney C∞-topology For each choice of k ≥ 0, the Whitney Ck-topology gives a topology for C∞(M,N); in other words the Whitney Ck-topology tells us which subsets of C∞(M,N) are open sets. Let us denote by Wk the set of open subsets of C∞(M,N) with respect to the Whitney Ck-topology. Then the Whitney C∞-topology is defined to be the topology whose basis is given by W, where:
W = ⋃ k = 0 ∞ W k . {\displaystyle W=\bigcup _{k=0}^{\infty }W^{k}.}
Dimensionality Notice that C∞(M,N) has infinite dimension, whereas Jk(M,N) has finite dimension. In fact, Jk(M,N) is a real, finite-dimensional manifold. 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
dim { R k [ x 1 , … , x m ] } = ∑ i = 1 k ( m + i − 1 ) ! ( m − 1 ) ! ⋅ i ! = ( ( m + k ) ! m ! ⋅ k ! − 1 ) . {\displaystyle \dim \left\{\mathbb {R} ^{k}[x_{1},\ldots ,x_{m}]\right\}=\sum _{i=1}^{k}{\frac {(m+i-1)!}{(m-1)!\cdot i!}}=\left({\frac {(m+k)!}{m!\cdot k!}}-1\right).}
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. Next, define:
B m , n k = ⨁ i = 1 n R k [ x 1 , … , x m ] , ⟹ dim { B m , n k } = n dim { A m k } = n ( ( m + k ) ! m ! ⋅ k ! − 1 ) . {\displaystyle B_{m,n}^{k}=\bigoplus _{i=1}^{n}\mathbb {R} ^{k}[x_{1},\ldots ,x_{m}],\implies \dim \left\{B_{m,n}^{k}\right\}=n\dim \left\{A_{m}^{k}\right\}=n\left({\frac {(m+k)!}{m!\cdot k!}}-1\right).}
Using b to denote the dimension Bkm,n, we see that Bkm,n ≅ ℝb, and so is a real, finite-dimensional manifold. In fact, if M and N have dimension m and n respectively then:
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