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 , 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 consider an open subset and denote by Sk(U) the following:
The sets Sk(U) form a basis for the Whitney Ck-topology on C∞(M,N)., p. 42.
Whitney C∞-topology For each choice of , 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:
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 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 } then, by the standard theory of vector spaces and so is a real, finite-dimensional manifold. Next, define:
Using b to denote the dimension Bkm,n, we see that , and so is a real, finite-dimensional manifold.
In fact, if M and N have dimension m and n respectively then:, p. 40.
Topology Given the Whitney C∞-topology, the space C∞(M,N) is a Baire space, i.e. every residual set is dense., p. 44.
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